# Where is the mistake in this formal proof?

#### solakis

##### Active member
we have the following formal proof:

i) $$\displaystyle \forall x[\forall y(xy=y)\Longrightarrow x=1]$$........................................theoren in real Nos

2)$$\displaystyle \forall y(xy=y)\Longrightarrow x=1$$.......................1,U.E ,x=x

3) $$\displaystyle (x0=0)\Longrightarrow x=1$$....................... 2,U.E ,y=0

4)$$\displaystyle \forall A[A.0=0]$$.....................................Theorem in Real Nos

5) )$$\displaystyle [x.0=0]$$.....................................4,U.E, A=x

6) x=1.........................................3,5 M.Ponens

7) )$$\displaystyle \forall A[A=1]$$..................................... 6,U.I

U.E=Universal Elimination

U.I = Universal Introduction

I am afraid to say i find no mistake

Last edited:

#### Deveno

##### Well-known member
MHB Math Scholar
The very first line is wrong, it should be:

1) $\forall x[\forall (y \neq 0): (xy = y ) \implies x = 1]$

One can see that your statement is clearly untrue by letting:

$x = 2, y = 0$

so it is not a theorem of the real numbers.

(Moral of the story: don't go proving stupid things by dividing by 0).

#### solakis

##### Active member
The very first line is wrong, it should be:

1) $\forall x[\forall (y \neq 0): (xy = y ) \implies x = 1]$

One can see that your statement is clearly untrue by letting:

$x = 2, y = 0$

so it is not a theorem of the real numbers.

(Moral of the story: don't go proving stupid things by dividing by 0).

1)$$\displaystyle \forall y(xy=y)$$.................................hypothesis

2)$$\displaystyle (x1=1)$$............................................1,U.E ,Y=1

3)$$\displaystyle \forall A[A.1=A]$$.................................Axiom in Real Nos

4))$$\displaystyle [x1=x]$$.................................3,U.E,A=x

5) x=1.....................................Substituting ( 4) into (2)

6)$$\displaystyle \forall y(xy=y)\Longrightarrow x=1$$................................. From (1) to (5) by using the rule of conditional proof

7)$$\displaystyle \forall x[\forall y(xy=y)\Longrightarrow x=1]$$.................................................6,U.I

Where is the mistake