Step |
Hyp |
Ref |
Expression |
1 |
|
df-ne |
|- ( A =/= 0 <-> -. A = 0 ) |
2 |
|
ax-rrecex |
|- ( ( A e. RR /\ A =/= 0 ) -> E. x e. RR ( A x. x ) = 1 ) |
3 |
|
simpll |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> A e. RR ) |
4 |
3
|
recnd |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> A e. CC ) |
5 |
|
simprl |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> x e. RR ) |
6 |
5
|
recnd |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> x e. CC ) |
7 |
4 6 4
|
mulassd |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( ( A x. x ) x. A ) = ( A x. ( x x. A ) ) ) |
8 |
|
simprr |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( A x. x ) = 1 ) |
9 |
8
|
oveq1d |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( ( A x. x ) x. A ) = ( 1 x. A ) ) |
10 |
3 5 8
|
remulinvcom |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( x x. A ) = 1 ) |
11 |
10
|
oveq2d |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( A x. ( x x. A ) ) = ( A x. 1 ) ) |
12 |
|
ax-1rid |
|- ( A e. RR -> ( A x. 1 ) = A ) |
13 |
3 12
|
syl |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( A x. 1 ) = A ) |
14 |
11 13
|
eqtrd |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( A x. ( x x. A ) ) = A ) |
15 |
7 9 14
|
3eqtr3d |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( 1 x. A ) = A ) |
16 |
2 15
|
rexlimddv |
|- ( ( A e. RR /\ A =/= 0 ) -> ( 1 x. A ) = A ) |
17 |
16
|
ex |
|- ( A e. RR -> ( A =/= 0 -> ( 1 x. A ) = A ) ) |
18 |
1 17
|
syl5bir |
|- ( A e. RR -> ( -. A = 0 -> ( 1 x. A ) = A ) ) |
19 |
|
1re |
|- 1 e. RR |
20 |
|
remul01 |
|- ( 1 e. RR -> ( 1 x. 0 ) = 0 ) |
21 |
19 20
|
mp1i |
|- ( A = 0 -> ( 1 x. 0 ) = 0 ) |
22 |
|
oveq2 |
|- ( A = 0 -> ( 1 x. A ) = ( 1 x. 0 ) ) |
23 |
|
id |
|- ( A = 0 -> A = 0 ) |
24 |
21 22 23
|
3eqtr4d |
|- ( A = 0 -> ( 1 x. A ) = A ) |
25 |
18 24
|
pm2.61d2 |
|- ( A e. RR -> ( 1 x. A ) = A ) |