Step |
Hyp |
Ref |
Expression |
1 |
|
remulcand.1 |
|- ( ph -> A e. RR ) |
2 |
|
remulcand.2 |
|- ( ph -> B e. RR ) |
3 |
|
remulcand.3 |
|- ( ph -> C e. RR ) |
4 |
|
remulcand.4 |
|- ( ph -> C =/= 0 ) |
5 |
|
ax-rrecex |
|- ( ( C e. RR /\ C =/= 0 ) -> E. x e. RR ( C x. x ) = 1 ) |
6 |
3 4 5
|
syl2anc |
|- ( ph -> E. x e. RR ( C x. x ) = 1 ) |
7 |
3
|
adantr |
|- ( ( ph /\ x e. RR ) -> C e. RR ) |
8 |
7
|
adantr |
|- ( ( ( ph /\ x e. RR ) /\ ( C x. x ) = 1 ) -> C e. RR ) |
9 |
|
simplr |
|- ( ( ( ph /\ x e. RR ) /\ ( C x. x ) = 1 ) -> x e. RR ) |
10 |
|
simpr |
|- ( ( ( ph /\ x e. RR ) /\ ( C x. x ) = 1 ) -> ( C x. x ) = 1 ) |
11 |
8 9 10
|
remulinvcom |
|- ( ( ( ph /\ x e. RR ) /\ ( C x. x ) = 1 ) -> ( x x. C ) = 1 ) |
12 |
11
|
ex |
|- ( ( ph /\ x e. RR ) -> ( ( C x. x ) = 1 -> ( x x. C ) = 1 ) ) |
13 |
|
oveq2 |
|- ( ( C x. A ) = ( C x. B ) -> ( x x. ( C x. A ) ) = ( x x. ( C x. B ) ) ) |
14 |
13
|
3ad2ant3 |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ( x x. ( C x. A ) ) = ( x x. ( C x. B ) ) ) |
15 |
|
simp2 |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ( x x. C ) = 1 ) |
16 |
15
|
oveq1d |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ( ( x x. C ) x. A ) = ( 1 x. A ) ) |
17 |
|
simp1r |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> x e. RR ) |
18 |
17
|
recnd |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> x e. CC ) |
19 |
7
|
3ad2ant1 |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> C e. RR ) |
20 |
19
|
recnd |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> C e. CC ) |
21 |
|
simp1l |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ph ) |
22 |
21 1
|
syl |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> A e. RR ) |
23 |
22
|
recnd |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> A e. CC ) |
24 |
18 20 23
|
mulassd |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ( ( x x. C ) x. A ) = ( x x. ( C x. A ) ) ) |
25 |
|
remulid2 |
|- ( A e. RR -> ( 1 x. A ) = A ) |
26 |
22 25
|
syl |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ( 1 x. A ) = A ) |
27 |
16 24 26
|
3eqtr3d |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ( x x. ( C x. A ) ) = A ) |
28 |
15
|
oveq1d |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ( ( x x. C ) x. B ) = ( 1 x. B ) ) |
29 |
21 2
|
syl |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> B e. RR ) |
30 |
29
|
recnd |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> B e. CC ) |
31 |
18 20 30
|
mulassd |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ( ( x x. C ) x. B ) = ( x x. ( C x. B ) ) ) |
32 |
|
remulid2 |
|- ( B e. RR -> ( 1 x. B ) = B ) |
33 |
29 32
|
syl |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ( 1 x. B ) = B ) |
34 |
28 31 33
|
3eqtr3d |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> ( x x. ( C x. B ) ) = B ) |
35 |
14 27 34
|
3eqtr3d |
|- ( ( ( ph /\ x e. RR ) /\ ( x x. C ) = 1 /\ ( C x. A ) = ( C x. B ) ) -> A = B ) |
36 |
35
|
3exp |
|- ( ( ph /\ x e. RR ) -> ( ( x x. C ) = 1 -> ( ( C x. A ) = ( C x. B ) -> A = B ) ) ) |
37 |
12 36
|
syld |
|- ( ( ph /\ x e. RR ) -> ( ( C x. x ) = 1 -> ( ( C x. A ) = ( C x. B ) -> A = B ) ) ) |
38 |
37
|
impr |
|- ( ( ph /\ ( x e. RR /\ ( C x. x ) = 1 ) ) -> ( ( C x. A ) = ( C x. B ) -> A = B ) ) |
39 |
6 38
|
rexlimddv |
|- ( ph -> ( ( C x. A ) = ( C x. B ) -> A = B ) ) |
40 |
|
oveq2 |
|- ( A = B -> ( C x. A ) = ( C x. B ) ) |
41 |
39 40
|
impbid1 |
|- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) ) |