Step |
Hyp |
Ref |
Expression |
1 |
|
remulinvcom.1 |
|- ( ph -> A e. RR ) |
2 |
|
remulinvcom.2 |
|- ( ph -> B e. RR ) |
3 |
|
remulinvcom.3 |
|- ( ph -> ( A x. B ) = 1 ) |
4 |
|
ax-1ne0 |
|- 1 =/= 0 |
5 |
4
|
a1i |
|- ( ph -> 1 =/= 0 ) |
6 |
3 5
|
eqnetrd |
|- ( ph -> ( A x. B ) =/= 0 ) |
7 |
|
simpr |
|- ( ( ph /\ B = 0 ) -> B = 0 ) |
8 |
7
|
oveq2d |
|- ( ( ph /\ B = 0 ) -> ( A x. B ) = ( A x. 0 ) ) |
9 |
1
|
adantr |
|- ( ( ph /\ B = 0 ) -> A e. RR ) |
10 |
|
remul01 |
|- ( A e. RR -> ( A x. 0 ) = 0 ) |
11 |
9 10
|
syl |
|- ( ( ph /\ B = 0 ) -> ( A x. 0 ) = 0 ) |
12 |
8 11
|
eqtrd |
|- ( ( ph /\ B = 0 ) -> ( A x. B ) = 0 ) |
13 |
6 12
|
mteqand |
|- ( ph -> B =/= 0 ) |
14 |
|
ax-rrecex |
|- ( ( B e. RR /\ B =/= 0 ) -> E. x e. RR ( B x. x ) = 1 ) |
15 |
2 13 14
|
syl2anc |
|- ( ph -> E. x e. RR ( B x. x ) = 1 ) |
16 |
|
simprl |
|- ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) -> x e. RR ) |
17 |
|
simprr |
|- ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) -> ( B x. x ) = 1 ) |
18 |
4
|
a1i |
|- ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) -> 1 =/= 0 ) |
19 |
17 18
|
eqnetrd |
|- ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) -> ( B x. x ) =/= 0 ) |
20 |
|
simpr |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ x = 0 ) -> x = 0 ) |
21 |
20
|
oveq2d |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ x = 0 ) -> ( B x. x ) = ( B x. 0 ) ) |
22 |
2
|
ad2antrr |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ x = 0 ) -> B e. RR ) |
23 |
|
remul01 |
|- ( B e. RR -> ( B x. 0 ) = 0 ) |
24 |
22 23
|
syl |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ x = 0 ) -> ( B x. 0 ) = 0 ) |
25 |
21 24
|
eqtrd |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ x = 0 ) -> ( B x. x ) = 0 ) |
26 |
19 25
|
mteqand |
|- ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) -> x =/= 0 ) |
27 |
|
ax-rrecex |
|- ( ( x e. RR /\ x =/= 0 ) -> E. y e. RR ( x x. y ) = 1 ) |
28 |
16 26 27
|
syl2anc |
|- ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) -> E. y e. RR ( x x. y ) = 1 ) |
29 |
|
simplrr |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( B x. x ) = 1 ) |
30 |
29
|
oveq2d |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( A x. ( B x. x ) ) = ( A x. 1 ) ) |
31 |
30
|
oveq1d |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( ( A x. ( B x. x ) ) x. y ) = ( ( A x. 1 ) x. y ) ) |
32 |
1
|
ad2antrr |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> A e. RR ) |
33 |
2
|
ad2antrr |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> B e. RR ) |
34 |
32 33
|
remulcld |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( A x. B ) e. RR ) |
35 |
34
|
recnd |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( A x. B ) e. CC ) |
36 |
|
simplrl |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> x e. RR ) |
37 |
36
|
recnd |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> x e. CC ) |
38 |
|
simprl |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> y e. RR ) |
39 |
38
|
recnd |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> y e. CC ) |
40 |
35 37 39
|
mulassd |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( ( ( A x. B ) x. x ) x. y ) = ( ( A x. B ) x. ( x x. y ) ) ) |
41 |
32
|
recnd |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> A e. CC ) |
42 |
33
|
recnd |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> B e. CC ) |
43 |
41 42 37
|
mulassd |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( ( A x. B ) x. x ) = ( A x. ( B x. x ) ) ) |
44 |
43
|
oveq1d |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( ( ( A x. B ) x. x ) x. y ) = ( ( A x. ( B x. x ) ) x. y ) ) |
45 |
3
|
ad2antrr |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( A x. B ) = 1 ) |
46 |
|
simprr |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( x x. y ) = 1 ) |
47 |
45 46
|
oveq12d |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( ( A x. B ) x. ( x x. y ) ) = ( 1 x. 1 ) ) |
48 |
|
1t1e1ALT |
|- ( 1 x. 1 ) = 1 |
49 |
47 48
|
eqtrdi |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( ( A x. B ) x. ( x x. y ) ) = 1 ) |
50 |
40 44 49
|
3eqtr3d |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( ( A x. ( B x. x ) ) x. y ) = 1 ) |
51 |
|
ax-1rid |
|- ( A e. RR -> ( A x. 1 ) = A ) |
52 |
32 51
|
syl |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( A x. 1 ) = A ) |
53 |
52
|
oveq1d |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( ( A x. 1 ) x. y ) = ( A x. y ) ) |
54 |
31 50 53
|
3eqtr3rd |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( A x. y ) = 1 ) |
55 |
54 46
|
eqtr4d |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( A x. y ) = ( x x. y ) ) |
56 |
4
|
a1i |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> 1 =/= 0 ) |
57 |
46 56
|
eqnetrd |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( x x. y ) =/= 0 ) |
58 |
|
simpr |
|- ( ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) /\ y = 0 ) -> y = 0 ) |
59 |
58
|
oveq2d |
|- ( ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) /\ y = 0 ) -> ( x x. y ) = ( x x. 0 ) ) |
60 |
36
|
adantr |
|- ( ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) /\ y = 0 ) -> x e. RR ) |
61 |
|
remul01 |
|- ( x e. RR -> ( x x. 0 ) = 0 ) |
62 |
60 61
|
syl |
|- ( ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) /\ y = 0 ) -> ( x x. 0 ) = 0 ) |
63 |
59 62
|
eqtrd |
|- ( ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) /\ y = 0 ) -> ( x x. y ) = 0 ) |
64 |
57 63
|
mteqand |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> y =/= 0 ) |
65 |
32 36 38 64
|
remulcan2d |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( ( A x. y ) = ( x x. y ) <-> A = x ) ) |
66 |
55 65
|
mpbid |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> A = x ) |
67 |
|
simpr |
|- ( ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) /\ A = x ) -> A = x ) |
68 |
67
|
oveq2d |
|- ( ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) /\ A = x ) -> ( B x. A ) = ( B x. x ) ) |
69 |
17
|
ad2antrr |
|- ( ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) /\ A = x ) -> ( B x. x ) = 1 ) |
70 |
68 69
|
eqtrd |
|- ( ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) /\ A = x ) -> ( B x. A ) = 1 ) |
71 |
66 70
|
mpdan |
|- ( ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) /\ ( y e. RR /\ ( x x. y ) = 1 ) ) -> ( B x. A ) = 1 ) |
72 |
28 71
|
rexlimddv |
|- ( ( ph /\ ( x e. RR /\ ( B x. x ) = 1 ) ) -> ( B x. A ) = 1 ) |
73 |
15 72
|
rexlimddv |
|- ( ph -> ( B x. A ) = 1 ) |