Step |
Hyp |
Ref |
Expression |
1 |
|
eqtr2 |
|- ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ x = ( ( 1st ` s ) |g ( 1st ` r ) ) ) -> ( ( 1st ` u ) |g ( 1st ` v ) ) = ( ( 1st ` s ) |g ( 1st ` r ) ) ) |
2 |
|
fvex |
|- ( 1st ` u ) e. _V |
3 |
|
fvex |
|- ( 1st ` v ) e. _V |
4 |
|
gonafv |
|- ( ( ( 1st ` u ) e. _V /\ ( 1st ` v ) e. _V ) -> ( ( 1st ` u ) |g ( 1st ` v ) ) = <. 1o , <. ( 1st ` u ) , ( 1st ` v ) >. >. ) |
5 |
2 3 4
|
mp2an |
|- ( ( 1st ` u ) |g ( 1st ` v ) ) = <. 1o , <. ( 1st ` u ) , ( 1st ` v ) >. >. |
6 |
|
fvex |
|- ( 1st ` s ) e. _V |
7 |
|
fvex |
|- ( 1st ` r ) e. _V |
8 |
|
gonafv |
|- ( ( ( 1st ` s ) e. _V /\ ( 1st ` r ) e. _V ) -> ( ( 1st ` s ) |g ( 1st ` r ) ) = <. 1o , <. ( 1st ` s ) , ( 1st ` r ) >. >. ) |
9 |
6 7 8
|
mp2an |
|- ( ( 1st ` s ) |g ( 1st ` r ) ) = <. 1o , <. ( 1st ` s ) , ( 1st ` r ) >. >. |
10 |
5 9
|
eqeq12i |
|- ( ( ( 1st ` u ) |g ( 1st ` v ) ) = ( ( 1st ` s ) |g ( 1st ` r ) ) <-> <. 1o , <. ( 1st ` u ) , ( 1st ` v ) >. >. = <. 1o , <. ( 1st ` s ) , ( 1st ` r ) >. >. ) |
11 |
|
1oex |
|- 1o e. _V |
12 |
|
opex |
|- <. ( 1st ` u ) , ( 1st ` v ) >. e. _V |
13 |
11 12
|
opth |
|- ( <. 1o , <. ( 1st ` u ) , ( 1st ` v ) >. >. = <. 1o , <. ( 1st ` s ) , ( 1st ` r ) >. >. <-> ( 1o = 1o /\ <. ( 1st ` u ) , ( 1st ` v ) >. = <. ( 1st ` s ) , ( 1st ` r ) >. ) ) |
14 |
2 3
|
opth |
|- ( <. ( 1st ` u ) , ( 1st ` v ) >. = <. ( 1st ` s ) , ( 1st ` r ) >. <-> ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) ) |
15 |
14
|
anbi2i |
|- ( ( 1o = 1o /\ <. ( 1st ` u ) , ( 1st ` v ) >. = <. ( 1st ` s ) , ( 1st ` r ) >. ) <-> ( 1o = 1o /\ ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) ) ) |
16 |
10 13 15
|
3bitri |
|- ( ( ( 1st ` u ) |g ( 1st ` v ) ) = ( ( 1st ` s ) |g ( 1st ` r ) ) <-> ( 1o = 1o /\ ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) ) ) |
17 |
|
funfv1st2nd |
|- ( ( Fun Z /\ s e. Z ) -> ( Z ` ( 1st ` s ) ) = ( 2nd ` s ) ) |
18 |
17
|
ex |
|- ( Fun Z -> ( s e. Z -> ( Z ` ( 1st ` s ) ) = ( 2nd ` s ) ) ) |
19 |
|
funfv1st2nd |
|- ( ( Fun Z /\ r e. Z ) -> ( Z ` ( 1st ` r ) ) = ( 2nd ` r ) ) |
20 |
19
|
ex |
|- ( Fun Z -> ( r e. Z -> ( Z ` ( 1st ` r ) ) = ( 2nd ` r ) ) ) |
21 |
18 20
|
anim12d |
|- ( Fun Z -> ( ( s e. Z /\ r e. Z ) -> ( ( Z ` ( 1st ` s ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` r ) ) = ( 2nd ` r ) ) ) ) |
22 |
|
funfv1st2nd |
|- ( ( Fun Z /\ u e. Z ) -> ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) ) |
23 |
22
|
ex |
|- ( Fun Z -> ( u e. Z -> ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) ) ) |
24 |
|
funfv1st2nd |
|- ( ( Fun Z /\ v e. Z ) -> ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) |
25 |
24
|
ex |
|- ( Fun Z -> ( v e. Z -> ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) ) |
26 |
23 25
|
anim12d |
|- ( Fun Z -> ( ( u e. Z /\ v e. Z ) -> ( ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) ) ) |
27 |
|
fveq2 |
|- ( ( 1st ` s ) = ( 1st ` u ) -> ( Z ` ( 1st ` s ) ) = ( Z ` ( 1st ` u ) ) ) |
28 |
27
|
eqcoms |
|- ( ( 1st ` u ) = ( 1st ` s ) -> ( Z ` ( 1st ` s ) ) = ( Z ` ( 1st ` u ) ) ) |
29 |
28
|
adantr |
|- ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( Z ` ( 1st ` s ) ) = ( Z ` ( 1st ` u ) ) ) |
30 |
29
|
eqeq1d |
|- ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( ( Z ` ( 1st ` s ) ) = ( 2nd ` s ) <-> ( Z ` ( 1st ` u ) ) = ( 2nd ` s ) ) ) |
31 |
|
fveq2 |
|- ( ( 1st ` r ) = ( 1st ` v ) -> ( Z ` ( 1st ` r ) ) = ( Z ` ( 1st ` v ) ) ) |
32 |
31
|
eqcoms |
|- ( ( 1st ` v ) = ( 1st ` r ) -> ( Z ` ( 1st ` r ) ) = ( Z ` ( 1st ` v ) ) ) |
33 |
32
|
adantl |
|- ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( Z ` ( 1st ` r ) ) = ( Z ` ( 1st ` v ) ) ) |
34 |
33
|
eqeq1d |
|- ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( ( Z ` ( 1st ` r ) ) = ( 2nd ` r ) <-> ( Z ` ( 1st ` v ) ) = ( 2nd ` r ) ) ) |
35 |
30 34
|
anbi12d |
|- ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( ( ( Z ` ( 1st ` s ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` r ) ) = ( 2nd ` r ) ) <-> ( ( Z ` ( 1st ` u ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` r ) ) ) ) |
36 |
35
|
anbi1d |
|- ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( ( ( ( Z ` ( 1st ` s ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` r ) ) = ( 2nd ` r ) ) /\ ( ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) ) <-> ( ( ( Z ` ( 1st ` u ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` r ) ) /\ ( ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) ) ) ) |
37 |
|
eqtr2 |
|- ( ( ( Z ` ( 1st ` u ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) ) -> ( 2nd ` s ) = ( 2nd ` u ) ) |
38 |
37
|
ad2ant2r |
|- ( ( ( ( Z ` ( 1st ` u ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` r ) ) /\ ( ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) ) -> ( 2nd ` s ) = ( 2nd ` u ) ) |
39 |
|
eqtr2 |
|- ( ( ( Z ` ( 1st ` v ) ) = ( 2nd ` r ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) -> ( 2nd ` r ) = ( 2nd ` v ) ) |
40 |
39
|
ad2ant2l |
|- ( ( ( ( Z ` ( 1st ` u ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` r ) ) /\ ( ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) ) -> ( 2nd ` r ) = ( 2nd ` v ) ) |
41 |
38 40
|
ineq12d |
|- ( ( ( ( Z ` ( 1st ` u ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` r ) ) /\ ( ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) ) -> ( ( 2nd ` s ) i^i ( 2nd ` r ) ) = ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) |
42 |
36 41
|
syl6bi |
|- ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( ( ( ( Z ` ( 1st ` s ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` r ) ) = ( 2nd ` r ) ) /\ ( ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) ) -> ( ( 2nd ` s ) i^i ( 2nd ` r ) ) = ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) |
43 |
42
|
com12 |
|- ( ( ( ( Z ` ( 1st ` s ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` r ) ) = ( 2nd ` r ) ) /\ ( ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) ) -> ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( ( 2nd ` s ) i^i ( 2nd ` r ) ) = ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) |
44 |
43
|
a1i |
|- ( Fun Z -> ( ( ( ( Z ` ( 1st ` s ) ) = ( 2nd ` s ) /\ ( Z ` ( 1st ` r ) ) = ( 2nd ` r ) ) /\ ( ( Z ` ( 1st ` u ) ) = ( 2nd ` u ) /\ ( Z ` ( 1st ` v ) ) = ( 2nd ` v ) ) ) -> ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( ( 2nd ` s ) i^i ( 2nd ` r ) ) = ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) |
45 |
21 26 44
|
syl2and |
|- ( Fun Z -> ( ( ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) -> ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( ( 2nd ` s ) i^i ( 2nd ` r ) ) = ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) |
46 |
45
|
expd |
|- ( Fun Z -> ( ( s e. Z /\ r e. Z ) -> ( ( u e. Z /\ v e. Z ) -> ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( ( 2nd ` s ) i^i ( 2nd ` r ) ) = ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) ) |
47 |
46
|
3imp1 |
|- ( ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) /\ ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) ) -> ( ( 2nd ` s ) i^i ( 2nd ` r ) ) = ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) |
48 |
47
|
difeq2d |
|- ( ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) /\ ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) ) -> ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) |
49 |
48
|
adantr |
|- ( ( ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) /\ ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) ) /\ ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) -> ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) |
50 |
|
eqeq12 |
|- ( ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) -> ( y = w <-> ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) |
51 |
50
|
adantl |
|- ( ( ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) /\ ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) ) /\ ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) -> ( y = w <-> ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) |
52 |
49 51
|
mpbird |
|- ( ( ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) /\ ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) ) /\ ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) -> y = w ) |
53 |
52
|
exp43 |
|- ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) -> ( ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) -> ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) -> ( w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) -> y = w ) ) ) ) |
54 |
53
|
adantld |
|- ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) -> ( ( 1o = 1o /\ ( ( 1st ` u ) = ( 1st ` s ) /\ ( 1st ` v ) = ( 1st ` r ) ) ) -> ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) -> ( w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) -> y = w ) ) ) ) |
55 |
16 54
|
syl5bi |
|- ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) -> ( ( ( 1st ` u ) |g ( 1st ` v ) ) = ( ( 1st ` s ) |g ( 1st ` r ) ) -> ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) -> ( w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) -> y = w ) ) ) ) |
56 |
1 55
|
syl5 |
|- ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ x = ( ( 1st ` s ) |g ( 1st ` r ) ) ) -> ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) -> ( w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) -> y = w ) ) ) ) |
57 |
56
|
expd |
|- ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) -> ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) -> ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) -> ( w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) -> y = w ) ) ) ) ) |
58 |
57
|
com35 |
|- ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) -> ( w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) -> ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) -> ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) -> y = w ) ) ) ) ) |
59 |
58
|
impd |
|- ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) -> ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) -> ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) -> y = w ) ) ) ) |
60 |
59
|
com24 |
|- ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) -> ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) -> ( y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) -> y = w ) ) ) ) |
61 |
60
|
impd |
|- ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) -> y = w ) ) ) |
62 |
61
|
3imp |
|- ( ( ( Fun Z /\ ( s e. Z /\ r e. Z ) /\ ( u e. Z /\ v e. Z ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) /\ ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) -> y = w ) |