Step |
Hyp |
Ref |
Expression |
1 |
|
xpssca.t |
|- T = ( R Xs. S ) |
2 |
|
xpssca.g |
|- G = ( Scalar ` R ) |
3 |
|
xpssca.1 |
|- ( ph -> R e. V ) |
4 |
|
xpssca.2 |
|- ( ph -> S e. W ) |
5 |
|
xpsvsca.x |
|- X = ( Base ` R ) |
6 |
|
xpsvsca.y |
|- Y = ( Base ` S ) |
7 |
|
xpsvsca.k |
|- K = ( Base ` G ) |
8 |
|
xpsvsca.m |
|- .x. = ( .s ` R ) |
9 |
|
xpsvsca.n |
|- .X. = ( .s ` S ) |
10 |
|
xpsvsca.p |
|- .xb = ( .s ` T ) |
11 |
|
xpsvsca.3 |
|- ( ph -> A e. K ) |
12 |
|
xpsvsca.4 |
|- ( ph -> B e. X ) |
13 |
|
xpsvsca.5 |
|- ( ph -> C e. Y ) |
14 |
|
xpsvsca.6 |
|- ( ph -> ( A .x. B ) e. X ) |
15 |
|
xpsvsca.7 |
|- ( ph -> ( A .X. C ) e. Y ) |
16 |
|
df-ov |
|- ( B ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) C ) = ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` <. B , C >. ) |
17 |
|
eqid |
|- ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) |
18 |
17
|
xpsfval |
|- ( ( B e. X /\ C e. Y ) -> ( B ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) C ) = { <. (/) , B >. , <. 1o , C >. } ) |
19 |
12 13 18
|
syl2anc |
|- ( ph -> ( B ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) C ) = { <. (/) , B >. , <. 1o , C >. } ) |
20 |
16 19
|
eqtr3id |
|- ( ph -> ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` <. B , C >. ) = { <. (/) , B >. , <. 1o , C >. } ) |
21 |
12 13
|
opelxpd |
|- ( ph -> <. B , C >. e. ( X X. Y ) ) |
22 |
17
|
xpsff1o2 |
|- ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) |
23 |
|
f1of |
|- ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -> ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) --> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ) |
24 |
22 23
|
ax-mp |
|- ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) --> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) |
25 |
24
|
ffvelrni |
|- ( <. B , C >. e. ( X X. Y ) -> ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` <. B , C >. ) e. ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ) |
26 |
21 25
|
syl |
|- ( ph -> ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` <. B , C >. ) e. ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ) |
27 |
20 26
|
eqeltrrd |
|- ( ph -> { <. (/) , B >. , <. 1o , C >. } e. ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ) |
28 |
|
eqid |
|- ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) = ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) |
29 |
1 5 6 3 4 17 2 28
|
xpsval |
|- ( ph -> T = ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) "s ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) ) |
30 |
1 5 6 3 4 17 2 28
|
xpsrnbas |
|- ( ph -> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) = ( Base ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) ) |
31 |
|
f1ocnv |
|- ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -1-1-onto-> ( X X. Y ) ) |
32 |
22 31
|
mp1i |
|- ( ph -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -1-1-onto-> ( X X. Y ) ) |
33 |
|
f1ofo |
|- ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -1-1-onto-> ( X X. Y ) -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -onto-> ( X X. Y ) ) |
34 |
32 33
|
syl |
|- ( ph -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -onto-> ( X X. Y ) ) |
35 |
|
ovexd |
|- ( ph -> ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) e. _V ) |
36 |
2
|
fvexi |
|- G e. _V |
37 |
36
|
a1i |
|- ( T. -> G e. _V ) |
38 |
|
prex |
|- { <. (/) , R >. , <. 1o , S >. } e. _V |
39 |
38
|
a1i |
|- ( T. -> { <. (/) , R >. , <. 1o , S >. } e. _V ) |
40 |
28 37 39
|
prdssca |
|- ( T. -> G = ( Scalar ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) ) |
41 |
40
|
mptru |
|- G = ( Scalar ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) |
42 |
|
eqid |
|- ( .s ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) = ( .s ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) |
43 |
32
|
f1ovscpbl |
|- ( ( ph /\ ( a e. K /\ b e. ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) /\ c e. ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ) ) -> ( ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` b ) = ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` c ) -> ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` ( a ( .s ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) b ) ) = ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` ( a ( .s ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) c ) ) ) ) |
44 |
29 30 34 35 41 7 42 10 43
|
imasvscaval |
|- ( ( ph /\ A e. K /\ { <. (/) , B >. , <. 1o , C >. } e. ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ) -> ( A .xb ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` { <. (/) , B >. , <. 1o , C >. } ) ) = ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` ( A ( .s ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) { <. (/) , B >. , <. 1o , C >. } ) ) ) |
45 |
11 27 44
|
mpd3an23 |
|- ( ph -> ( A .xb ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` { <. (/) , B >. , <. 1o , C >. } ) ) = ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` ( A ( .s ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) { <. (/) , B >. , <. 1o , C >. } ) ) ) |
46 |
|
f1ocnvfv |
|- ( ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) /\ <. B , C >. e. ( X X. Y ) ) -> ( ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` <. B , C >. ) = { <. (/) , B >. , <. 1o , C >. } -> ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` { <. (/) , B >. , <. 1o , C >. } ) = <. B , C >. ) ) |
47 |
22 21 46
|
sylancr |
|- ( ph -> ( ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` <. B , C >. ) = { <. (/) , B >. , <. 1o , C >. } -> ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` { <. (/) , B >. , <. 1o , C >. } ) = <. B , C >. ) ) |
48 |
20 47
|
mpd |
|- ( ph -> ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` { <. (/) , B >. , <. 1o , C >. } ) = <. B , C >. ) |
49 |
48
|
oveq2d |
|- ( ph -> ( A .xb ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` { <. (/) , B >. , <. 1o , C >. } ) ) = ( A .xb <. B , C >. ) ) |
50 |
|
iftrue |
|- ( k = (/) -> if ( k = (/) , R , S ) = R ) |
51 |
50
|
fveq2d |
|- ( k = (/) -> ( .s ` if ( k = (/) , R , S ) ) = ( .s ` R ) ) |
52 |
51 8
|
eqtr4di |
|- ( k = (/) -> ( .s ` if ( k = (/) , R , S ) ) = .x. ) |
53 |
|
eqidd |
|- ( k = (/) -> A = A ) |
54 |
|
iftrue |
|- ( k = (/) -> if ( k = (/) , B , C ) = B ) |
55 |
52 53 54
|
oveq123d |
|- ( k = (/) -> ( A ( .s ` if ( k = (/) , R , S ) ) if ( k = (/) , B , C ) ) = ( A .x. B ) ) |
56 |
|
iftrue |
|- ( k = (/) -> if ( k = (/) , ( A .x. B ) , ( A .X. C ) ) = ( A .x. B ) ) |
57 |
55 56
|
eqtr4d |
|- ( k = (/) -> ( A ( .s ` if ( k = (/) , R , S ) ) if ( k = (/) , B , C ) ) = if ( k = (/) , ( A .x. B ) , ( A .X. C ) ) ) |
58 |
|
iffalse |
|- ( -. k = (/) -> if ( k = (/) , R , S ) = S ) |
59 |
58
|
fveq2d |
|- ( -. k = (/) -> ( .s ` if ( k = (/) , R , S ) ) = ( .s ` S ) ) |
60 |
59 9
|
eqtr4di |
|- ( -. k = (/) -> ( .s ` if ( k = (/) , R , S ) ) = .X. ) |
61 |
|
eqidd |
|- ( -. k = (/) -> A = A ) |
62 |
|
iffalse |
|- ( -. k = (/) -> if ( k = (/) , B , C ) = C ) |
63 |
60 61 62
|
oveq123d |
|- ( -. k = (/) -> ( A ( .s ` if ( k = (/) , R , S ) ) if ( k = (/) , B , C ) ) = ( A .X. C ) ) |
64 |
|
iffalse |
|- ( -. k = (/) -> if ( k = (/) , ( A .x. B ) , ( A .X. C ) ) = ( A .X. C ) ) |
65 |
63 64
|
eqtr4d |
|- ( -. k = (/) -> ( A ( .s ` if ( k = (/) , R , S ) ) if ( k = (/) , B , C ) ) = if ( k = (/) , ( A .x. B ) , ( A .X. C ) ) ) |
66 |
57 65
|
pm2.61i |
|- ( A ( .s ` if ( k = (/) , R , S ) ) if ( k = (/) , B , C ) ) = if ( k = (/) , ( A .x. B ) , ( A .X. C ) ) |
67 |
3
|
adantr |
|- ( ( ph /\ k e. 2o ) -> R e. V ) |
68 |
4
|
adantr |
|- ( ( ph /\ k e. 2o ) -> S e. W ) |
69 |
|
simpr |
|- ( ( ph /\ k e. 2o ) -> k e. 2o ) |
70 |
|
fvprif |
|- ( ( R e. V /\ S e. W /\ k e. 2o ) -> ( { <. (/) , R >. , <. 1o , S >. } ` k ) = if ( k = (/) , R , S ) ) |
71 |
67 68 69 70
|
syl3anc |
|- ( ( ph /\ k e. 2o ) -> ( { <. (/) , R >. , <. 1o , S >. } ` k ) = if ( k = (/) , R , S ) ) |
72 |
71
|
fveq2d |
|- ( ( ph /\ k e. 2o ) -> ( .s ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) = ( .s ` if ( k = (/) , R , S ) ) ) |
73 |
|
eqidd |
|- ( ( ph /\ k e. 2o ) -> A = A ) |
74 |
12
|
adantr |
|- ( ( ph /\ k e. 2o ) -> B e. X ) |
75 |
13
|
adantr |
|- ( ( ph /\ k e. 2o ) -> C e. Y ) |
76 |
|
fvprif |
|- ( ( B e. X /\ C e. Y /\ k e. 2o ) -> ( { <. (/) , B >. , <. 1o , C >. } ` k ) = if ( k = (/) , B , C ) ) |
77 |
74 75 69 76
|
syl3anc |
|- ( ( ph /\ k e. 2o ) -> ( { <. (/) , B >. , <. 1o , C >. } ` k ) = if ( k = (/) , B , C ) ) |
78 |
72 73 77
|
oveq123d |
|- ( ( ph /\ k e. 2o ) -> ( A ( .s ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) ( { <. (/) , B >. , <. 1o , C >. } ` k ) ) = ( A ( .s ` if ( k = (/) , R , S ) ) if ( k = (/) , B , C ) ) ) |
79 |
14
|
adantr |
|- ( ( ph /\ k e. 2o ) -> ( A .x. B ) e. X ) |
80 |
15
|
adantr |
|- ( ( ph /\ k e. 2o ) -> ( A .X. C ) e. Y ) |
81 |
|
fvprif |
|- ( ( ( A .x. B ) e. X /\ ( A .X. C ) e. Y /\ k e. 2o ) -> ( { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ` k ) = if ( k = (/) , ( A .x. B ) , ( A .X. C ) ) ) |
82 |
79 80 69 81
|
syl3anc |
|- ( ( ph /\ k e. 2o ) -> ( { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ` k ) = if ( k = (/) , ( A .x. B ) , ( A .X. C ) ) ) |
83 |
66 78 82
|
3eqtr4a |
|- ( ( ph /\ k e. 2o ) -> ( A ( .s ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) ( { <. (/) , B >. , <. 1o , C >. } ` k ) ) = ( { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ` k ) ) |
84 |
83
|
mpteq2dva |
|- ( ph -> ( k e. 2o |-> ( A ( .s ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) ( { <. (/) , B >. , <. 1o , C >. } ` k ) ) ) = ( k e. 2o |-> ( { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ` k ) ) ) |
85 |
|
eqid |
|- ( Base ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) = ( Base ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) |
86 |
36
|
a1i |
|- ( ph -> G e. _V ) |
87 |
|
2on |
|- 2o e. On |
88 |
87
|
a1i |
|- ( ph -> 2o e. On ) |
89 |
|
fnpr2o |
|- ( ( R e. V /\ S e. W ) -> { <. (/) , R >. , <. 1o , S >. } Fn 2o ) |
90 |
3 4 89
|
syl2anc |
|- ( ph -> { <. (/) , R >. , <. 1o , S >. } Fn 2o ) |
91 |
27 30
|
eleqtrd |
|- ( ph -> { <. (/) , B >. , <. 1o , C >. } e. ( Base ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) ) |
92 |
28 85 42 7 86 88 90 11 91
|
prdsvscaval |
|- ( ph -> ( A ( .s ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) { <. (/) , B >. , <. 1o , C >. } ) = ( k e. 2o |-> ( A ( .s ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) ( { <. (/) , B >. , <. 1o , C >. } ` k ) ) ) ) |
93 |
|
fnpr2o |
|- ( ( ( A .x. B ) e. X /\ ( A .X. C ) e. Y ) -> { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } Fn 2o ) |
94 |
14 15 93
|
syl2anc |
|- ( ph -> { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } Fn 2o ) |
95 |
|
dffn5 |
|- ( { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } Fn 2o <-> { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } = ( k e. 2o |-> ( { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ` k ) ) ) |
96 |
94 95
|
sylib |
|- ( ph -> { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } = ( k e. 2o |-> ( { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ` k ) ) ) |
97 |
84 92 96
|
3eqtr4d |
|- ( ph -> ( A ( .s ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) { <. (/) , B >. , <. 1o , C >. } ) = { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ) |
98 |
97
|
fveq2d |
|- ( ph -> ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` ( A ( .s ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) { <. (/) , B >. , <. 1o , C >. } ) ) = ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ) ) |
99 |
|
df-ov |
|- ( ( A .x. B ) ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ( A .X. C ) ) = ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` <. ( A .x. B ) , ( A .X. C ) >. ) |
100 |
17
|
xpsfval |
|- ( ( ( A .x. B ) e. X /\ ( A .X. C ) e. Y ) -> ( ( A .x. B ) ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ( A .X. C ) ) = { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ) |
101 |
14 15 100
|
syl2anc |
|- ( ph -> ( ( A .x. B ) ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ( A .X. C ) ) = { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ) |
102 |
99 101
|
eqtr3id |
|- ( ph -> ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` <. ( A .x. B ) , ( A .X. C ) >. ) = { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ) |
103 |
14 15
|
opelxpd |
|- ( ph -> <. ( A .x. B ) , ( A .X. C ) >. e. ( X X. Y ) ) |
104 |
|
f1ocnvfv |
|- ( ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) /\ <. ( A .x. B ) , ( A .X. C ) >. e. ( X X. Y ) ) -> ( ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` <. ( A .x. B ) , ( A .X. C ) >. ) = { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } -> ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ) = <. ( A .x. B ) , ( A .X. C ) >. ) ) |
105 |
22 103 104
|
sylancr |
|- ( ph -> ( ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` <. ( A .x. B ) , ( A .X. C ) >. ) = { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } -> ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ) = <. ( A .x. B ) , ( A .X. C ) >. ) ) |
106 |
102 105
|
mpd |
|- ( ph -> ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` { <. (/) , ( A .x. B ) >. , <. 1o , ( A .X. C ) >. } ) = <. ( A .x. B ) , ( A .X. C ) >. ) |
107 |
98 106
|
eqtrd |
|- ( ph -> ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) ` ( A ( .s ` ( G Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) { <. (/) , B >. , <. 1o , C >. } ) ) = <. ( A .x. B ) , ( A .X. C ) >. ) |
108 |
45 49 107
|
3eqtr3d |
|- ( ph -> ( A .xb <. B , C >. ) = <. ( A .x. B ) , ( A .X. C ) >. ) |