Step |
Hyp |
Ref |
Expression |
1 |
|
xpsval.t |
|- T = ( R Xs. S ) |
2 |
|
xpsval.x |
|- X = ( Base ` R ) |
3 |
|
xpsval.y |
|- Y = ( Base ` S ) |
4 |
|
xpsval.1 |
|- ( ph -> R e. V ) |
5 |
|
xpsval.2 |
|- ( ph -> S e. W ) |
6 |
|
xpsadd.3 |
|- ( ph -> A e. X ) |
7 |
|
xpsadd.4 |
|- ( ph -> B e. Y ) |
8 |
|
xpsadd.5 |
|- ( ph -> C e. X ) |
9 |
|
xpsadd.6 |
|- ( ph -> D e. Y ) |
10 |
|
xpsadd.7 |
|- ( ph -> ( A .x. C ) e. X ) |
11 |
|
xpsadd.8 |
|- ( ph -> ( B .X. D ) e. Y ) |
12 |
|
xpsaddlem.m |
|- .x. = ( E ` R ) |
13 |
|
xpsaddlem.n |
|- .X. = ( E ` S ) |
14 |
|
xpsaddlem.p |
|- .xb = ( E ` T ) |
15 |
|
xpsaddlem.f |
|- F = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) |
16 |
|
xpsaddlem.u |
|- U = ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } ) |
17 |
|
xpsaddlem.1 |
|- ( ( ph /\ { <. (/) , A >. , <. 1o , B >. } e. ran F /\ { <. (/) , C >. , <. 1o , D >. } e. ran F ) -> ( ( `' F ` { <. (/) , A >. , <. 1o , B >. } ) .xb ( `' F ` { <. (/) , C >. , <. 1o , D >. } ) ) = ( `' F ` ( { <. (/) , A >. , <. 1o , B >. } ( E ` U ) { <. (/) , C >. , <. 1o , D >. } ) ) ) |
18 |
|
xpsaddlem.2 |
|- ( ( { <. (/) , R >. , <. 1o , S >. } Fn 2o /\ { <. (/) , A >. , <. 1o , B >. } e. ( Base ` U ) /\ { <. (/) , C >. , <. 1o , D >. } e. ( Base ` U ) ) -> ( { <. (/) , A >. , <. 1o , B >. } ( E ` U ) { <. (/) , C >. , <. 1o , D >. } ) = ( k e. 2o |-> ( ( { <. (/) , A >. , <. 1o , B >. } ` k ) ( E ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) ( { <. (/) , C >. , <. 1o , D >. } ` k ) ) ) ) |
19 |
|
df-ov |
|- ( A F B ) = ( F ` <. A , B >. ) |
20 |
15
|
xpsfval |
|- ( ( A e. X /\ B e. Y ) -> ( A F B ) = { <. (/) , A >. , <. 1o , B >. } ) |
21 |
6 7 20
|
syl2anc |
|- ( ph -> ( A F B ) = { <. (/) , A >. , <. 1o , B >. } ) |
22 |
19 21
|
eqtr3id |
|- ( ph -> ( F ` <. A , B >. ) = { <. (/) , A >. , <. 1o , B >. } ) |
23 |
6 7
|
opelxpd |
|- ( ph -> <. A , B >. e. ( X X. Y ) ) |
24 |
15
|
xpsff1o2 |
|- F : ( X X. Y ) -1-1-onto-> ran F |
25 |
|
f1of |
|- ( F : ( X X. Y ) -1-1-onto-> ran F -> F : ( X X. Y ) --> ran F ) |
26 |
24 25
|
ax-mp |
|- F : ( X X. Y ) --> ran F |
27 |
26
|
ffvelrni |
|- ( <. A , B >. e. ( X X. Y ) -> ( F ` <. A , B >. ) e. ran F ) |
28 |
23 27
|
syl |
|- ( ph -> ( F ` <. A , B >. ) e. ran F ) |
29 |
22 28
|
eqeltrrd |
|- ( ph -> { <. (/) , A >. , <. 1o , B >. } e. ran F ) |
30 |
|
df-ov |
|- ( C F D ) = ( F ` <. C , D >. ) |
31 |
15
|
xpsfval |
|- ( ( C e. X /\ D e. Y ) -> ( C F D ) = { <. (/) , C >. , <. 1o , D >. } ) |
32 |
8 9 31
|
syl2anc |
|- ( ph -> ( C F D ) = { <. (/) , C >. , <. 1o , D >. } ) |
33 |
30 32
|
eqtr3id |
|- ( ph -> ( F ` <. C , D >. ) = { <. (/) , C >. , <. 1o , D >. } ) |
34 |
8 9
|
opelxpd |
|- ( ph -> <. C , D >. e. ( X X. Y ) ) |
35 |
26
|
ffvelrni |
|- ( <. C , D >. e. ( X X. Y ) -> ( F ` <. C , D >. ) e. ran F ) |
36 |
34 35
|
syl |
|- ( ph -> ( F ` <. C , D >. ) e. ran F ) |
37 |
33 36
|
eqeltrrd |
|- ( ph -> { <. (/) , C >. , <. 1o , D >. } e. ran F ) |
38 |
29 37 17
|
mpd3an23 |
|- ( ph -> ( ( `' F ` { <. (/) , A >. , <. 1o , B >. } ) .xb ( `' F ` { <. (/) , C >. , <. 1o , D >. } ) ) = ( `' F ` ( { <. (/) , A >. , <. 1o , B >. } ( E ` U ) { <. (/) , C >. , <. 1o , D >. } ) ) ) |
39 |
|
f1ocnvfv |
|- ( ( F : ( X X. Y ) -1-1-onto-> ran F /\ <. A , B >. e. ( X X. Y ) ) -> ( ( F ` <. A , B >. ) = { <. (/) , A >. , <. 1o , B >. } -> ( `' F ` { <. (/) , A >. , <. 1o , B >. } ) = <. A , B >. ) ) |
40 |
24 23 39
|
sylancr |
|- ( ph -> ( ( F ` <. A , B >. ) = { <. (/) , A >. , <. 1o , B >. } -> ( `' F ` { <. (/) , A >. , <. 1o , B >. } ) = <. A , B >. ) ) |
41 |
22 40
|
mpd |
|- ( ph -> ( `' F ` { <. (/) , A >. , <. 1o , B >. } ) = <. A , B >. ) |
42 |
|
f1ocnvfv |
|- ( ( F : ( X X. Y ) -1-1-onto-> ran F /\ <. C , D >. e. ( X X. Y ) ) -> ( ( F ` <. C , D >. ) = { <. (/) , C >. , <. 1o , D >. } -> ( `' F ` { <. (/) , C >. , <. 1o , D >. } ) = <. C , D >. ) ) |
43 |
24 34 42
|
sylancr |
|- ( ph -> ( ( F ` <. C , D >. ) = { <. (/) , C >. , <. 1o , D >. } -> ( `' F ` { <. (/) , C >. , <. 1o , D >. } ) = <. C , D >. ) ) |
44 |
33 43
|
mpd |
|- ( ph -> ( `' F ` { <. (/) , C >. , <. 1o , D >. } ) = <. C , D >. ) |
45 |
41 44
|
oveq12d |
|- ( ph -> ( ( `' F ` { <. (/) , A >. , <. 1o , B >. } ) .xb ( `' F ` { <. (/) , C >. , <. 1o , D >. } ) ) = ( <. A , B >. .xb <. C , D >. ) ) |
46 |
|
iftrue |
|- ( k = (/) -> if ( k = (/) , R , S ) = R ) |
47 |
46
|
fveq2d |
|- ( k = (/) -> ( E ` if ( k = (/) , R , S ) ) = ( E ` R ) ) |
48 |
47 12
|
eqtr4di |
|- ( k = (/) -> ( E ` if ( k = (/) , R , S ) ) = .x. ) |
49 |
|
iftrue |
|- ( k = (/) -> if ( k = (/) , A , B ) = A ) |
50 |
|
iftrue |
|- ( k = (/) -> if ( k = (/) , C , D ) = C ) |
51 |
48 49 50
|
oveq123d |
|- ( k = (/) -> ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) = ( A .x. C ) ) |
52 |
|
iftrue |
|- ( k = (/) -> if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) = ( A .x. C ) ) |
53 |
51 52
|
eqtr4d |
|- ( k = (/) -> ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) = if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) ) |
54 |
|
iffalse |
|- ( -. k = (/) -> if ( k = (/) , R , S ) = S ) |
55 |
54
|
fveq2d |
|- ( -. k = (/) -> ( E ` if ( k = (/) , R , S ) ) = ( E ` S ) ) |
56 |
55 13
|
eqtr4di |
|- ( -. k = (/) -> ( E ` if ( k = (/) , R , S ) ) = .X. ) |
57 |
|
iffalse |
|- ( -. k = (/) -> if ( k = (/) , A , B ) = B ) |
58 |
|
iffalse |
|- ( -. k = (/) -> if ( k = (/) , C , D ) = D ) |
59 |
56 57 58
|
oveq123d |
|- ( -. k = (/) -> ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) = ( B .X. D ) ) |
60 |
|
iffalse |
|- ( -. k = (/) -> if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) = ( B .X. D ) ) |
61 |
59 60
|
eqtr4d |
|- ( -. k = (/) -> ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) = if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) ) |
62 |
53 61
|
pm2.61i |
|- ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) = if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) |
63 |
4
|
adantr |
|- ( ( ph /\ k e. 2o ) -> R e. V ) |
64 |
5
|
adantr |
|- ( ( ph /\ k e. 2o ) -> S e. W ) |
65 |
|
simpr |
|- ( ( ph /\ k e. 2o ) -> k e. 2o ) |
66 |
|
fvprif |
|- ( ( R e. V /\ S e. W /\ k e. 2o ) -> ( { <. (/) , R >. , <. 1o , S >. } ` k ) = if ( k = (/) , R , S ) ) |
67 |
63 64 65 66
|
syl3anc |
|- ( ( ph /\ k e. 2o ) -> ( { <. (/) , R >. , <. 1o , S >. } ` k ) = if ( k = (/) , R , S ) ) |
68 |
67
|
fveq2d |
|- ( ( ph /\ k e. 2o ) -> ( E ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) = ( E ` if ( k = (/) , R , S ) ) ) |
69 |
6
|
adantr |
|- ( ( ph /\ k e. 2o ) -> A e. X ) |
70 |
7
|
adantr |
|- ( ( ph /\ k e. 2o ) -> B e. Y ) |
71 |
|
fvprif |
|- ( ( A e. X /\ B e. Y /\ k e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` k ) = if ( k = (/) , A , B ) ) |
72 |
69 70 65 71
|
syl3anc |
|- ( ( ph /\ k e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` k ) = if ( k = (/) , A , B ) ) |
73 |
8
|
adantr |
|- ( ( ph /\ k e. 2o ) -> C e. X ) |
74 |
9
|
adantr |
|- ( ( ph /\ k e. 2o ) -> D e. Y ) |
75 |
|
fvprif |
|- ( ( C e. X /\ D e. Y /\ k e. 2o ) -> ( { <. (/) , C >. , <. 1o , D >. } ` k ) = if ( k = (/) , C , D ) ) |
76 |
73 74 65 75
|
syl3anc |
|- ( ( ph /\ k e. 2o ) -> ( { <. (/) , C >. , <. 1o , D >. } ` k ) = if ( k = (/) , C , D ) ) |
77 |
68 72 76
|
oveq123d |
|- ( ( ph /\ k e. 2o ) -> ( ( { <. (/) , A >. , <. 1o , B >. } ` k ) ( E ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) ( { <. (/) , C >. , <. 1o , D >. } ` k ) ) = ( if ( k = (/) , A , B ) ( E ` if ( k = (/) , R , S ) ) if ( k = (/) , C , D ) ) ) |
78 |
10
|
adantr |
|- ( ( ph /\ k e. 2o ) -> ( A .x. C ) e. X ) |
79 |
11
|
adantr |
|- ( ( ph /\ k e. 2o ) -> ( B .X. D ) e. Y ) |
80 |
|
fvprif |
|- ( ( ( A .x. C ) e. X /\ ( B .X. D ) e. Y /\ k e. 2o ) -> ( { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ` k ) = if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) ) |
81 |
78 79 65 80
|
syl3anc |
|- ( ( ph /\ k e. 2o ) -> ( { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ` k ) = if ( k = (/) , ( A .x. C ) , ( B .X. D ) ) ) |
82 |
62 77 81
|
3eqtr4a |
|- ( ( ph /\ k e. 2o ) -> ( ( { <. (/) , A >. , <. 1o , B >. } ` k ) ( E ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) ( { <. (/) , C >. , <. 1o , D >. } ` k ) ) = ( { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ` k ) ) |
83 |
82
|
mpteq2dva |
|- ( ph -> ( k e. 2o |-> ( ( { <. (/) , A >. , <. 1o , B >. } ` k ) ( E ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) ( { <. (/) , C >. , <. 1o , D >. } ` k ) ) ) = ( k e. 2o |-> ( { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ` k ) ) ) |
84 |
|
fnpr2o |
|- ( ( R e. V /\ S e. W ) -> { <. (/) , R >. , <. 1o , S >. } Fn 2o ) |
85 |
4 5 84
|
syl2anc |
|- ( ph -> { <. (/) , R >. , <. 1o , S >. } Fn 2o ) |
86 |
|
eqid |
|- ( Scalar ` R ) = ( Scalar ` R ) |
87 |
1 2 3 4 5 15 86 16
|
xpsrnbas |
|- ( ph -> ran F = ( Base ` U ) ) |
88 |
29 87
|
eleqtrd |
|- ( ph -> { <. (/) , A >. , <. 1o , B >. } e. ( Base ` U ) ) |
89 |
37 87
|
eleqtrd |
|- ( ph -> { <. (/) , C >. , <. 1o , D >. } e. ( Base ` U ) ) |
90 |
85 88 89 18
|
syl3anc |
|- ( ph -> ( { <. (/) , A >. , <. 1o , B >. } ( E ` U ) { <. (/) , C >. , <. 1o , D >. } ) = ( k e. 2o |-> ( ( { <. (/) , A >. , <. 1o , B >. } ` k ) ( E ` ( { <. (/) , R >. , <. 1o , S >. } ` k ) ) ( { <. (/) , C >. , <. 1o , D >. } ` k ) ) ) ) |
91 |
|
fnpr2o |
|- ( ( ( A .x. C ) e. X /\ ( B .X. D ) e. Y ) -> { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } Fn 2o ) |
92 |
10 11 91
|
syl2anc |
|- ( ph -> { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } Fn 2o ) |
93 |
|
dffn5 |
|- ( { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } Fn 2o <-> { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } = ( k e. 2o |-> ( { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ` k ) ) ) |
94 |
92 93
|
sylib |
|- ( ph -> { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } = ( k e. 2o |-> ( { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ` k ) ) ) |
95 |
83 90 94
|
3eqtr4d |
|- ( ph -> ( { <. (/) , A >. , <. 1o , B >. } ( E ` U ) { <. (/) , C >. , <. 1o , D >. } ) = { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ) |
96 |
95
|
fveq2d |
|- ( ph -> ( `' F ` ( { <. (/) , A >. , <. 1o , B >. } ( E ` U ) { <. (/) , C >. , <. 1o , D >. } ) ) = ( `' F ` { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ) ) |
97 |
|
df-ov |
|- ( ( A .x. C ) F ( B .X. D ) ) = ( F ` <. ( A .x. C ) , ( B .X. D ) >. ) |
98 |
15
|
xpsfval |
|- ( ( ( A .x. C ) e. X /\ ( B .X. D ) e. Y ) -> ( ( A .x. C ) F ( B .X. D ) ) = { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ) |
99 |
10 11 98
|
syl2anc |
|- ( ph -> ( ( A .x. C ) F ( B .X. D ) ) = { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ) |
100 |
97 99
|
eqtr3id |
|- ( ph -> ( F ` <. ( A .x. C ) , ( B .X. D ) >. ) = { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ) |
101 |
10 11
|
opelxpd |
|- ( ph -> <. ( A .x. C ) , ( B .X. D ) >. e. ( X X. Y ) ) |
102 |
|
f1ocnvfv |
|- ( ( F : ( X X. Y ) -1-1-onto-> ran F /\ <. ( A .x. C ) , ( B .X. D ) >. e. ( X X. Y ) ) -> ( ( F ` <. ( A .x. C ) , ( B .X. D ) >. ) = { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } -> ( `' F ` { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ) = <. ( A .x. C ) , ( B .X. D ) >. ) ) |
103 |
24 101 102
|
sylancr |
|- ( ph -> ( ( F ` <. ( A .x. C ) , ( B .X. D ) >. ) = { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } -> ( `' F ` { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ) = <. ( A .x. C ) , ( B .X. D ) >. ) ) |
104 |
100 103
|
mpd |
|- ( ph -> ( `' F ` { <. (/) , ( A .x. C ) >. , <. 1o , ( B .X. D ) >. } ) = <. ( A .x. C ) , ( B .X. D ) >. ) |
105 |
96 104
|
eqtrd |
|- ( ph -> ( `' F ` ( { <. (/) , A >. , <. 1o , B >. } ( E ` U ) { <. (/) , C >. , <. 1o , D >. } ) ) = <. ( A .x. C ) , ( B .X. D ) >. ) |
106 |
38 45 105
|
3eqtr3d |
|- ( ph -> ( <. A , B >. .xb <. C , D >. ) = <. ( A .x. C ) , ( B .X. D ) >. ) |