Theorem xpsval 13184

Description: Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)

Hypotheses

Ref	Expression
xpsval.t	|- T = ( R X.s S )
xpsval.x	|- X = ( Base ` R )
xpsval.y	|- Y = ( Base ` S )
xpsval.1	|- ( ph -> R e. V )
xpsval.2	|- ( ph -> S e. W )
xpsval.f	|- F = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )
xpsval.k	|- G = (Scalar ` R )
xpsval.u	|- U = ( G X_s { <. (/) , R >. , <. 1o , S >. } )

Assertion

Ref	Expression
xpsval	|- ( ph -> T = ( `' F "s U ) )

Distinct variable groups:   x, y   x, W   x, X, y   x, R   x, Y, y

Allowed substitution hints:   ph( x, y)   R( y)   S( x, y)   T( x, y)   U( x, y)   F( x, y)   G( x, y)   V( x, y)   W( y)

Proof of Theorem xpsval

Dummy variables s r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsval.t	. 2 |- T = ( R X.s S )
2		xpsval.1	. . . 4 |- ( ph -> R e. V )
3	2	elexd 2785	. . 3 |- ( ph -> R e. _V )
4		xpsval.2	. . . 4 |- ( ph -> S e. W )
5	4	elexd 2785	. . 3 |- ( ph -> S e. _V )
6		xpsval.x	. . . . . . 7 |- X = ( Base ` R )
7		basfn 12890	. . . . . . . 8 |- Base Fn _V
8		funfvex 5593	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5376	. . . . . . . 8 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
10	7, 3, 9	sylancr 414	. . . . . . 7 |- ( ph -> ( Base ` R ) e. _V )
11	6, 10	eqeltrid 2292	. . . . . 6 |- ( ph -> X e. _V )
12		xpsval.y	. . . . . . 7 |- Y = ( Base ` S )
13		funfvex 5593	. . . . . . . . 9 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
14	13	funfni 5376	. . . . . . . 8 |- ( ( Base Fn _V /\ S e. _V ) -> ( Base ` S ) e. _V )
15	7, 5, 14	sylancr 414	. . . . . . 7 |- ( ph -> ( Base ` S ) e. _V )
16	12, 15	eqeltrid 2292	. . . . . 6 |- ( ph -> Y e. _V )
17		xpsval.f	. . . . . . 7 |- F = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )
18	17	mpoexg 6297	. . . . . 6 |- ( ( X e. _V /\ Y e. _V ) -> F e. _V )
19	11, 16, 18	syl2anc 411	. . . . 5 |- ( ph -> F e. _V )
20		cnvexg 5220	. . . . 5 |- ( F e. _V -> `' F e. _V )
21	19, 20	syl 14	. . . 4 |- ( ph -> `' F e. _V )
22		xpsval.u	. . . . 5 |- U = ( G X_s { <. (/) , R >. , <. 1o , S >. } )
23		xpsval.k	. . . . . . 7 |- G = (Scalar ` R )
24		scaslid 12985	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
25	24	slotex 12859	. . . . . . . 8 |- ( R e. V -> (Scalar ` R ) e. _V )
26	2, 25	syl 14	. . . . . . 7 |- ( ph -> (Scalar ` R ) e. _V )
27	23, 26	eqeltrid 2292	. . . . . 6 |- ( ph -> G e. _V )
28		0lt2o 6527	. . . . . . . 8 |- (/) e. 2o
29		opexg 4272	. . . . . . . 8 |- ( ( (/) e. 2o /\ R e. V ) -> <. (/) , R >. e. _V )
30	28, 2, 29	sylancr 414	. . . . . . 7 |- ( ph -> <. (/) , R >. e. _V )
31		1lt2o 6528	. . . . . . . 8 |- 1o e. 2o
32		opexg 4272	. . . . . . . 8 |- ( ( 1o e. 2o /\ S e. W ) -> <. 1o , S >. e. _V )
33	31, 4, 32	sylancr 414	. . . . . . 7 |- ( ph -> <. 1o , S >. e. _V )
34		prexg 4255	. . . . . . 7 |- ( ( <. (/) , R >. e. _V /\ <. 1o , S >. e. _V ) -> { <. (/) , R >. , <. 1o , S >. } e. _V )
35	30, 33, 34	syl2anc 411	. . . . . 6 |- ( ph -> { <. (/) , R >. , <. 1o , S >. } e. _V )
36		prdsex 13101	. . . . . 6 |- ( ( G e. _V /\ { <. (/) , R >. , <. 1o , S >. } e. _V ) -> ( G X_s { <. (/) , R >. , <. 1o , S >. } ) e. _V )
37	27, 35, 36	syl2anc 411	. . . . 5 |- ( ph -> ( G X_s { <. (/) , R >. , <. 1o , S >. } ) e. _V )
38	22, 37	eqeltrid 2292	. . . 4 |- ( ph -> U e. _V )
39		imasex 13137	. . . 4 |- ( ( `' F e. _V /\ U e. _V ) -> ( `' F "s U ) e. _V )
40	21, 38, 39	syl2anc 411	. . 3 |- ( ph -> ( `' F "s U ) e. _V )
41		fveq2 5576	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
42	41, 6	eqtr4di 2256	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = X )
43		fveq2 5576	. . . . . . . . 9 |- ( s = S -> ( Base ` s ) = ( Base ` S ) )
44	43, 12	eqtr4di 2256	. . . . . . . 8 |- ( s = S -> ( Base ` s ) = Y )
45		mpoeq12 6005	. . . . . . . 8 |- ( ( ( Base ` r ) = X /\ ( Base ` s ) = Y ) -> ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) )
46	42, 44, 45	syl2an 289	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) )
47	46, 17	eqtr4di 2256	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = F )
48	47	cnveqd 4854	. . . . 5 |- ( ( r = R /\ s = S ) -> `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = `' F )
49		fveq2 5576	. . . . . . . . 9 |- ( r = R -> (Scalar ` r ) = (Scalar ` R ) )
50	49	adantr 276	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> (Scalar ` r ) = (Scalar ` R ) )
51	50, 23	eqtr4di 2256	. . . . . . 7 |- ( ( r = R /\ s = S ) -> (Scalar ` r ) = G )
52		simpl 109	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> r = R )
53	52	opeq2d 3826	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> <. (/) , r >. = <. (/) , R >. )
54		simpr 110	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> s = S )
55	54	opeq2d 3826	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> <. 1o , s >. = <. 1o , S >. )
56	53, 55	preq12d 3718	. . . . . . 7 |- ( ( r = R /\ s = S ) -> { <. (/) , r >. , <. 1o , s >. } = { <. (/) , R >. , <. 1o , S >. } )
57	51, 56	oveq12d 5962	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) = ( G X_s { <. (/) , R >. , <. 1o , S >. } ) )
58	57, 22	eqtr4di 2256	. . . . 5 |- ( ( r = R /\ s = S ) -> ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) = U )
59	48, 58	oveq12d 5962	. . . 4 |- ( ( r = R /\ s = S ) -> ( `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) "s ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) ) = ( `' F "s U ) )
60		df-xps 13136	. . . 4 |- X.s = ( r e. _V , s e. _V |-> ( `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) "s ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) ) )
61	59, 60	ovmpoga 6075	. . 3 |- ( ( R e. _V /\ S e. _V /\ ( `' F "s U ) e. _V ) -> ( R X.s S ) = ( `' F "s U ) )
62	3, 5, 40, 61	syl3anc 1250	. 2 |- ( ph -> ( R X.s S ) = ( `' F "s U ) )
63	1, 62	eqtrid 2250	1 |- ( ph -> T = ( `' F "s U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   (/)c0 3460   {cpr 3634   <.cop 3636   `'ccnv 4674   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   e. cmpo 5946   1oc1o 6495   2oc2o 6496   Basecbs 12832  Scalarcsca 12912   X__scprds 13097   "_s cimas 13131   X._s cxps 13133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-tp 3641  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-1o 6502  df-2o 6503  df-map 6737  df-ixp 6786  df-sup 7086  df-sub 8245  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-7 9100  df-8 9101  df-9 9102  df-n0 9296  df-dec 9505  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-mulr 12923  df-sca 12925  df-vsca 12926  df-ip 12927  df-tset 12928  df-ple 12929  df-ds 12931  df-hom 12933  df-cco 12934  df-rest 13073  df-topn 13074  df-topgen 13092  df-pt 13093  df-prds 13099  df-iimas 13134  df-xps 13136

This theorem is referenced by: (None)