Theorem xpsval 13455

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 2815	. . 3 |- ( ph -> R e. _V )
4		xpsval.2	. . . 4 |- ( ph -> S e. W )
5	4	elexd 2815	. . 3 |- ( ph -> S e. _V )
6		xpsval.x	. . . . . . 7 |- X = ( Base ` R )
7		basfn 13161	. . . . . . . 8 |- Base Fn _V
8		funfvex 5656	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5431	. . . . . . . 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 2317	. . . . . 6 |- ( ph -> X e. _V )
12		xpsval.y	. . . . . . 7 |- Y = ( Base ` S )
13		funfvex 5656	. . . . . . . . 9 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
14	13	funfni 5431	. . . . . . . 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 2317	. . . . . 6 |- ( ph -> Y e. _V )
17		xpsval.f	. . . . . . 7 |- F = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )
18	17	mpoexg 6378	. . . . . 6 |- ( ( X e. _V /\ Y e. _V ) -> F e. _V )
19	11, 16, 18	syl2anc 411	. . . . 5 |- ( ph -> F e. _V )
20		cnvexg 5273	. . . . 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 13256	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
25	24	slotex 13129	. . . . . . . 8 |- ( R e. V -> (Scalar ` R ) e. _V )
26	2, 25	syl 14	. . . . . . 7 |- ( ph -> (Scalar ` R ) e. _V )
27	23, 26	eqeltrid 2317	. . . . . 6 |- ( ph -> G e. _V )
28		0lt2o 6611	. . . . . . . 8 |- (/) e. 2o
29		opexg 4319	. . . . . . . 8 |- ( ( (/) e. 2o /\ R e. V ) -> <. (/) , R >. e. _V )
30	28, 2, 29	sylancr 414	. . . . . . 7 |- ( ph -> <. (/) , R >. e. _V )
31		1lt2o 6612	. . . . . . . 8 |- 1o e. 2o
32		opexg 4319	. . . . . . . 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 4300	. . . . . . 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 13372	. . . . . 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 2317	. . . 4 |- ( ph -> U e. _V )
39		imasex 13408	. . . 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 5639	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
42	41, 6	eqtr4di 2281	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = X )
43		fveq2 5639	. . . . . . . . 9 |- ( s = S -> ( Base ` s ) = ( Base ` S ) )
44	43, 12	eqtr4di 2281	. . . . . . . 8 |- ( s = S -> ( Base ` s ) = Y )
45		mpoeq12 6083	. . . . . . . 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 2281	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = F )
48	47	cnveqd 4905	. . . . 5 |- ( ( r = R /\ s = S ) -> `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = `' F )
49		fveq2 5639	. . . . . . . . 9 |- ( r = R -> (Scalar ` r ) = (Scalar ` R ) )
50	49	adantr 276	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> (Scalar ` r ) = (Scalar ` R ) )
51	50, 23	eqtr4di 2281	. . . . . . 7 |- ( ( r = R /\ s = S ) -> (Scalar ` r ) = G )
52		simpl 109	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> r = R )
53	52	opeq2d 3868	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> <. (/) , r >. = <. (/) , R >. )
54		simpr 110	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> s = S )
55	54	opeq2d 3868	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> <. 1o , s >. = <. 1o , S >. )
56	53, 55	preq12d 3755	. . . . . . 7 |- ( ( r = R /\ s = S ) -> { <. (/) , r >. , <. 1o , s >. } = { <. (/) , R >. , <. 1o , S >. } )
57	51, 56	oveq12d 6038	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) = ( G X_s { <. (/) , R >. , <. 1o , S >. } ) )
58	57, 22	eqtr4di 2281	. . . . 5 |- ( ( r = R /\ s = S ) -> ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) = U )
59	48, 58	oveq12d 6038	. . . 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 13407	. . . 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 6153	. . 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 1273	. 2 |- ( ph -> ( R X.s S ) = ( `' F "s U ) )
63	1, 62	eqtrid 2275	1 |- ( ph -> T = ( `' F "s U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2201   _Vcvv 2801   (/)c0 3493   {cpr 3669   <.cop 3671   `'ccnv 4723   Fn wfn 5320   ` cfv 5325  (class class class)co 6020   e. cmpo 6022   1oc1o 6577   2oc2o 6578   Basecbs 13102  Scalarcsca 13183   X__scprds 13368   "_s cimas 13402   X._s cxps 13404

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-addcom 8134  ax-mulcom 8135  ax-addass 8136  ax-mulass 8137  ax-distr 8138  ax-i2m1 8139  ax-1rid 8141  ax-0id 8142  ax-rnegex 8143  ax-cnre 8145

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3653  df-sn 3674  df-pr 3675  df-tp 3676  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-id 4389  df-iord 4462  df-on 4464  df-suc 4467  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-1o 6584  df-2o 6585  df-map 6821  df-ixp 6870  df-sup 7185  df-sub 8354  df-inn 9146  df-2 9204  df-3 9205  df-4 9206  df-5 9207  df-6 9208  df-7 9209  df-8 9210  df-9 9211  df-n0 9405  df-dec 9614  df-ndx 13105  df-slot 13106  df-base 13108  df-plusg 13193  df-mulr 13194  df-sca 13196  df-vsca 13197  df-ip 13198  df-tset 13199  df-ple 13200  df-ds 13202  df-hom 13204  df-cco 13205  df-rest 13344  df-topn 13345  df-topgen 13363  df-pt 13364  df-prds 13370  df-iimas 13405  df-xps 13407

This theorem is referenced by: (None)