Theorem xpsval 13498

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 2817	. . 3 |- ( ph -> R e. _V )
4		xpsval.2	. . . 4 |- ( ph -> S e. W )
5	4	elexd 2817	. . 3 |- ( ph -> S e. _V )
6		xpsval.x	. . . . . . 7 |- X = ( Base ` R )
7		basfn 13204	. . . . . . . 8 |- Base Fn _V
8		funfvex 5665	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5439	. . . . . . . 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 2318	. . . . . 6 |- ( ph -> X e. _V )
12		xpsval.y	. . . . . . 7 |- Y = ( Base ` S )
13		funfvex 5665	. . . . . . . . 9 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
14	13	funfni 5439	. . . . . . . 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 2318	. . . . . 6 |- ( ph -> Y e. _V )
17		xpsval.f	. . . . . . 7 |- F = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )
18	17	mpoexg 6385	. . . . . 6 |- ( ( X e. _V /\ Y e. _V ) -> F e. _V )
19	11, 16, 18	syl2anc 411	. . . . 5 |- ( ph -> F e. _V )
20		cnvexg 5281	. . . . 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 13299	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
25	24	slotex 13172	. . . . . . . 8 |- ( R e. V -> (Scalar ` R ) e. _V )
26	2, 25	syl 14	. . . . . . 7 |- ( ph -> (Scalar ` R ) e. _V )
27	23, 26	eqeltrid 2318	. . . . . 6 |- ( ph -> G e. _V )
28		0lt2o 6652	. . . . . . . 8 |- (/) e. 2o
29		opexg 4326	. . . . . . . 8 |- ( ( (/) e. 2o /\ R e. V ) -> <. (/) , R >. e. _V )
30	28, 2, 29	sylancr 414	. . . . . . 7 |- ( ph -> <. (/) , R >. e. _V )
31		1lt2o 6653	. . . . . . . 8 |- 1o e. 2o
32		opexg 4326	. . . . . . . 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 4307	. . . . . . 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 13415	. . . . . 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 2318	. . . 4 |- ( ph -> U e. _V )
39		imasex 13451	. . . 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 5648	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
42	41, 6	eqtr4di 2282	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = X )
43		fveq2 5648	. . . . . . . . 9 |- ( s = S -> ( Base ` s ) = ( Base ` S ) )
44	43, 12	eqtr4di 2282	. . . . . . . 8 |- ( s = S -> ( Base ` s ) = Y )
45		mpoeq12 6091	. . . . . . . 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 2282	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = F )
48	47	cnveqd 4912	. . . . 5 |- ( ( r = R /\ s = S ) -> `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = `' F )
49		fveq2 5648	. . . . . . . . 9 |- ( r = R -> (Scalar ` r ) = (Scalar ` R ) )
50	49	adantr 276	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> (Scalar ` r ) = (Scalar ` R ) )
51	50, 23	eqtr4di 2282	. . . . . . 7 |- ( ( r = R /\ s = S ) -> (Scalar ` r ) = G )
52		simpl 109	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> r = R )
53	52	opeq2d 3874	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> <. (/) , r >. = <. (/) , R >. )
54		simpr 110	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> s = S )
55	54	opeq2d 3874	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> <. 1o , s >. = <. 1o , S >. )
56	53, 55	preq12d 3760	. . . . . . 7 |- ( ( r = R /\ s = S ) -> { <. (/) , r >. , <. 1o , s >. } = { <. (/) , R >. , <. 1o , S >. } )
57	51, 56	oveq12d 6046	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) = ( G X_s { <. (/) , R >. , <. 1o , S >. } ) )
58	57, 22	eqtr4di 2282	. . . . 5 |- ( ( r = R /\ s = S ) -> ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) = U )
59	48, 58	oveq12d 6046	. . . 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 13450	. . . 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 6161	. . 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 1274	. 2 |- ( ph -> ( R X.s S ) = ( `' F "s U ) )
63	1, 62	eqtrid 2276	1 |- ( ph -> T = ( `' F "s U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   (/)c0 3496   {cpr 3674   <.cop 3676   `'ccnv 4730   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   1oc1o 6618   2oc2o 6619   Basecbs 13145  Scalarcsca 13226   X__scprds 13411   "_s cimas 13445   X._s cxps 13447

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-map 6862  df-ixp 6911  df-sup 7226  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-mulr 13237  df-sca 13239  df-vsca 13240  df-ip 13241  df-tset 13242  df-ple 13243  df-ds 13245  df-hom 13247  df-cco 13248  df-rest 13387  df-topn 13388  df-topgen 13406  df-pt 13407  df-prds 13413  df-iimas 13448  df-xps 13450

This theorem is referenced by: (None)