Theorem xpsval 12935

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 2773	. . 3 |- ( ph -> R e. _V )
4		xpsval.2	. . . 4 |- ( ph -> S e. W )
5	4	elexd 2773	. . 3 |- ( ph -> S e. _V )
6		xpsval.x	. . . . . . 7 |- X = ( Base ` R )
7		basfn 12676	. . . . . . . 8 |- Base Fn _V
8		funfvex 5571	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5354	. . . . . . . 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 2280	. . . . . 6 |- ( ph -> X e. _V )
12		xpsval.y	. . . . . . 7 |- Y = ( Base ` S )
13		funfvex 5571	. . . . . . . . 9 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
14	13	funfni 5354	. . . . . . . 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 2280	. . . . . 6 |- ( ph -> Y e. _V )
17		xpsval.f	. . . . . . 7 |- F = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )
18	17	mpoexg 6264	. . . . . 6 |- ( ( X e. _V /\ Y e. _V ) -> F e. _V )
19	11, 16, 18	syl2anc 411	. . . . 5 |- ( ph -> F e. _V )
20		cnvexg 5203	. . . . 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 12770	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
25	24	slotex 12645	. . . . . . . 8 |- ( R e. V -> (Scalar ` R ) e. _V )
26	2, 25	syl 14	. . . . . . 7 |- ( ph -> (Scalar ` R ) e. _V )
27	23, 26	eqeltrid 2280	. . . . . 6 |- ( ph -> G e. _V )
28		0lt2o 6494	. . . . . . . 8 |- (/) e. 2o
29		opexg 4257	. . . . . . . 8 |- ( ( (/) e. 2o /\ R e. V ) -> <. (/) , R >. e. _V )
30	28, 2, 29	sylancr 414	. . . . . . 7 |- ( ph -> <. (/) , R >. e. _V )
31		1lt2o 6495	. . . . . . . 8 |- 1o e. 2o
32		opexg 4257	. . . . . . . 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 4240	. . . . . . 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 12880	. . . . . 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 2280	. . . 4 |- ( ph -> U e. _V )
39		imasex 12888	. . . 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 5554	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
42	41, 6	eqtr4di 2244	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = X )
43		fveq2 5554	. . . . . . . . 9 |- ( s = S -> ( Base ` s ) = ( Base ` S ) )
44	43, 12	eqtr4di 2244	. . . . . . . 8 |- ( s = S -> ( Base ` s ) = Y )
45		mpoeq12 5978	. . . . . . . 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 2244	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = F )
48	47	cnveqd 4838	. . . . 5 |- ( ( r = R /\ s = S ) -> `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = `' F )
49		fveq2 5554	. . . . . . . . 9 |- ( r = R -> (Scalar ` r ) = (Scalar ` R ) )
50	49	adantr 276	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> (Scalar ` r ) = (Scalar ` R ) )
51	50, 23	eqtr4di 2244	. . . . . . 7 |- ( ( r = R /\ s = S ) -> (Scalar ` r ) = G )
52		simpl 109	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> r = R )
53	52	opeq2d 3811	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> <. (/) , r >. = <. (/) , R >. )
54		simpr 110	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> s = S )
55	54	opeq2d 3811	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> <. 1o , s >. = <. 1o , S >. )
56	53, 55	preq12d 3703	. . . . . . 7 |- ( ( r = R /\ s = S ) -> { <. (/) , r >. , <. 1o , s >. } = { <. (/) , R >. , <. 1o , S >. } )
57	51, 56	oveq12d 5936	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) = ( G X_s { <. (/) , R >. , <. 1o , S >. } ) )
58	57, 22	eqtr4di 2244	. . . . 5 |- ( ( r = R /\ s = S ) -> ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) = U )
59	48, 58	oveq12d 5936	. . . 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 12887	. . . 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 6048	. . 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 1249	. 2 |- ( ph -> ( R X.s S ) = ( `' F "s U ) )
63	1, 62	eqtrid 2238	1 |- ( ph -> T = ( `' F "s U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   (/)c0 3446   {cpr 3619   <.cop 3621   `'ccnv 4658   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   1oc1o 6462   2oc2o 6463   Basecbs 12618  Scalarcsca 12698   X__scprds 12876   "_s cimas 12882   X._s cxps 12884

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-1o 6469  df-2o 6470  df-map 6704  df-ixp 6753  df-sup 7043  df-sub 8192  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-9 9048  df-n0 9241  df-dec 9449  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-tset 12714  df-ple 12715  df-ds 12717  df-hom 12719  df-cco 12720  df-rest 12852  df-topn 12853  df-topgen 12871  df-pt 12872  df-prds 12878  df-iimas 12885  df-xps 12887

This theorem is referenced by: (None)