Theorem xpsval 14178

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 2835	. . 3 |- ( ph -> R e. _V )
4		xpsval.2	. . . 4 |- ( ph -> S e. W )
5	4	elexd 2835	. . 3 |- ( ph -> S e. _V )
6		xpsval.x	. . . . . . 7 |- X = ( Base ` R )
7		basfn 13389	. . . . . . . 8 |- Base Fn _V
8		funfvex 5707	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5478	. . . . . . . 8 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
10	7, 3, 9	sylancr 418	. . . . . . 7 |- ( ph -> ( Base ` R ) e. _V )
11	6, 10	eqeltrid 2325	. . . . . 6 |- ( ph -> X e. _V )
12		xpsval.y	. . . . . . 7 |- Y = ( Base ` S )
13		funfvex 5707	. . . . . . . . 9 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
14	13	funfni 5478	. . . . . . . 8 |- ( ( Base Fn _V /\ S e. _V ) -> ( Base ` S ) e. _V )
15	7, 5, 14	sylancr 418	. . . . . . 7 |- ( ph -> ( Base ` S ) e. _V )
16	12, 15	eqeltrid 2325	. . . . . 6 |- ( ph -> Y e. _V )
17		xpsval.f	. . . . . . 7 |- F = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )
18	17	mpoexg 6437	. . . . . 6 |- ( ( X e. _V /\ Y e. _V ) -> F e. _V )
19	11, 16, 18	syl2anc 415	. . . . 5 |- ( ph -> F e. _V )
20		cnvexg 5320	. . . . 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 13484	. . . . . . . . 9 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
25	24	slotex 13357	. . . . . . . 8 |- ( R e. V -> (Scalar ` R ) e. _V )
26	2, 25	syl 14	. . . . . . 7 |- ( ph -> (Scalar ` R ) e. _V )
27	23, 26	eqeltrid 2325	. . . . . 6 |- ( ph -> G e. _V )
28		0lt2o 6704	. . . . . . . 8 |- (/) e. 2o
29		opexg 4363	. . . . . . . 8 |- ( ( (/) e. 2o /\ R e. V ) -> <. (/) , R >. e. _V )
30	28, 2, 29	sylancr 418	. . . . . . 7 |- ( ph -> <. (/) , R >. e. _V )
31		1lt2o 6705	. . . . . . . 8 |- 1o e. 2o
32		opexg 4363	. . . . . . . 8 |- ( ( 1o e. 2o /\ S e. W ) -> <. 1o , S >. e. _V )
33	31, 4, 32	sylancr 418	. . . . . . 7 |- ( ph -> <. 1o , S >. e. _V )
34		prexg 4344	. . . . . . 7 |- ( ( <. (/) , R >. e. _V /\ <. 1o , S >. e. _V ) -> { <. (/) , R >. , <. 1o , S >. } e. _V )
35	30, 33, 34	syl2anc 415	. . . . . 6 |- ( ph -> { <. (/) , R >. , <. 1o , S >. } e. _V )
36		prdsex 14149	. . . . . 6 |- ( ( G e. _V /\ { <. (/) , R >. , <. 1o , S >. } e. _V ) -> ( G X_s { <. (/) , R >. , <. 1o , S >. } ) e. _V )
37	27, 35, 36	syl2anc 415	. . . . 5 |- ( ph -> ( G X_s { <. (/) , R >. , <. 1o , S >. } ) e. _V )
38	22, 37	eqeltrid 2325	. . . 4 |- ( ph -> U e. _V )
39		imasex 13603	. . . 4 |- ( ( `' F e. _V /\ U e. _V ) -> ( `' F "s U ) e. _V )
40	21, 38, 39	syl2anc 415	. . 3 |- ( ph -> ( `' F "s U ) e. _V )
41		fveq2 5690	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
42	41, 6	eqtr4di 2289	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = X )
43		fveq2 5690	. . . . . . . . 9 |- ( s = S -> ( Base ` s ) = ( Base ` S ) )
44	43, 12	eqtr4di 2289	. . . . . . . 8 |- ( s = S -> ( Base ` s ) = Y )
45		mpoeq12 6138	. . . . . . . 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 2289	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = F )
48	47	cnveqd 4951	. . . . 5 |- ( ( r = R /\ s = S ) -> `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) = `' F )
49		fveq2 5690	. . . . . . . . 9 |- ( r = R -> (Scalar ` r ) = (Scalar ` R ) )
50	49	adantr 276	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> (Scalar ` r ) = (Scalar ` R ) )
51	50, 23	eqtr4di 2289	. . . . . . 7 |- ( ( r = R /\ s = S ) -> (Scalar ` r ) = G )
52		simpl 109	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> r = R )
53	52	opeq2d 3906	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> <. (/) , r >. = <. (/) , R >. )
54		simpr 110	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> s = S )
55	54	opeq2d 3906	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> <. 1o , s >. = <. 1o , S >. )
56	53, 55	preq12d 3792	. . . . . . 7 |- ( ( r = R /\ s = S ) -> { <. (/) , r >. , <. 1o , s >. } = { <. (/) , R >. , <. 1o , S >. } )
57	51, 56	oveq12d 6093	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) = ( G X_s { <. (/) , R >. , <. 1o , S >. } ) )
58	57, 22	eqtr4di 2289	. . . . 5 |- ( ( r = R /\ s = S ) -> ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) = U )
59	48, 58	oveq12d 6093	. . . 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 14177	. . . 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 6208	. . 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 1278	. 2 |- ( ph -> ( R X.s S ) = ( `' F "s U ) )
63	1, 62	eqtrid 2283	1 |- ( ph -> T = ( `' F "s U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   (/)c0 3520   {cpr 3706   <.cop 3708   `'ccnv 4768   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   1oc1o 6670   2oc2o 6671   Basecbs 13330  Scalarcsca 13411   "_s cimas 13599   X__scprds 14146   X._s cxps 14176

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-2o 6678  df-map 6914  df-ixp 6971  df-sup 7314  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-mulr 13422  df-sca 13424  df-vsca 13425  df-ip 13426  df-tset 13427  df-ple 13428  df-ds 13430  df-hom 13432  df-cco 13433  df-rest 13572  df-topn 13573  df-topgen 13591  df-pt 13592  df-iimas 13601  df-prds 14147  df-xps 14177

This theorem is referenced by: (None)