Theorem sraipg 14408

Description: The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.)

Hypotheses

Ref	Expression
srapart.a	|- ( ph -> A = ( (subringAlg ` W ) ` S ) )
srapart.s	|- ( ph -> S C_ ( Base ` W ) )
srapart.ex	|- ( ph -> W e. X )

Assertion

Ref	Expression
sraipg	|- ( ph -> ( .r ` W ) = ( .i ` A ) )

Proof of Theorem sraipg

Step	Hyp	Ref	Expression
1		srapart.ex	. . . . 5 |- ( ph -> W e. X )
2		scaslid 13186	. . . . . . 7 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
3	2	simpri 113	. . . . . 6 |- (Scalar ` ndx ) e. NN
4	3	a1i 9	. . . . 5 |- ( ph -> (Scalar ` ndx ) e. NN )
5		basfn 13091	. . . . . . . 8 |- Base Fn _V
6	1	elexd 2813	. . . . . . . 8 |- ( ph -> W e. _V )
7		funfvex 5644	. . . . . . . . 9 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
8	7	funfni 5423	. . . . . . . 8 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
9	5, 6, 8	sylancr 414	. . . . . . 7 |- ( ph -> ( Base ` W ) e. _V )
10		srapart.s	. . . . . . 7 |- ( ph -> S C_ ( Base ` W ) )
11	9, 10	ssexd 4224	. . . . . 6 |- ( ph -> S e. _V )
12		ressex 13098	. . . . . 6 |- ( ( W e. X /\ S e. _V ) -> ( W ↾s S ) e. _V )
13	1, 11, 12	syl2anc 411	. . . . 5 |- ( ph -> ( W ↾s S ) e. _V )
14		setsex 13064	. . . . 5 |- ( ( W e. X /\ (Scalar ` ndx ) e. NN /\ ( W ↾s S ) e. _V ) -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
15	1, 4, 13, 14	syl3anc 1271	. . . 4 |- ( ph -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
16		vscaslid 13196	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
17	16	simpri 113	. . . . 5 |- ( .s ` ndx ) e. NN
18	17	a1i 9	. . . 4 |- ( ph -> ( .s ` ndx ) e. NN )
19		mulrslid 13165	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
20	19	slotex 13059	. . . . 5 |- ( W e. X -> ( .r ` W ) e. _V )
21	1, 20	syl 14	. . . 4 |- ( ph -> ( .r ` W ) e. _V )
22		setsex 13064	. . . 4 |- ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V /\ ( .s ` ndx ) e. NN /\ ( .r ` W ) e. _V ) -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
23	15, 18, 21, 22	syl3anc 1271	. . 3 |- ( ph -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
24		ipslid 13204	. . . 4 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
25	24	setsslid 13083	. . 3 |- ( ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V /\ ( .r ` W ) e. _V ) -> ( .r ` W ) = ( .i ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
26	23, 21, 25	syl2anc 411	. 2 |- ( ph -> ( .r ` W ) = ( .i ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
27		srapart.a	. . . 4 |- ( ph -> A = ( (subringAlg ` W ) ` S ) )
28		sraval 14401	. . . . 5 |- ( ( W e. _V /\ S C_ ( Base ` W ) ) -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
29	6, 10, 28	syl2anc 411	. . . 4 |- ( ph -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
30	27, 29	eqtrd 2262	. . 3 |- ( ph -> A = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
31	30	fveq2d 5631	. 2 |- ( ph -> ( .i ` A ) = ( .i ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
32	26, 31	eqtr4d 2265	1 |- ( ph -> ( .r ` W ) = ( .i ` A ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   <.cop 3669   Fn wfn 5313   ` cfv 5318  (class class class)co 6001   NNcn 9110   ndxcnx 13029   sSet csts 13030  Slot cslot 13031   Basecbs 13032   ↾_s cress 13033   .rcmulr 13111  Scalarcsca 13113   .scvsca 13114   .icip 13115  subringAlg csra 14397

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-iress 13040  df-mulr 13124  df-sca 13126  df-vsca 13127  df-ip 13128  df-sra 14399

This theorem is referenced by: (None)