Theorem sravscag 14780

Description: The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)

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
sravscag	|- ( ph -> ( .r ` W ) = ( .s ` A ) )

Proof of Theorem sravscag

Step	Hyp	Ref	Expression
1		srapart.ex	. . . . 5 |- ( ph -> W e. X )
2		scaslid 13507	. . . . . . 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 13411	. . . . . . . 8 |- Base Fn _V
6	1	elexd 2835	. . . . . . . 8 |- ( ph -> W e. _V )
7		funfvex 5712	. . . . . . . . 9 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
8	7	funfni 5483	. . . . . . . 8 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
9	5, 6, 8	sylancr 418	. . . . . . 7 |- ( ph -> ( Base ` W ) e. _V )
10		srapart.s	. . . . . . 7 |- ( ph -> S C_ ( Base ` W ) )
11	9, 10	ssexd 4273	. . . . . 6 |- ( ph -> S e. _V )
12		ressex 13419	. . . . . 6 |- ( ( W e. X /\ S e. _V ) -> ( W ↾s S ) e. _V )
13	1, 11, 12	syl2anc 415	. . . . 5 |- ( ph -> ( W ↾s S ) e. _V )
14		setsex 13384	. . . . 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 1278	. . . 4 |- ( ph -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
16		vscaslid 13517	. . . . . 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 13486	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
20	19	slotex 13379	. . . . 5 |- ( W e. X -> ( .r ` W ) e. _V )
21	1, 20	syl 14	. . . 4 |- ( ph -> ( .r ` W ) e. _V )
22		setsex 13384	. . . 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 1278	. . 3 |- ( ph -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
24		slotsdifipndx 13529	. . . . 5 |- ( ( .s ` ndx ) =/= ( .i ` ndx ) /\ (Scalar ` ndx ) =/= ( .i ` ndx ) )
25	24	simpli 111	. . . 4 |- ( .s ` ndx ) =/= ( .i ` ndx )
26		ipslid 13525	. . . . 5 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
27	26	simpri 113	. . . 4 |- ( .i ` ndx ) e. NN
28	16, 25, 27	setsslnid 13404	. . 3 |- ( ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V /\ ( .r ` W ) e. _V ) -> ( .s ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) = ( .s ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
29	23, 21, 28	syl2anc 415	. 2 |- ( ph -> ( .s ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) = ( .s ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
30	16	setsslid 13403	. . 3 |- ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V /\ ( .r ` W ) e. _V ) -> ( .r ` W ) = ( .s ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) )
31	15, 21, 30	syl2anc 415	. 2 |- ( ph -> ( .r ` W ) = ( .s ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) )
32		srapart.a	. . . 4 |- ( ph -> A = ( (subringAlg ` W ) ` S ) )
33		sraval 14774	. . . . 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 ) >. ) )
34	6, 10, 33	syl2anc 415	. . . 4 |- ( ph -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
35	32, 34	eqtrd 2271	. . 3 |- ( ph -> A = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
36	35	fveq2d 5699	. 2 |- ( ph -> ( .s ` A ) = ( .s ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
37	29, 31, 36	3eqtr4d 2281	1 |- ( ph -> ( .r ` W ) = ( .s ` A ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   C_ wss 3220   <.cop 3712   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   NNcn 9304   ndxcnx 13349   sSet csts 13350  Slot cslot 13351   Basecbs 13352   ↾_s cress 13353   .rcmulr 13432  Scalarcsca 13434   .scvsca 13435   .icip 13436  subringAlg csra 14770

This proof depends on 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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof 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-nel 2516  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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-mulr 13445  df-sca 13447  df-vsca 13448  df-ip 13449  df-sra 14772

This theorem is used by:  sralmod  14787  rlmvscag  14798