Theorem srascag 13941

Description: The set of scalars 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
srascag	|- ( ph -> ( W ↾s S ) = (Scalar ` A ) )

Proof of Theorem srascag

Step	Hyp	Ref	Expression
1		srapart.ex	. . . . 5 |- ( ph -> W e. X )
2		scaslid 12773	. . . . . . 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 12679	. . . . . . . 8 |- Base Fn _V
6	1	elexd 2773	. . . . . . . 8 |- ( ph -> W e. _V )
7		funfvex 5572	. . . . . . . . 9 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
8	7	funfni 5355	. . . . . . . 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 4170	. . . . . 6 |- ( ph -> S e. _V )
12		ressex 12686	. . . . . 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 12653	. . . . 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 1249	. . . 4 |- ( ph -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
16		mulrslid 12752	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
17	16	slotex 12648	. . . . 5 |- ( W e. X -> ( .r ` W ) e. _V )
18	1, 17	syl 14	. . . 4 |- ( ph -> ( .r ` W ) e. _V )
19		vscandxnscandx 12782	. . . . . 6 |- ( .s ` ndx ) =/= (Scalar ` ndx )
20	19	necomi 2449	. . . . 5 |- (Scalar ` ndx ) =/= ( .s ` ndx )
21		vscaslid 12783	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
22	21	simpri 113	. . . . 5 |- ( .s ` ndx ) e. NN
23	2, 20, 22	setsslnid 12673	. . . 4 |- ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V /\ ( .r ` W ) e. _V ) -> (Scalar ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) ) = (Scalar ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) )
24	15, 18, 23	syl2anc 411	. . 3 |- ( ph -> (Scalar ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) ) = (Scalar ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) )
25	22	a1i 9	. . . . 5 |- ( ph -> ( .s ` ndx ) e. NN )
26		setsex 12653	. . . . 5 |- ( ( ( 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 )
27	15, 25, 18, 26	syl3anc 1249	. . . 4 |- ( ph -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
28		slotsdifipndx 12795	. . . . . 6 |- ( ( .s ` ndx ) =/= ( .i ` ndx ) /\ (Scalar ` ndx ) =/= ( .i ` ndx ) )
29	28	simpri 113	. . . . 5 |- (Scalar ` ndx ) =/= ( .i ` ndx )
30		ipslid 12791	. . . . . 6 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
31	30	simpri 113	. . . . 5 |- ( .i ` ndx ) e. NN
32	2, 29, 31	setsslnid 12673	. . . 4 |- ( ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V /\ ( .r ` W ) e. _V ) -> (Scalar ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) = (Scalar ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
33	27, 18, 32	syl2anc 411	. . 3 |- ( ph -> (Scalar ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) = (Scalar ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
34	24, 33	eqtrd 2226	. 2 |- ( ph -> (Scalar ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) ) = (Scalar ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
35	2	setsslid 12672	. . 3 |- ( ( W e. X /\ ( W ↾s S ) e. _V ) -> ( W ↾s S ) = (Scalar ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) ) )
36	1, 13, 35	syl2anc 411	. 2 |- ( ph -> ( W ↾s S ) = (Scalar ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) ) )
37		srapart.a	. . . 4 |- ( ph -> A = ( (subringAlg ` W ) ` S ) )
38		sraval 13936	. . . . 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 ) >. ) )
39	6, 10, 38	syl2anc 411	. . . 4 |- ( ph -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
40	37, 39	eqtrd 2226	. . 3 |- ( ph -> A = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
41	40	fveq2d 5559	. 2 |- ( ph -> (Scalar ` A ) = (Scalar ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
42	34, 36, 41	3eqtr4d 2236	1 |- ( ph -> ( W ↾s S ) = (Scalar ` A ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   =/= wne 2364   _Vcvv 2760   C_ wss 3154   <.cop 3622   Fn wfn 5250   ` cfv 5255  (class class class)co 5919   NNcn 8984   ndxcnx 12618   sSet csts 12619  Slot cslot 12620   Basecbs 12621   ↾_s cress 12622   .rcmulr 12699  Scalarcsca 12701   .scvsca 12702   .icip 12703  subringAlg csra 13932

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 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

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-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-mulr 12712  df-sca 12714  df-vsca 12715  df-ip 12716  df-sra 13934

This theorem is referenced by:  sralmod  13949  rlmscabas  13959