Theorem srascag 13938

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 12770	. . . . . . 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 12676	. . . . . . . 8 |- Base Fn _V
6	1	elexd 2773	. . . . . . . 8 |- ( ph -> W e. _V )
7		funfvex 5571	. . . . . . . . 9 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
8	7	funfni 5354	. . . . . . . 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 4169	. . . . . 6 |- ( ph -> S e. _V )
12		ressex 12683	. . . . . 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 12650	. . . . 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 12749	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
17	16	slotex 12645	. . . . 5 |- ( W e. X -> ( .r ` W ) e. _V )
18	1, 17	syl 14	. . . 4 |- ( ph -> ( .r ` W ) e. _V )
19		vscandxnscandx 12779	. . . . . 6 |- ( .s ` ndx ) =/= (Scalar ` ndx )
20	19	necomi 2449	. . . . 5 |- (Scalar ` ndx ) =/= ( .s ` ndx )
21		vscaslid 12780	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
22	21	simpri 113	. . . . 5 |- ( .s ` ndx ) e. NN
23	2, 20, 22	setsslnid 12670	. . . 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 12650	. . . . 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 12792	. . . . . 6 |- ( ( .s ` ndx ) =/= ( .i ` ndx ) /\ (Scalar ` ndx ) =/= ( .i ` ndx ) )
29	28	simpri 113	. . . . 5 |- (Scalar ` ndx ) =/= ( .i ` ndx )
30		ipslid 12788	. . . . . 6 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
31	30	simpri 113	. . . . 5 |- ( .i ` ndx ) e. NN
32	2, 29, 31	setsslnid 12670	. . . 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 12669	. . 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 13933	. . . . 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 5558	. 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 3153   <.cop 3621   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   NNcn 8982   ndxcnx 12615   sSet csts 12616  Slot cslot 12617   Basecbs 12618   ↾_s cress 12619   .rcmulr 12696  Scalarcsca 12698   .scvsca 12699   .icip 12700  subringAlg csra 13929

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-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-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 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-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  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-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  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-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-sra 13931

This theorem is referenced by:  sralmod  13946  rlmscabas  13956