Theorem srascag 13998

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 12830	. . . . . . 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 12736	. . . . . . . 8 |- Base Fn _V
6	1	elexd 2776	. . . . . . . 8 |- ( ph -> W e. _V )
7		funfvex 5575	. . . . . . . . 9 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
8	7	funfni 5358	. . . . . . . 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 4173	. . . . . 6 |- ( ph -> S e. _V )
12		ressex 12743	. . . . . 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 12710	. . . . 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 12809	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
17	16	slotex 12705	. . . . 5 |- ( W e. X -> ( .r ` W ) e. _V )
18	1, 17	syl 14	. . . 4 |- ( ph -> ( .r ` W ) e. _V )
19		vscandxnscandx 12839	. . . . . 6 |- ( .s ` ndx ) =/= (Scalar ` ndx )
20	19	necomi 2452	. . . . 5 |- (Scalar ` ndx ) =/= ( .s ` ndx )
21		vscaslid 12840	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
22	21	simpri 113	. . . . 5 |- ( .s ` ndx ) e. NN
23	2, 20, 22	setsslnid 12730	. . . 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 12710	. . . . 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 12852	. . . . . 6 |- ( ( .s ` ndx ) =/= ( .i ` ndx ) /\ (Scalar ` ndx ) =/= ( .i ` ndx ) )
29	28	simpri 113	. . . . 5 |- (Scalar ` ndx ) =/= ( .i ` ndx )
30		ipslid 12848	. . . . . 6 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
31	30	simpri 113	. . . . 5 |- ( .i ` ndx ) e. NN
32	2, 29, 31	setsslnid 12730	. . . 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 2229	. 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 12729	. . 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 13993	. . . . 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 2229	. . 3 |- ( ph -> A = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
41	40	fveq2d 5562	. 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 2239	1 |- ( ph -> ( W ↾s S ) = (Scalar ` A ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   =/= wne 2367   _Vcvv 2763   C_ wss 3157   <.cop 3625   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   NNcn 8990   ndxcnx 12675   sSet csts 12676  Slot cslot 12677   Basecbs 12678   ↾_s cress 12679   .rcmulr 12756  Scalarcsca 12758   .scvsca 12759   .icip 12760  subringAlg csra 13989

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-mulr 12769  df-sca 12771  df-vsca 12772  df-ip 12773  df-sra 13991

This theorem is referenced by:  sralmod  14006  rlmscabas  14016