Theorem sraex 14080

Description: Existence of a subring algebra. (Contributed by Jim Kingdon, 16-Apr-2025.)

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
sraex	|- ( ph -> A e. _V )

Proof of Theorem sraex

Step	Hyp	Ref	Expression
1		srapart.a	. 2 |- ( ph -> A = ( (subringAlg ` W ) ` S ) )
2		srapart.ex	. . . 4 |- ( ph -> W e. X )
3		srapart.s	. . . 4 |- ( ph -> S C_ ( Base ` W ) )
4		sraval 14071	. . . 4 |- ( ( W e. X /\ S C_ ( Base ` W ) ) -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
5	2, 3, 4	syl2anc 411	. . 3 |- ( ph -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
6		scaslid 12857	. . . . . . . 8 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
7	6	simpri 113	. . . . . . 7 |- (Scalar ` ndx ) e. NN
8	7	a1i 9	. . . . . 6 |- ( ph -> (Scalar ` ndx ) e. NN )
9		basfn 12763	. . . . . . . . 9 |- Base Fn _V
10	2	elexd 2776	. . . . . . . . 9 |- ( ph -> W e. _V )
11		funfvex 5578	. . . . . . . . . 10 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
12	11	funfni 5361	. . . . . . . . 9 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
13	9, 10, 12	sylancr 414	. . . . . . . 8 |- ( ph -> ( Base ` W ) e. _V )
14	13, 3	ssexd 4174	. . . . . . 7 |- ( ph -> S e. _V )
15		ressex 12770	. . . . . . 7 |- ( ( W e. X /\ S e. _V ) -> ( W ↾s S ) e. _V )
16	2, 14, 15	syl2anc 411	. . . . . 6 |- ( ph -> ( W ↾s S ) e. _V )
17		setsex 12737	. . . . . 6 |- ( ( W e. X /\ (Scalar ` ndx ) e. NN /\ ( W ↾s S ) e. _V ) -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
18	2, 8, 16, 17	syl3anc 1249	. . . . 5 |- ( ph -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
19		vscaslid 12867	. . . . . . 7 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
20	19	simpri 113	. . . . . 6 |- ( .s ` ndx ) e. NN
21	20	a1i 9	. . . . 5 |- ( ph -> ( .s ` ndx ) e. NN )
22		mulrslid 12836	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
23	22	slotex 12732	. . . . . 6 |- ( W e. X -> ( .r ` W ) e. _V )
24	2, 23	syl 14	. . . . 5 |- ( ph -> ( .r ` W ) e. _V )
25		setsex 12737	. . . . 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 )
26	18, 21, 24, 25	syl3anc 1249	. . . 4 |- ( ph -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
27		ipslid 12875	. . . . . 6 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
28	27	simpri 113	. . . . 5 |- ( .i ` ndx ) e. NN
29	28	a1i 9	. . . 4 |- ( ph -> ( .i ` ndx ) e. NN )
30		setsex 12737	. . . 4 |- ( ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V /\ ( .i ` ndx ) e. NN /\ ( .r ` W ) e. _V ) -> ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) e. _V )
31	26, 29, 24, 30	syl3anc 1249	. . 3 |- ( ph -> ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) e. _V )
32	5, 31	eqeltrd 2273	. 2 |- ( ph -> ( (subringAlg ` W ) ` S ) e. _V )
33	1, 32	eqeltrd 2273	1 |- ( ph -> A e. _V )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   <.cop 3626   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   NNcn 9009   ndxcnx 12702   sSet csts 12703  Slot cslot 12704   Basecbs 12705   ↾_s cress 12706   .rcmulr 12783  Scalarcsca 12785   .scvsca 12786   .icip 12787  subringAlg csra 14067

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 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1re 7992  ax-addrcl 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-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-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-5 9071  df-6 9072  df-7 9073  df-8 9074  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-iress 12713  df-mulr 12796  df-sca 12798  df-vsca 12799  df-ip 12800  df-sra 14069

This theorem is referenced by:  sratopng  14081  sralmod0g  14085  rlmfn  14087  rlmvalg  14088