Theorem sraex 13635

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 13626	. . . 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 12626	. . . . . . . 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 12534	. . . . . . . . 9 |- Base Fn _V
10	2	elexd 2762	. . . . . . . . 9 |- ( ph -> W e. _V )
11		funfvex 5544	. . . . . . . . . 10 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
12	11	funfni 5328	. . . . . . . . 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 4155	. . . . . . 7 |- ( ph -> S e. _V )
15		ressex 12539	. . . . . . 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 12508	. . . . . 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 1248	. . . . 5 |- ( ph -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
19		vscaslid 12636	. . . . . . 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 12605	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
23	22	slotex 12503	. . . . . 6 |- ( W e. X -> ( .r ` W ) e. _V )
24	2, 23	syl 14	. . . . 5 |- ( ph -> ( .r ` W ) e. _V )
25		setsex 12508	. . . . 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 1248	. . . 4 |- ( ph -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
27		ipslid 12644	. . . . . 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 12508	. . . 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 1248	. . 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 2264	. 2 |- ( ph -> ( (subringAlg ` W ) ` S ) e. _V )
33	1, 32	eqeltrd 2264	1 |- ( ph -> A e. _V )

Syntax hints:   -> wi 4   = wceq 1363   e. wcel 2158   _Vcvv 2749   C_ wss 3141   <.cop 3607   Fn wfn 5223   ` cfv 5228  (class class class)co 5888   NNcn 8933   ndxcnx 12473   sSet csts 12474  Slot cslot 12475   Basecbs 12476   ↾_s cress 12477   .rcmulr 12552  Scalarcsca 12554   .scvsca 12555   .icip 12556  subringAlg csra 13622

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1re 7919  ax-addrcl 7922

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-5 8995  df-6 8996  df-7 8997  df-8 8998  df-ndx 12479  df-slot 12480  df-base 12482  df-sets 12483  df-iress 12484  df-mulr 12565  df-sca 12567  df-vsca 12568  df-ip 12569  df-sra 13624

This theorem is referenced by:  sratopng  13636  sralmod0g  13640  rlmfn  13642  rlmvalg  13643