Theorem sralemg 13627

Description: Lemma for srabaseg 13628 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)

Hypotheses

Ref	Expression
srapart.a	|- ( ph -> A = ( (subringAlg ` W ) ` S ) )
srapart.s	|- ( ph -> S C_ ( Base ` W ) )
srapart.ex	|- ( ph -> W e. X )
sralemg.1	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
sralem.2	|- (Scalar ` ndx ) =/= ( E ` ndx )
sralem.3	|- ( .s ` ndx ) =/= ( E ` ndx )
sralem.4	|- ( .i ` ndx ) =/= ( E ` ndx )

Assertion

Ref	Expression
sralemg	|- ( ph -> ( E ` W ) = ( E ` A ) )

Proof of Theorem sralemg

Step	Hyp	Ref	Expression
1		srapart.ex	. . . 4 |- ( ph -> W e. X )
2		basfn 12534	. . . . . . 7 |- Base Fn _V
3	1	elexd 2762	. . . . . . 7 |- ( ph -> W e. _V )
4		funfvex 5544	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
5	4	funfni 5328	. . . . . . 7 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
6	2, 3, 5	sylancr 414	. . . . . 6 |- ( ph -> ( Base ` W ) e. _V )
7		srapart.s	. . . . . 6 |- ( ph -> S C_ ( Base ` W ) )
8	6, 7	ssexd 4155	. . . . 5 |- ( ph -> S e. _V )
9		ressex 12539	. . . . 5 |- ( ( W e. X /\ S e. _V ) -> ( W ↾s S ) e. _V )
10	1, 8, 9	syl2anc 411	. . . 4 |- ( ph -> ( W ↾s S ) e. _V )
11		sralemg.1	. . . . 5 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
12		sralem.2	. . . . . 6 |- (Scalar ` ndx ) =/= ( E ` ndx )
13	12	necomi 2442	. . . . 5 |- ( E ` ndx ) =/= (Scalar ` ndx )
14		scaslid 12626	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
15	14	simpri 113	. . . . 5 |- (Scalar ` ndx ) e. NN
16	11, 13, 15	setsslnid 12528	. . . 4 |- ( ( W e. X /\ ( W ↾s S ) e. _V ) -> ( E ` W ) = ( E ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) ) )
17	1, 10, 16	syl2anc 411	. . 3 |- ( ph -> ( E ` W ) = ( E ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) ) )
18	15	a1i 9	. . . . 5 |- ( ph -> (Scalar ` ndx ) e. NN )
19		setsex 12508	. . . . 5 |- ( ( W e. X /\ (Scalar ` ndx ) e. NN /\ ( W ↾s S ) e. _V ) -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
20	1, 18, 10, 19	syl3anc 1248	. . . 4 |- ( ph -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
21		mulrslid 12605	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
22	21	slotex 12503	. . . . 5 |- ( W e. X -> ( .r ` W ) e. _V )
23	1, 22	syl 14	. . . 4 |- ( ph -> ( .r ` W ) e. _V )
24		sralem.3	. . . . . 6 |- ( .s ` ndx ) =/= ( E ` ndx )
25	24	necomi 2442	. . . . 5 |- ( E ` ndx ) =/= ( .s ` ndx )
26		vscaslid 12636	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
27	26	simpri 113	. . . . 5 |- ( .s ` ndx ) e. NN
28	11, 25, 27	setsslnid 12528	. . . 4 |- ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V /\ ( .r ` W ) e. _V ) -> ( E ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) ) = ( E ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) )
29	20, 23, 28	syl2anc 411	. . 3 |- ( ph -> ( E ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) ) = ( E ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) )
30	27	a1i 9	. . . . 5 |- ( ph -> ( .s ` ndx ) e. NN )
31		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 )
32	20, 30, 23, 31	syl3anc 1248	. . . 4 |- ( ph -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
33		sralem.4	. . . . . 6 |- ( .i ` ndx ) =/= ( E ` ndx )
34	33	necomi 2442	. . . . 5 |- ( E ` ndx ) =/= ( .i ` ndx )
35		ipslid 12644	. . . . . 6 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
36	35	simpri 113	. . . . 5 |- ( .i ` ndx ) e. NN
37	11, 34, 36	setsslnid 12528	. . . 4 |- ( ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V /\ ( .r ` W ) e. _V ) -> ( E ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) = ( E ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
38	32, 23, 37	syl2anc 411	. . 3 |- ( ph -> ( E ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) = ( E ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
39	17, 29, 38	3eqtrd 2224	. 2 |- ( ph -> ( E ` W ) = ( E ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
40		srapart.a	. . . 4 |- ( ph -> A = ( (subringAlg ` W ) ` S ) )
41		sraval 13626	. . . . 5 |- ( ( 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 ) >. ) )
42	1, 7, 41	syl2anc 411	. . . 4 |- ( ph -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
43	40, 42	eqtrd 2220	. . 3 |- ( ph -> A = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
44	43	fveq2d 5531	. 2 |- ( ph -> ( E ` A ) = ( E ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
45	39, 44	eqtr4d 2223	1 |- ( ph -> ( E ` W ) = ( E ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   =/= wne 2357   _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-nul 3435  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:  srabaseg  13628  sraaddgg  13629  sramulrg  13630  sratsetg  13634  sradsg  13637