Theorem sralemg 13970

Description: Lemma for srabaseg 13971 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 12712	. . . . . . 7 |- Base Fn _V
3	1	elexd 2776	. . . . . . 7 |- ( ph -> W e. _V )
4		funfvex 5575	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
5	4	funfni 5358	. . . . . . 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 4173	. . . . 5 |- ( ph -> S e. _V )
9		ressex 12719	. . . . 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 2452	. . . . 5 |- ( E ` ndx ) =/= (Scalar ` ndx )
14		scaslid 12806	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
15	14	simpri 113	. . . . 5 |- (Scalar ` ndx ) e. NN
16	11, 13, 15	setsslnid 12706	. . . 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 12686	. . . . 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 1249	. . . 4 |- ( ph -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
21		mulrslid 12785	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
22	21	slotex 12681	. . . . 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 2452	. . . . 5 |- ( E ` ndx ) =/= ( .s ` ndx )
26		vscaslid 12816	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
27	26	simpri 113	. . . . 5 |- ( .s ` ndx ) e. NN
28	11, 25, 27	setsslnid 12706	. . . 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 12686	. . . . 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 1249	. . . 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 2452	. . . . 5 |- ( E ` ndx ) =/= ( .i ` ndx )
35		ipslid 12824	. . . . . 6 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
36	35	simpri 113	. . . . 5 |- ( .i ` ndx ) e. NN
37	11, 34, 36	setsslnid 12706	. . . 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 2233	. 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 13969	. . . . 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 2229	. . 3 |- ( ph -> A = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
44	43	fveq2d 5562	. 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 2232	1 |- ( ph -> ( E ` W ) = ( E ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = 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 8987   ndxcnx 12651   sSet csts 12652  Slot cslot 12653   Basecbs 12654   ↾_s cress 12655   .rcmulr 12732  Scalarcsca 12734   .scvsca 12735   .icip 12736  subringAlg csra 13965

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 7968  ax-resscn 7969  ax-1re 7971  ax-addrcl 7974

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-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-inn 8988  df-2 9046  df-3 9047  df-4 9048  df-5 9049  df-6 9050  df-7 9051  df-8 9052  df-ndx 12657  df-slot 12658  df-base 12660  df-sets 12661  df-iress 12662  df-mulr 12745  df-sca 12747  df-vsca 12748  df-ip 12749  df-sra 13967

This theorem is referenced by:  srabaseg  13971  sraaddgg  13972  sramulrg  13973  sratsetg  13977  sradsg  13980