Theorem sralemg 13934

Description: Lemma for srabaseg 13935 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 12676	. . . . . . 7 |- Base Fn _V
3	1	elexd 2773	. . . . . . 7 |- ( ph -> W e. _V )
4		funfvex 5571	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
5	4	funfni 5354	. . . . . . 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 4169	. . . . 5 |- ( ph -> S e. _V )
9		ressex 12683	. . . . 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 2449	. . . . 5 |- ( E ` ndx ) =/= (Scalar ` ndx )
14		scaslid 12770	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
15	14	simpri 113	. . . . 5 |- (Scalar ` ndx ) e. NN
16	11, 13, 15	setsslnid 12670	. . . 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 12650	. . . . 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 12749	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
22	21	slotex 12645	. . . . 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 2449	. . . . 5 |- ( E ` ndx ) =/= ( .s ` ndx )
26		vscaslid 12780	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
27	26	simpri 113	. . . . 5 |- ( .s ` ndx ) e. NN
28	11, 25, 27	setsslnid 12670	. . . 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 12650	. . . . 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 2449	. . . . 5 |- ( E ` ndx ) =/= ( .i ` ndx )
35		ipslid 12788	. . . . . 6 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
36	35	simpri 113	. . . . 5 |- ( .i ` ndx ) e. NN
37	11, 34, 36	setsslnid 12670	. . . 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 2230	. 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 13933	. . . . 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 2226	. . 3 |- ( ph -> A = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
44	43	fveq2d 5558	. 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 2229	1 |- ( ph -> ( E ` W ) = ( E ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   =/= wne 2364   _Vcvv 2760   C_ wss 3153   <.cop 3621   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   NNcn 8982   ndxcnx 12615   sSet csts 12616  Slot cslot 12617   Basecbs 12618   ↾_s cress 12619   .rcmulr 12696  Scalarcsca 12698   .scvsca 12699   .icip 12700  subringAlg csra 13929

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-sra 13931

This theorem is referenced by:  srabaseg  13935  sraaddgg  13936  sramulrg  13937  sratsetg  13941  sradsg  13944