Theorem sralemg 14534

Description: Lemma for srabaseg 14535 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 13221	. . . . . . 7 |- Base Fn _V
3	1	elexd 2817	. . . . . . 7 |- ( ph -> W e. _V )
4		funfvex 5665	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
5	4	funfni 5439	. . . . . . 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 4234	. . . . 5 |- ( ph -> S e. _V )
9		ressex 13228	. . . . 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 2488	. . . . 5 |- ( E ` ndx ) =/= (Scalar ` ndx )
14		scaslid 13316	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
15	14	simpri 113	. . . . 5 |- (Scalar ` ndx ) e. NN
16	11, 13, 15	setsslnid 13214	. . . 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 13194	. . . . 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 1274	. . . 4 |- ( ph -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
21		mulrslid 13295	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
22	21	slotex 13189	. . . . 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 2488	. . . . 5 |- ( E ` ndx ) =/= ( .s ` ndx )
26		vscaslid 13326	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
27	26	simpri 113	. . . . 5 |- ( .s ` ndx ) e. NN
28	11, 25, 27	setsslnid 13214	. . . 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 13194	. . . . 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 1274	. . . 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 2488	. . . . 5 |- ( E ` ndx ) =/= ( .i ` ndx )
35		ipslid 13334	. . . . . 6 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
36	35	simpri 113	. . . . 5 |- ( .i ` ndx ) e. NN
37	11, 34, 36	setsslnid 13214	. . . 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 2268	. 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 14533	. . . . 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 2264	. . 3 |- ( ph -> A = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
44	43	fveq2d 5652	. 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 2267	1 |- ( ph -> ( E ` W ) = ( E ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   =/= wne 2403   _Vcvv 2803   C_ wss 3201   <.cop 3676   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   NNcn 9202   ndxcnx 13159   sSet csts 13160  Slot cslot 13161   Basecbs 13162   ↾_s cress 13163   .rcmulr 13241  Scalarcsca 13243   .scvsca 13244   .icip 13245  subringAlg csra 14529

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-mulr 13254  df-sca 13256  df-vsca 13257  df-ip 13258  df-sra 14531

This theorem is referenced by:  srabaseg  14535  sraaddgg  14536  sramulrg  14537  sratsetg  14541  sradsg  14544