Theorem sraval 14416

Description: Lemma for srabaseg 14418 through sravscag 14422. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)

Assertion

Ref	Expression
sraval	|- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )

Proof of Theorem sraval

Dummy variables s w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2811	. . . 4 |- ( W e. V -> W e. _V )
2	1	adantr 276	. . 3 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> W e. _V )
3		df-sra 14414	. . . 4 |- subringAlg = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> ( ( ( w sSet <. (Scalar ` ndx ) , ( w ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` w ) >. ) sSet <. ( .i ` ndx ) , ( .r ` w ) >. ) ) )
4		fveq2 5629	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
5	4	pweqd 3654	. . . . 5 |- ( w = W -> ~P ( Base ` w ) = ~P ( Base ` W ) )
6		id 19	. . . . . . . 8 |- ( w = W -> w = W )
7		oveq1 6014	. . . . . . . . 9 |- ( w = W -> ( w ↾s s ) = ( W ↾s s ) )
8	7	opeq2d 3864	. . . . . . . 8 |- ( w = W -> <. (Scalar ` ndx ) , ( w ↾s s ) >. = <. (Scalar ` ndx ) , ( W ↾s s ) >. )
9	6, 8	oveq12d 6025	. . . . . . 7 |- ( w = W -> ( w sSet <. (Scalar ` ndx ) , ( w ↾s s ) >. ) = ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) )
10		fveq2 5629	. . . . . . . 8 |- ( w = W -> ( .r ` w ) = ( .r ` W ) )
11	10	opeq2d 3864	. . . . . . 7 |- ( w = W -> <. ( .s ` ndx ) , ( .r ` w ) >. = <. ( .s ` ndx ) , ( .r ` W ) >. )
12	9, 11	oveq12d 6025	. . . . . 6 |- ( w = W -> ( ( w sSet <. (Scalar ` ndx ) , ( w ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` w ) >. ) = ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) )
13	10	opeq2d 3864	. . . . . 6 |- ( w = W -> <. ( .i ` ndx ) , ( .r ` w ) >. = <. ( .i ` ndx ) , ( .r ` W ) >. )
14	12, 13	oveq12d 6025	. . . . 5 |- ( w = W -> ( ( ( w sSet <. (Scalar ` ndx ) , ( w ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` w ) >. ) sSet <. ( .i ` ndx ) , ( .r ` w ) >. ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
15	5, 14	mpteq12dv 4166	. . . 4 |- ( w = W -> ( s e. ~P ( Base ` w ) |-> ( ( ( w sSet <. (Scalar ` ndx ) , ( w ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` w ) >. ) sSet <. ( .i ` ndx ) , ( .r ` w ) >. ) ) = ( s e. ~P ( Base ` W ) |-> ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
16		elex 2811	. . . 4 |- ( W e. _V -> W e. _V )
17		basfn 13106	. . . . . . 7 |- Base Fn _V
18		funfvex 5646	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
19	18	funfni 5423	. . . . . . 7 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
20	17, 19	mpan 424	. . . . . 6 |- ( W e. _V -> ( Base ` W ) e. _V )
21	20	pwexd 4265	. . . . 5 |- ( W e. _V -> ~P ( Base ` W ) e. _V )
22	21	mptexd 5870	. . . 4 |- ( W e. _V -> ( s e. ~P ( Base ` W ) |-> ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) e. _V )
23	3, 15, 16, 22	fvmptd3 5730	. . 3 |- ( W e. _V -> (subringAlg ` W ) = ( s e. ~P ( Base ` W ) |-> ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
24	2, 23	syl 14	. 2 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> (subringAlg ` W ) = ( s e. ~P ( Base ` W ) |-> ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
25		simpr 110	. . . . . . 7 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> s = S )
26	25	oveq2d 6023	. . . . . 6 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> ( W ↾s s ) = ( W ↾s S ) )
27	26	opeq2d 3864	. . . . 5 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> <. (Scalar ` ndx ) , ( W ↾s s ) >. = <. (Scalar ` ndx ) , ( W ↾s S ) >. )
28	27	oveq2d 6023	. . . 4 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) = ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) )
29	28	oveq1d 6022	. . 3 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) = ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) )
30	29	oveq1d 6022	. 2 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
31		simpr 110	. . 3 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> S C_ ( Base ` W ) )
32		elpw2g 4240	. . . 4 |- ( ( Base ` W ) e. _V -> ( S e. ~P ( Base ` W ) <-> S C_ ( Base ` W ) ) )
33	2, 20, 32	3syl 17	. . 3 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( S e. ~P ( Base ` W ) <-> S C_ ( Base ` W ) ) )
34	31, 33	mpbird 167	. 2 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> S e. ~P ( Base ` W ) )
35		simpl 109	. . . . 5 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> W e. V )
36		scaslid 13201	. . . . . . 7 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
37	36	simpri 113	. . . . . 6 |- (Scalar ` ndx ) e. NN
38	37	a1i 9	. . . . 5 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> (Scalar ` ndx ) e. NN )
39	34	elexd 2813	. . . . . 6 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> S e. _V )
40		ressex 13113	. . . . . 6 |- ( ( W e. V /\ S e. _V ) -> ( W ↾s S ) e. _V )
41	39, 40	syldan 282	. . . . 5 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( W ↾s S ) e. _V )
42		setsex 13079	. . . . 5 |- ( ( W e. V /\ (Scalar ` ndx ) e. NN /\ ( W ↾s S ) e. _V ) -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
43	35, 38, 41, 42	syl3anc 1271	. . . 4 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
44		vscaslid 13211	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
45	44	simpri 113	. . . . 5 |- ( .s ` ndx ) e. NN
46	45	a1i 9	. . . 4 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( .s ` ndx ) e. NN )
47		mulrslid 13180	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
48	47	slotex 13074	. . . . 5 |- ( W e. V -> ( .r ` W ) e. _V )
49	48	adantr 276	. . . 4 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( .r ` W ) e. _V )
50		setsex 13079	. . . 4 |- ( ( ( 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 )
51	43, 46, 49, 50	syl3anc 1271	. . 3 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
52		ipslid 13219	. . . . 5 |- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
53	52	simpri 113	. . . 4 |- ( .i ` ndx ) e. NN
54	53	a1i 9	. . 3 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( .i ` ndx ) e. NN )
55		setsex 13079	. . 3 |- ( ( ( ( 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 )
56	51, 54, 49, 55	syl3anc 1271	. 2 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) e. _V )
57	24, 30, 34, 56	fvmptd 5717	1 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   <.cop 3669   |-> cmpt 4145   Fn wfn 5313   ` cfv 5318  (class class class)co 6007   NNcn 9121   ndxcnx 13044   sSet csts 13045  Slot cslot 13046   Basecbs 13047   ↾_s cress 13048   .rcmulr 13126  Scalarcsca 13128   .scvsca 13129   .icip 13130  subringAlg csra 14412

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-iress 13055  df-mulr 13139  df-sca 13141  df-vsca 13142  df-ip 13143  df-sra 14414

This theorem is referenced by:  sralemg  14417  srascag  14421  sravscag  14422  sraipg  14423  sraex  14425