Theorem sraval 13993

Description: Lemma for srabaseg 13995 through sravscag 13999. (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 2774	. . . 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 13991	. . . 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 5558	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
5	4	pweqd 3610	. . . . 5 |- ( w = W -> ~P ( Base ` w ) = ~P ( Base ` W ) )
6		id 19	. . . . . . . 8 |- ( w = W -> w = W )
7		oveq1 5929	. . . . . . . . 9 |- ( w = W -> ( w ↾s s ) = ( W ↾s s ) )
8	7	opeq2d 3815	. . . . . . . 8 |- ( w = W -> <. (Scalar ` ndx ) , ( w ↾s s ) >. = <. (Scalar ` ndx ) , ( W ↾s s ) >. )
9	6, 8	oveq12d 5940	. . . . . . 7 |- ( w = W -> ( w sSet <. (Scalar ` ndx ) , ( w ↾s s ) >. ) = ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) )
10		fveq2 5558	. . . . . . . 8 |- ( w = W -> ( .r ` w ) = ( .r ` W ) )
11	10	opeq2d 3815	. . . . . . 7 |- ( w = W -> <. ( .s ` ndx ) , ( .r ` w ) >. = <. ( .s ` ndx ) , ( .r ` W ) >. )
12	9, 11	oveq12d 5940	. . . . . 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 3815	. . . . . 6 |- ( w = W -> <. ( .i ` ndx ) , ( .r ` w ) >. = <. ( .i ` ndx ) , ( .r ` W ) >. )
14	12, 13	oveq12d 5940	. . . . 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 4115	. . . 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 2774	. . . 4 |- ( W e. _V -> W e. _V )
17		basfn 12736	. . . . . . 7 |- Base Fn _V
18		funfvex 5575	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
19	18	funfni 5358	. . . . . . 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 4214	. . . . 5 |- ( W e. _V -> ~P ( Base ` W ) e. _V )
22	21	mptexd 5789	. . . 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 5655	. . 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 5938	. . . . . 6 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> ( W ↾s s ) = ( W ↾s S ) )
27	26	opeq2d 3815	. . . . 5 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> <. (Scalar ` ndx ) , ( W ↾s s ) >. = <. (Scalar ` ndx ) , ( W ↾s S ) >. )
28	27	oveq2d 5938	. . . 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 5937	. . 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 5937	. 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 4189	. . . 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 12830	. . . . . . 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 2776	. . . . . 6 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> S e. _V )
40		ressex 12743	. . . . . 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 12710	. . . . 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 1249	. . . 4 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
44		vscaslid 12840	. . . . . 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 12809	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
48	47	slotex 12705	. . . . 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 12710	. . . 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 1249	. . 3 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
52		ipslid 12848	. . . . 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 12710	. . 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 1249	. 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 5642	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 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   ~Pcpw 3605   <.cop 3625   |-> cmpt 4094   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   NNcn 8990   ndxcnx 12675   sSet csts 12676  Slot cslot 12677   Basecbs 12678   ↾_s cress 12679   .rcmulr 12756  Scalarcsca 12758   .scvsca 12759   .icip 12760  subringAlg csra 13989

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 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

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-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 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-mulr 12769  df-sca 12771  df-vsca 12772  df-ip 12773  df-sra 13991

This theorem is referenced by:  sralemg  13994  srascag  13998  sravscag  13999  sraipg  14000  sraex  14002