Theorem sraval 14199

Description: Lemma for srabaseg 14201 through sravscag 14205. (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 2783	. . . 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 14197	. . . 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 5576	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
5	4	pweqd 3621	. . . . 5 |- ( w = W -> ~P ( Base ` w ) = ~P ( Base ` W ) )
6		id 19	. . . . . . . 8 |- ( w = W -> w = W )
7		oveq1 5951	. . . . . . . . 9 |- ( w = W -> ( w ↾s s ) = ( W ↾s s ) )
8	7	opeq2d 3826	. . . . . . . 8 |- ( w = W -> <. (Scalar ` ndx ) , ( w ↾s s ) >. = <. (Scalar ` ndx ) , ( W ↾s s ) >. )
9	6, 8	oveq12d 5962	. . . . . . 7 |- ( w = W -> ( w sSet <. (Scalar ` ndx ) , ( w ↾s s ) >. ) = ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) )
10		fveq2 5576	. . . . . . . 8 |- ( w = W -> ( .r ` w ) = ( .r ` W ) )
11	10	opeq2d 3826	. . . . . . 7 |- ( w = W -> <. ( .s ` ndx ) , ( .r ` w ) >. = <. ( .s ` ndx ) , ( .r ` W ) >. )
12	9, 11	oveq12d 5962	. . . . . 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 3826	. . . . . 6 |- ( w = W -> <. ( .i ` ndx ) , ( .r ` w ) >. = <. ( .i ` ndx ) , ( .r ` W ) >. )
14	12, 13	oveq12d 5962	. . . . 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 4126	. . . 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 2783	. . . 4 |- ( W e. _V -> W e. _V )
17		basfn 12890	. . . . . . 7 |- Base Fn _V
18		funfvex 5593	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
19	18	funfni 5376	. . . . . . 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 4225	. . . . 5 |- ( W e. _V -> ~P ( Base ` W ) e. _V )
22	21	mptexd 5811	. . . 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 5673	. . 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 5960	. . . . . 6 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> ( W ↾s s ) = ( W ↾s S ) )
27	26	opeq2d 3826	. . . . 5 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> <. (Scalar ` ndx ) , ( W ↾s s ) >. = <. (Scalar ` ndx ) , ( W ↾s S ) >. )
28	27	oveq2d 5960	. . . 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 5959	. . 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 5959	. 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 4200	. . . 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 12985	. . . . . . 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 2785	. . . . . 6 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> S e. _V )
40		ressex 12897	. . . . . 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 12864	. . . . 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 1250	. . . 4 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
44		vscaslid 12995	. . . . . 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 12964	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
48	47	slotex 12859	. . . . 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 12864	. . . 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 1250	. . 3 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
52		ipslid 13003	. . . . 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 12864	. . 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 1250	. 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 5660	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 1373   e. wcel 2176   _Vcvv 2772   C_ wss 3166   ~Pcpw 3616   <.cop 3636   |-> cmpt 4105   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   NNcn 9036   ndxcnx 12829   sSet csts 12830  Slot cslot 12831   Basecbs 12832   ↾_s cress 12833   .rcmulr 12910  Scalarcsca 12912   .scvsca 12913   .icip 12914  subringAlg csra 14195

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-7 9100  df-8 9101  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-mulr 12923  df-sca 12925  df-vsca 12926  df-ip 12927  df-sra 14197

This theorem is referenced by:  sralemg  14200  srascag  14204  sravscag  14205  sraipg  14206  sraex  14208