Theorem sraval 14585

Description: Lemma for srabaseg 14587 through sravscag 14591. (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 2825	. . . 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 14583	. . . 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 5670	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
5	4	pweqd 3674	. . . . 5 |- ( w = W -> ~P ( Base ` w ) = ~P ( Base ` W ) )
6		id 19	. . . . . . . 8 |- ( w = W -> w = W )
7		oveq1 6057	. . . . . . . . 9 |- ( w = W -> ( w ↾s s ) = ( W ↾s s ) )
8	7	opeq2d 3890	. . . . . . . 8 |- ( w = W -> <. (Scalar ` ndx ) , ( w ↾s s ) >. = <. (Scalar ` ndx ) , ( W ↾s s ) >. )
9	6, 8	oveq12d 6068	. . . . . . 7 |- ( w = W -> ( w sSet <. (Scalar ` ndx ) , ( w ↾s s ) >. ) = ( W sSet <. (Scalar ` ndx ) , ( W ↾s s ) >. ) )
10		fveq2 5670	. . . . . . . 8 |- ( w = W -> ( .r ` w ) = ( .r ` W ) )
11	10	opeq2d 3890	. . . . . . 7 |- ( w = W -> <. ( .s ` ndx ) , ( .r ` w ) >. = <. ( .s ` ndx ) , ( .r ` W ) >. )
12	9, 11	oveq12d 6068	. . . . . 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 3890	. . . . . 6 |- ( w = W -> <. ( .i ` ndx ) , ( .r ` w ) >. = <. ( .i ` ndx ) , ( .r ` W ) >. )
14	12, 13	oveq12d 6068	. . . . 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 4192	. . . 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 2825	. . . 4 |- ( W e. _V -> W e. _V )
17		basfn 13271	. . . . . . 7 |- Base Fn _V
18		funfvex 5687	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
19	18	funfni 5458	. . . . . . 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 4294	. . . . 5 |- ( W e. _V -> ~P ( Base ` W ) e. _V )
22	21	mptexd 5913	. . . 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 5771	. . 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 6066	. . . . . 6 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> ( W ↾s s ) = ( W ↾s S ) )
27	26	opeq2d 3890	. . . . 5 |- ( ( ( W e. V /\ S C_ ( Base ` W ) ) /\ s = S ) -> <. (Scalar ` ndx ) , ( W ↾s s ) >. = <. (Scalar ` ndx ) , ( W ↾s S ) >. )
28	27	oveq2d 6066	. . . 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 6065	. . 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 6065	. 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 4268	. . . 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 13366	. . . . . . 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 2827	. . . . . 6 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> S e. _V )
40		ressex 13278	. . . . . 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 13244	. . . . 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 1274	. . . 4 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V )
44		vscaslid 13376	. . . . . 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 13345	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
48	47	slotex 13239	. . . . 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 13244	. . . 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 1274	. . 3 |- ( ( W e. V /\ S C_ ( Base ` W ) ) -> ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V )
52		ipslid 13384	. . . . 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 13244	. . 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 1274	. 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 5758	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 1398   e. wcel 2203   _Vcvv 2813   C_ wss 3211   ~Pcpw 3669   <.cop 3692   |-> cmpt 4171   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   NNcn 9237   ndxcnx 13209   sSet csts 13210  Slot cslot 13211   Basecbs 13212   ↾_s cress 13213   .rcmulr 13291  Scalarcsca 13293   .scvsca 13294   .icip 13295  subringAlg csra 14581

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-mulr 13304  df-sca 13306  df-vsca 13307  df-ip 13308  df-sra 14583

This theorem is referenced by:  sralemg  14586  srascag  14590  sravscag  14591  sraipg  14592  sraex  14594