Theorem aspval 14998

Description: Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
aspval.a	|- A = (AlgSpan ` W )
aspval.v	|- V = ( Base ` W )
aspval.l	|- L = ( LSubSp ` W )

Assertion

Ref	Expression
aspval	|- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } )

Distinct variable groups:   t, L   t, S   t, V   t, W

Allowed substitution hint:   A( t)

Proof of Theorem aspval

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

Step	Hyp	Ref	Expression
1		aspval.a	. . . . 5 |- A = (AlgSpan ` W )
2		df-asp 14983	. . . . . 6 |- AlgSpan = ( w e. AssAlg |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( (SubRing ` w ) i^i ( LSubSp ` w ) ) | s C_ t } ) )
3		fveq2 5693	. . . . . . . . 9 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		aspval.v	. . . . . . . . 9 |- V = ( Base ` W )
5	3, 4	eqtr4di 2289	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = V )
6	5	pweqd 3693	. . . . . . 7 |- ( w = W -> ~P ( Base ` w ) = ~P V )
7		fveq2 5693	. . . . . . . . . 10 |- ( w = W -> (SubRing ` w ) = (SubRing ` W ) )
8		fveq2 5693	. . . . . . . . . . 11 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
9		aspval.l	. . . . . . . . . . 11 |- L = ( LSubSp ` W )
10	8, 9	eqtr4di 2289	. . . . . . . . . 10 |- ( w = W -> ( LSubSp ` w ) = L )
11	7, 10	ineq12d 3433	. . . . . . . . 9 |- ( w = W -> ( (SubRing ` w ) i^i ( LSubSp ` w ) ) = ( (SubRing ` W ) i^i L ) )
12	11	rabeqdv 2815	. . . . . . . 8 |- ( w = W -> { t e. ( (SubRing ` w ) i^i ( LSubSp ` w ) ) | s C_ t } = { t e. ( (SubRing ` W ) i^i L ) | s C_ t } )
13	12	inteqd 3973	. . . . . . 7 |- ( w = W -> |^| { t e. ( (SubRing ` w ) i^i ( LSubSp ` w ) ) | s C_ t } = |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } )
14	6, 13	mpteq12dv 4211	. . . . . 6 |- ( w = W -> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( (SubRing ` w ) i^i ( LSubSp ` w ) ) | s C_ t } ) = ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) )
15		id 19	. . . . . 6 |- ( W e. AssAlg -> W e. AssAlg )
16		basfn 13394	. . . . . . . . . 10 |- Base Fn _V
17		elex 2833	. . . . . . . . . 10 |- ( W e. AssAlg -> W e. _V )
18		funfvex 5710	. . . . . . . . . . 11 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
19	18	funfni 5481	. . . . . . . . . 10 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
20	16, 17, 19	sylancr 418	. . . . . . . . 9 |- ( W e. AssAlg -> ( Base ` W ) e. _V )
21	4, 20	eqeltrid 2325	. . . . . . . 8 |- ( W e. AssAlg -> V e. _V )
22	21	pwexd 4316	. . . . . . 7 |- ( W e. AssAlg -> ~P V e. _V )
23	22	mptexd 5938	. . . . . 6 |- ( W e. AssAlg -> ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) e. _V )
24	2, 14, 15, 23	fvmptd3 5796	. . . . 5 |- ( W e. AssAlg -> (AlgSpan ` W ) = ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) )
25	1, 24	eqtrid 2283	. . . 4 |- ( W e. AssAlg -> A = ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) )
26	25	fveq1d 5695	. . 3 |- ( W e. AssAlg -> ( A ` S ) = ( ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) ` S ) )
27	26	adantr 276	. 2 |- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = ( ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) ` S ) )
28		eqid 2238	. . 3 |- ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) = ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } )
29		sseq1 3271	. . . . 5 |- ( s = S -> ( s C_ t <-> S C_ t ) )
30	29	rabbidv 2810	. . . 4 |- ( s = S -> { t e. ( (SubRing ` W ) i^i L ) | s C_ t } = { t e. ( (SubRing ` W ) i^i L ) | S C_ t } )
31	30	inteqd 3973	. . 3 |- ( s = S -> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } = |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } )
32		elpw2g 4290	. . . . 5 |- ( V e. _V -> ( S e. ~P V <-> S C_ V ) )
33	21, 32	syl 14	. . . 4 |- ( W e. AssAlg -> ( S e. ~P V <-> S C_ V ) )
34	33	biimpar 297	. . 3 |- ( ( W e. AssAlg /\ S C_ V ) -> S e. ~P V )
35		assaring 14990	. . . . . . 7 |- ( W e. AssAlg -> W e. Ring )
36	4	subrgid 14514	. . . . . . 7 |- ( W e. Ring -> V e. (SubRing ` W ) )
37	35, 36	syl 14	. . . . . 6 |- ( W e. AssAlg -> V e. (SubRing ` W ) )
38		assalmod 14989	. . . . . . 7 |- ( W e. AssAlg -> W e. LMod )
39	4, 9	lss1 14682	. . . . . . 7 |- ( W e. LMod -> V e. L )
40	38, 39	syl 14	. . . . . 6 |- ( W e. AssAlg -> V e. L )
41	37, 40	elind 3414	. . . . 5 |- ( W e. AssAlg -> V e. ( (SubRing ` W ) i^i L ) )
42		sseq2 3272	. . . . . 6 |- ( t = V -> ( S C_ t <-> S C_ V ) )
43	42	rspcev 2929	. . . . 5 |- ( ( V e. ( (SubRing ` W ) i^i L ) /\ S C_ V ) -> E. t e. ( (SubRing ` W ) i^i L ) S C_ t )
44	41, 43	sylan 283	. . . 4 |- ( ( W e. AssAlg /\ S C_ V ) -> E. t e. ( (SubRing ` W ) i^i L ) S C_ t )
45		intexrabim 4287	. . . 4 |- ( E. t e. ( (SubRing ` W ) i^i L ) S C_ t -> |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } e. _V )
46	44, 45	syl 14	. . 3 |- ( ( W e. AssAlg /\ S C_ V ) -> |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } e. _V )
47	28, 31, 34, 46	fvmptd3 5796	. 2 |- ( ( W e. AssAlg /\ S C_ V ) -> ( ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) ` S ) = |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } )
48	27, 47	eqtrd 2271	1 |- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   _Vcvv 2821   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   |^|cint 3968   |-> cmpt 4190   Fn wfn 5370   ` cfv 5375   Basecbs 13335   Ringcrg 14283  SubRingcsubrg 14508   LModclmod 14606   LSubSpclss 14672  AssAlgcasa 14979  AlgSpancasp 14980

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-mgp 14201  df-ur 14246  df-ring 14285  df-subrg 14510  df-lmod 14608  df-lssm 14673  df-assa 14982  df-asp 14983

This theorem is referenced by:  asplss  14999  aspid  15000  aspsubrg  15001  aspss  15002  aspssid  15003