Theorem aspval 19328

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		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
3		aspval.v	. . . . . . . . 9 |- V = ( Base ` W )
4	2, 3	syl6eqr 2674	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = V )
5	4	pweqd 4163	. . . . . . 7 |- ( w = W -> ~P ( Base ` w ) = ~P V )
6		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> (SubRing ` w ) = (SubRing ` W ) )
7		fveq2 6191	. . . . . . . . . . 11 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
8		aspval.l	. . . . . . . . . . 11 |- L = ( LSubSp ` W )
9	7, 8	syl6eqr 2674	. . . . . . . . . 10 |- ( w = W -> ( LSubSp ` w ) = L )
10	6, 9	ineq12d 3815	. . . . . . . . 9 |- ( w = W -> ( (SubRing ` w ) i^i ( LSubSp ` w ) ) = ( (SubRing ` W ) i^i L ) )
11		rabeq 3192	. . . . . . . . 9 |- ( ( (SubRing ` w ) i^i ( LSubSp ` w ) ) = ( (SubRing ` W ) i^i L ) -> { t e. ( (SubRing ` w ) i^i ( LSubSp ` w ) ) | s C_ t } = { t e. ( (SubRing ` W ) i^i L ) | s C_ t } )
12	10, 11	syl 17	. . . . . . . 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 4480	. . . . . . 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	5, 13	mpteq12dv 4733	. . . . . 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		df-asp 19313	. . . . . 6 |- AlgSpan = ( w e. AssAlg |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( (SubRing ` w ) i^i ( LSubSp ` w ) ) | s C_ t } ) )
16		fvex 6201	. . . . . . . . 9 |- ( Base ` W ) e. _V
17	3, 16	eqeltri 2697	. . . . . . . 8 |- V e. _V
18	17	pwex 4848	. . . . . . 7 |- ~P V e. _V
19	18	mptex 6486	. . . . . 6 |- ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) e. _V
20	14, 15, 19	fvmpt 6282	. . . . 5 |- ( W e. AssAlg -> (AlgSpan ` W ) = ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) )
21	1, 20	syl5eq 2668	. . . 4 |- ( W e. AssAlg -> A = ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) )
22	21	fveq1d 6193	. . 3 |- ( W e. AssAlg -> ( A ` S ) = ( ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) ` S ) )
23	22	adantr 481	. 2 |- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = ( ( s e. ~P V |-> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } ) ` S ) )
24		simpr 477	. . . 4 |- ( ( W e. AssAlg /\ S C_ V ) -> S C_ V )
25	17	elpw2 4828	. . . 4 |- ( S e. ~P V <-> S C_ V )
26	24, 25	sylibr 224	. . 3 |- ( ( W e. AssAlg /\ S C_ V ) -> S e. ~P V )
27		assaring 19320	. . . . . . 7 |- ( W e. AssAlg -> W e. Ring )
28	3	subrgid 18782	. . . . . . 7 |- ( W e. Ring -> V e. (SubRing ` W ) )
29	27, 28	syl 17	. . . . . 6 |- ( W e. AssAlg -> V e. (SubRing ` W ) )
30		assalmod 19319	. . . . . . 7 |- ( W e. AssAlg -> W e. LMod )
31	3, 8	lss1 18939	. . . . . . 7 |- ( W e. LMod -> V e. L )
32	30, 31	syl 17	. . . . . 6 |- ( W e. AssAlg -> V e. L )
33	29, 32	elind 3798	. . . . 5 |- ( W e. AssAlg -> V e. ( (SubRing ` W ) i^i L ) )
34		sseq2 3627	. . . . . 6 |- ( t = V -> ( S C_ t <-> S C_ V ) )
35	34	rspcev 3309	. . . . 5 |- ( ( V e. ( (SubRing ` W ) i^i L ) /\ S C_ V ) -> E. t e. ( (SubRing ` W ) i^i L ) S C_ t )
36	33, 35	sylan 488	. . . 4 |- ( ( W e. AssAlg /\ S C_ V ) -> E. t e. ( (SubRing ` W ) i^i L ) S C_ t )
37		intexrab 4823	. . . 4 |- ( E. t e. ( (SubRing ` W ) i^i L ) S C_ t <-> |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } e. _V )
38	36, 37	sylib 208	. . 3 |- ( ( W e. AssAlg /\ S C_ V ) -> |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } e. _V )
39		sseq1 3626	. . . . . 6 |- ( s = S -> ( s C_ t <-> S C_ t ) )
40	39	rabbidv 3189	. . . . 5 |- ( s = S -> { t e. ( (SubRing ` W ) i^i L ) | s C_ t } = { t e. ( (SubRing ` W ) i^i L ) | S C_ t } )
41	40	inteqd 4480	. . . 4 |- ( s = S -> |^| { t e. ( (SubRing ` W ) i^i L ) | s C_ t } = |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } )
42		eqid 2622	. . . 4 |- ( 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 } )
43	41, 42	fvmptg 6280	. . 3 |- ( ( S e. ~P V /\ |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } e. _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 } )
44	26, 38, 43	syl2anc 693	. 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 } )
45	23, 44	eqtrd 2656	1 |- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = |^| { t e. ( (SubRing ` W ) i^i L ) | S C_ t } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   |^|cint 4475   |-> cmpt 4729   ` cfv 5888   Basecbs 15857   Ringcrg 18547  SubRingcsubrg 18776   LModclmod 18863   LSubSpclss 18932  AssAlgcasa 19309  AlgSpancasp 19310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-assa 19312  df-asp 19313

This theorem is referenced by:  asplss  19329  aspid  19330  aspsubrg  19331  aspss  19332  aspssid  19333  aspval2  19347