Theorem aspval2 19347

Description: The algebraic closure is the ring closure when the generating set is expanded to include all scalars. EDITORIAL : In light of this, is AlgSpan independently needed? (Contributed by Stefan O'Rear, 9-Mar-2015.)

Hypotheses

Ref	Expression
aspval2.a	|- A = (AlgSpan ` W )
aspval2.c	|- C = (algSc ` W )
aspval2.r	|- R = (mrCls ` (SubRing ` W ) )
aspval2.v	|- V = ( Base ` W )

Assertion

Ref	Expression
aspval2	|- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = ( R ` ( ran C u. S ) ) )

Proof of Theorem aspval2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . . . . . . . 9 |- ( x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) <-> ( x e. (SubRing ` W ) /\ x e. ( LSubSp ` W ) ) )
2	1	anbi1i 731	. . . . . . . 8 |- ( ( x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) /\ S C_ x ) <-> ( ( x e. (SubRing ` W ) /\ x e. ( LSubSp ` W ) ) /\ S C_ x ) )
3		anass 681	. . . . . . . 8 |- ( ( ( x e. (SubRing ` W ) /\ x e. ( LSubSp ` W ) ) /\ S C_ x ) <-> ( x e. (SubRing ` W ) /\ ( x e. ( LSubSp ` W ) /\ S C_ x ) ) )
4	2, 3	bitri 264	. . . . . . 7 |- ( ( x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) /\ S C_ x ) <-> ( x e. (SubRing ` W ) /\ ( x e. ( LSubSp ` W ) /\ S C_ x ) ) )
5		aspval2.c	. . . . . . . . . . 11 |- C = (algSc ` W )
6		eqid 2622	. . . . . . . . . . 11 |- ( LSubSp ` W ) = ( LSubSp ` W )
7	5, 6	issubassa2 19345	. . . . . . . . . 10 |- ( ( W e. AssAlg /\ x e. (SubRing ` W ) ) -> ( x e. ( LSubSp ` W ) <-> ran C C_ x ) )
8	7	anbi1d 741	. . . . . . . . 9 |- ( ( W e. AssAlg /\ x e. (SubRing ` W ) ) -> ( ( x e. ( LSubSp ` W ) /\ S C_ x ) <-> ( ran C C_ x /\ S C_ x ) ) )
9		unss 3787	. . . . . . . . 9 |- ( ( ran C C_ x /\ S C_ x ) <-> ( ran C u. S ) C_ x )
10	8, 9	syl6bb 276	. . . . . . . 8 |- ( ( W e. AssAlg /\ x e. (SubRing ` W ) ) -> ( ( x e. ( LSubSp ` W ) /\ S C_ x ) <-> ( ran C u. S ) C_ x ) )
11	10	pm5.32da 673	. . . . . . 7 |- ( W e. AssAlg -> ( ( x e. (SubRing ` W ) /\ ( x e. ( LSubSp ` W ) /\ S C_ x ) ) <-> ( x e. (SubRing ` W ) /\ ( ran C u. S ) C_ x ) ) )
12	4, 11	syl5bb 272	. . . . . 6 |- ( W e. AssAlg -> ( ( x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) /\ S C_ x ) <-> ( x e. (SubRing ` W ) /\ ( ran C u. S ) C_ x ) ) )
13	12	abbidv 2741	. . . . 5 |- ( W e. AssAlg -> { x | ( x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) /\ S C_ x ) } = { x | ( x e. (SubRing ` W ) /\ ( ran C u. S ) C_ x ) } )
14	13	adantr 481	. . . 4 |- ( ( W e. AssAlg /\ S C_ V ) -> { x | ( x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) /\ S C_ x ) } = { x | ( x e. (SubRing ` W ) /\ ( ran C u. S ) C_ x ) } )
15		df-rab 2921	. . . 4 |- { x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ x } = { x | ( x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) /\ S C_ x ) }
16		df-rab 2921	. . . 4 |- { x e. (SubRing ` W ) | ( ran C u. S ) C_ x } = { x | ( x e. (SubRing ` W ) /\ ( ran C u. S ) C_ x ) }
17	14, 15, 16	3eqtr4g 2681	. . 3 |- ( ( W e. AssAlg /\ S C_ V ) -> { x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ x } = { x e. (SubRing ` W ) | ( ran C u. S ) C_ x } )
18	17	inteqd 4480	. 2 |- ( ( W e. AssAlg /\ S C_ V ) -> |^| { x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ x } = |^| { x e. (SubRing ` W ) | ( ran C u. S ) C_ x } )
19		aspval2.a	. . 3 |- A = (AlgSpan ` W )
20		aspval2.v	. . 3 |- V = ( Base ` W )
21	19, 20, 6	aspval 19328	. 2 |- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = |^| { x e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ x } )
22		assaring 19320	. . . . 5 |- ( W e. AssAlg -> W e. Ring )
23	20	subrgmre 18804	. . . . 5 |- ( W e. Ring -> (SubRing ` W ) e. (Moore ` V ) )
24	22, 23	syl 17	. . . 4 |- ( W e. AssAlg -> (SubRing ` W ) e. (Moore ` V ) )
25	24	adantr 481	. . 3 |- ( ( W e. AssAlg /\ S C_ V ) -> (SubRing ` W ) e. (Moore ` V ) )
26		eqid 2622	. . . . . . 7 |- (Scalar ` W ) = (Scalar ` W )
27		assalmod 19319	. . . . . . 7 |- ( W e. AssAlg -> W e. LMod )
28		eqid 2622	. . . . . . 7 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
29	5, 26, 22, 27, 28, 20	asclf 19337	. . . . . 6 |- ( W e. AssAlg -> C : ( Base ` (Scalar ` W ) ) --> V )
30		frn 6053	. . . . . 6 |- ( C : ( Base ` (Scalar ` W ) ) --> V -> ran C C_ V )
31	29, 30	syl 17	. . . . 5 |- ( W e. AssAlg -> ran C C_ V )
32	31	adantr 481	. . . 4 |- ( ( W e. AssAlg /\ S C_ V ) -> ran C C_ V )
33		simpr 477	. . . 4 |- ( ( W e. AssAlg /\ S C_ V ) -> S C_ V )
34	32, 33	unssd 3789	. . 3 |- ( ( W e. AssAlg /\ S C_ V ) -> ( ran C u. S ) C_ V )
35		aspval2.r	. . . 4 |- R = (mrCls ` (SubRing ` W ) )
36	35	mrcval 16270	. . 3 |- ( ( (SubRing ` W ) e. (Moore ` V ) /\ ( ran C u. S ) C_ V ) -> ( R ` ( ran C u. S ) ) = |^| { x e. (SubRing ` W ) | ( ran C u. S ) C_ x } )
37	25, 34, 36	syl2anc 693	. 2 |- ( ( W e. AssAlg /\ S C_ V ) -> ( R ` ( ran C u. S ) ) = |^| { x e. (SubRing ` W ) | ( ran C u. S ) C_ x } )
38	18, 21, 37	3eqtr4d 2666	1 |- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = ( R ` ( ran C u. S ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   u. cun 3572   i^i cin 3573   C_ wss 3574   |^|cint 4475   ran crn 5115   -->wf 5884   ` cfv 5888   Basecbs 15857  Scalarcsca 15944  Moorecmre 16242  mrClscmrc 16243   Ringcrg 18547  SubRingcsubrg 18776   LSubSpclss 18932  AssAlgcasa 19309  AlgSpancasp 19310  algSccascl 19311

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-1st 7168  df-2nd 7169  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-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mre 16246  df-mrc 16247  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-assa 19312  df-asp 19313  df-ascl 19314

This theorem is referenced by:  evlseu  19516