Theorem aspval2 19853

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 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = (𝑅‘(ran 𝐶 ∪ 𝑆)))

Proof of Theorem aspval2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 4060	. . . . . . . . 9 ⊢ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)))
2	1	anbi1i 615	. . . . . . . 8 ⊢ ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ ((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥))
3		anass 461	. . . . . . . 8 ⊢ (((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥)))
4	2, 3	bitri 267	. . . . . . 7 ⊢ ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥)))
5		aspval2.c	. . . . . . . . . . 11 ⊢ 𝐶 = (algSc‘𝑊)
6		eqid 2780	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
7	5, 6	issubassa2 19851	. . . . . . . . . 10 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → (𝑥 ∈ (LSubSp‘𝑊) ↔ ran 𝐶 ⊆ 𝑥))
8	7	anbi1d 621	. . . . . . . . 9 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥)))
9		unss 4051	. . . . . . . . 9 ⊢ ((ran 𝐶 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)
10	8, 9	syl6bb 279	. . . . . . . 8 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥))
11	10	pm5.32da 571	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → ((𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)))
12	4, 11	syl5bb 275	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)))
13	12	abbidv 2845	. . . . 5 ⊢ (𝑊 ∈ AssAlg → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)})
14	13	adantr 473	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)})
15		df-rab 3099	. . . 4 ⊢ {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)}
16		df-rab 3099	. . . 4 ⊢ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)}
17	14, 15, 16	3eqtr4g 2841	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥} = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
18	17	inteqd 4759	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ∩ {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥} = ∩ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
19		aspval2.a	. . 3 ⊢ 𝐴 = (AlgSpan‘𝑊)
20		aspval2.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
21	19, 20, 6	aspval 19834	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥})
22		assaring 19826	. . . 4 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
23	20	subrgmre 19294	. . . 4 ⊢ (𝑊 ∈ Ring → (SubRing‘𝑊) ∈ (Moore‘𝑉))
24	22, 23	syl 17	. . 3 ⊢ (𝑊 ∈ AssAlg → (SubRing‘𝑊) ∈ (Moore‘𝑉))
25		eqid 2780	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
26		assalmod 19825	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
27		eqid 2780	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
28	5, 25, 22, 26, 27, 20	asclf 19843	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝐶:(Base‘(Scalar‘𝑊))⟶𝑉)
29	28	frnd 6356	. . . . 5 ⊢ (𝑊 ∈ AssAlg → ran 𝐶 ⊆ 𝑉)
30	29	adantr 473	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ran 𝐶 ⊆ 𝑉)
31		simpr 477	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
32	30, 31	unssd 4053	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (ran 𝐶 ∪ 𝑆) ⊆ 𝑉)
33		aspval2.r	. . . 4 ⊢ 𝑅 = (mrCls‘(SubRing‘𝑊))
34	33	mrcval 16751	. . 3 ⊢ (((SubRing‘𝑊) ∈ (Moore‘𝑉) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑉) → (𝑅‘(ran 𝐶 ∪ 𝑆)) = ∩ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
35	24, 32, 34	syl2an2r 673	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝑅‘(ran 𝐶 ∪ 𝑆)) = ∩ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
36	18, 21, 35	3eqtr4d 2826	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = (𝑅‘(ran 𝐶 ∪ 𝑆)))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  {cab 2760  {crab 3094   ∪ cun 3829   ∩ cin 3830   ⊆ wss 3831  ∩ cint 4754  ran crn 5412  ‘cfv 6193  Basecbs 16345  Scalarcsca 16430  Moorecmre 16723  mrClscmrc 16724  Ringcrg 19032  SubRingcsubrg 19266  LSubSpclss 19437  AssAlgcasa 19815  AlgSpancasp 19816  algSccascl 19817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-rep 5053  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285  ax-cnex 10397  ax-resscn 10398  ax-1cn 10399  ax-icn 10400  ax-addcl 10401  ax-addrcl 10402  ax-mulcl 10403  ax-mulrcl 10404  ax-mulcom 10405  ax-addass 10406  ax-mulass 10407  ax-distr 10408  ax-i2m1 10409  ax-1ne0 10410  ax-1rid 10411  ax-rnegex 10412  ax-rrecex 10413  ax-cnre 10414  ax-pre-lttri 10415  ax-pre-lttrn 10416  ax-pre-ltadd 10417  ax-pre-mulgt0 10418

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-tp 4449  df-op 4451  df-uni 4718  df-int 4755  df-iun 4799  df-br 4935  df-opab 4997  df-mpt 5014  df-tr 5036  df-id 5316  df-eprel 5321  df-po 5330  df-so 5331  df-fr 5370  df-we 5372  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-pred 5991  df-ord 6037  df-on 6038  df-lim 6039  df-suc 6040  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-riota 6943  df-ov 6985  df-oprab 6986  df-mpo 6987  df-om 7403  df-1st 7507  df-2nd 7508  df-wrecs 7756  df-recs 7818  df-rdg 7856  df-er 8095  df-en 8313  df-dom 8314  df-sdom 8315  df-pnf 10482  df-mnf 10483  df-xr 10484  df-ltxr 10485  df-le 10486  df-sub 10678  df-neg 10679  df-nn 11446  df-2 11509  df-3 11510  df-ndx 16348  df-slot 16349  df-base 16351  df-sets 16352  df-ress 16353  df-plusg 16440  df-mulr 16441  df-0g 16577  df-mre 16727  df-mrc 16728  df-mgm 17722  df-sgrp 17764  df-mnd 17775  df-grp 17906  df-minusg 17907  df-sbg 17908  df-subg 18072  df-mgp 18975  df-ur 18987  df-ring 19034  df-subrg 19268  df-lmod 19370  df-lss 19438  df-lsp 19478  df-assa 19818  df-asp 19819  df-ascl 19820

This theorem is referenced by:  evlseu  20021