Theorem aspval2 21855

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 3906	. . . . . . . . 9 ⊢ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)))
2	1	anbi1i 625	. . . . . . . 8 ⊢ ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ ((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥))
3		anass 468	. . . . . . . 8 ⊢ (((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥)))
4	2, 3	bitri 275	. . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
7	5, 6	issubassa2 21849	. . . . . . . . . 10 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → (𝑥 ∈ (LSubSp‘𝑊) ↔ ran 𝐶 ⊆ 𝑥))
8	7	anbi1d 632	. . . . . . . . 9 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥)))
9		unss 4131	. . . . . . . . 9 ⊢ ((ran 𝐶 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)
10	8, 9	bitrdi 287	. . . . . . . 8 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥))
11	10	pm5.32da 579	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → ((𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)))
12	4, 11	bitrid 283	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)))
13	12	abbidv 2803	. . . . 5 ⊢ (𝑊 ∈ AssAlg → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)})
14	13	adantr 480	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)})
15		df-rab 3391	. . . 4 ⊢ {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)}
16		df-rab 3391	. . . 4 ⊢ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)}
17	14, 15, 16	3eqtr4g 2797	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥} = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
18	17	inteqd 4895	. 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 21829	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥})
22		assaring 21818	. . . 4 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
23	20	subrgmre 20532	. . . 4 ⊢ (𝑊 ∈ Ring → (SubRing‘𝑊) ∈ (Moore‘𝑉))
24	22, 23	syl 17	. . 3 ⊢ (𝑊 ∈ AssAlg → (SubRing‘𝑊) ∈ (Moore‘𝑉))
25		eqid 2737	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
26		assalmod 21817	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
27		eqid 2737	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
28	5, 25, 22, 26, 27, 20	asclf 21838	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝐶:(Base‘(Scalar‘𝑊))⟶𝑉)
29	28	frnd 6668	. . . . 5 ⊢ (𝑊 ∈ AssAlg → ran 𝐶 ⊆ 𝑉)
30	29	adantr 480	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ran 𝐶 ⊆ 𝑉)
31		simpr 484	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
32	30, 31	unssd 4133	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (ran 𝐶 ∪ 𝑆) ⊆ 𝑉)
33		aspval2.r	. . . 4 ⊢ 𝑅 = (mrCls‘(SubRing‘𝑊))
34	33	mrcval 17534	. . 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 686	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝑅‘(ran 𝐶 ∪ 𝑆)) = ∩ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
36	18, 21, 35	3eqtr4d 2782	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = (𝑅‘(ran 𝐶 ∪ 𝑆)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  {crab 3390   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∩ cint 4890  ran crn 5623  ‘cfv 6490  Basecbs 17137  Scalarcsca 17181  Moorecmre 17502  mrClscmrc 17503  Ringcrg 20172  SubRingcsubrg 20504  LSubSpclss 20884  AssAlgcasa 21807  AlgSpancasp 21808  algSccascl 21809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17138  df-ress 17159  df-plusg 17191  df-mulr 17192  df-0g 17362  df-mre 17506  df-mrc 17507  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18870  df-minusg 18871  df-sbg 18872  df-subg 19057  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-ring 20174  df-subrng 20481  df-subrg 20505  df-lmod 20815  df-lss 20885  df-lsp 20925  df-assa 21810  df-asp 21811  df-ascl 21812

This theorem is referenced by:  evlseu  22039