Theorem aspval2 22206

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 4093	. . . . . . . . 9 ⊢ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)))
2	1	anbi1i 636	. . . . . . . 8 ⊢ ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ ((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥))
3		anass 474	. . . . . . . 8 ⊢ (((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥)))
4	2, 3	bitri 278	. . . . . . 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 2760	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
7	5, 6	issubassa2 22200	. . . . . . . . . 10 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → (𝑥 ∈ (LSubSp‘𝑊) ↔ ran 𝐶 ⊆ 𝑥))
8	7	anbi1d 643	. . . . . . . . 9 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥)))
9		unss 4188	. . . . . . . . 9 ⊢ ((ran 𝐶 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)
10	8, 9	bitrdi 290	. . . . . . . 8 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥))
11	10	pm5.32da 590	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → ((𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)))
12	4, 11	bitrid 286	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)))
13	12	abbidv 2826	. . . . 5 ⊢ (𝑊 ∈ AssAlg → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)})
14	13	adantr 486	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)})
15		df-rab 3413	. . . 4 ⊢ {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)}
16		df-rab 3413	. . . 4 ⊢ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)}
17	14, 15, 16	3eqtr4g 2820	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥} = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
18	17	inteqd 4911	. 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 22180	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥})
22		assaring 22169	. . . 4 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
23	20	subrgmre 20849	. . . 4 ⊢ (𝑊 ∈ Ring → (SubRing‘𝑊) ∈ (Moore‘𝑉))
24	22, 23	syl 18	. . 3 ⊢ (𝑊 ∈ AssAlg → (SubRing‘𝑊) ∈ (Moore‘𝑉))
25		eqid 2760	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
26		assalmod 22168	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
27		eqid 2760	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
28	5, 25, 22, 26, 27, 20	asclf 22189	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝐶:(Base‘(Scalar‘𝑊))⟶𝑉)
29	28	frnd 6718	. . . . 5 ⊢ (𝑊 ∈ AssAlg → ran 𝐶 ⊆ 𝑉)
30	29	adantr 486	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ran 𝐶 ⊆ 𝑉)
31		simpr 490	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
32	30, 31	unssd 4190	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (ran 𝐶 ∪ 𝑆) ⊆ 𝑉)
33		aspval2.r	. . . 4 ⊢ 𝑅 = (mrCls‘(SubRing‘𝑊))
34	33	mrcval 17784	. . 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 698	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝑅‘(ran 𝐶 ∪ 𝑆)) = ∩ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
36	18, 21, 35	3eqtr4d 2805	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = (𝑅‘(ran 𝐶 ∪ 𝑆)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  {crab 3412   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  ∩ cint 4906  ran crn 5651  ‘cfv 6538  Basecbs 17387  Scalarcsca 17431  Moorecmre 17752  mrClscmrc 17753  Ringcrg 20459  SubRingcsubrg 20821  LSubSpclss 21206  AssAlgcasa 22158  AlgSpancasp 22159  algSccascl 22160

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-0g 17612  df-mre 17756  df-mrc 17757  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-sbg 19149  df-subg 19333  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-subrng 20798  df-subrg 20822  df-lmod 21137  df-lss 21207  df-lsp 21247  df-assa 22161  df-asp 22162  df-ascl 22163

This theorem is used by:  evlseu  22392