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Theorem aspval 21418

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 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 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 6888	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		aspval.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
4	2, 3	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
5	4	pweqd 4618	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
6		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (SubRing‘𝑤) = (SubRing‘𝑊))
7		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8		aspval.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LSubSp‘𝑊)
9	7, 8	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
10	6, 9	ineq12d 4212	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) = ((SubRing‘𝑊) ∩ 𝐿))
11	10	rabeqdv 3447	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
12	11	inteqd 4954	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
13	5, 12	mpteq12dv 5238	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}))
14		df-asp 21400	. . . . . 6 ⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
15	3	fvexi 6902	. . . . . . . 8 ⊢ 𝑉 ∈ V
16	15	pwex 5377	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
17	16	mptex 7221	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}) ∈ V
18	13, 14, 17	fvmpt 6995	. . . . 5 ⊢ (𝑊 ∈ AssAlg → (AlgSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}))
19	1, 18	eqtrid 2784	. . . 4 ⊢ (𝑊 ∈ AssAlg → 𝐴 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}))
20	19	fveq1d 6890	. . 3 ⊢ (𝑊 ∈ AssAlg → (𝐴‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆))
21	20	adantr 481	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆))
22		eqid 2732	. . 3 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
23		sseq1 4006	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑡 ↔ 𝑆 ⊆ 𝑡))
24	23	rabbidv 3440	. . . 4 ⊢ (𝑠 = 𝑆 → {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
25	24	inteqd 4954	. . 3 ⊢ (𝑠 = 𝑆 → ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
26		simpr 485	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
27	15	elpw2 5344	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 ⊆ 𝑉)
28	26, 27	sylibr 233	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ∈ 𝒫 𝑉)
29		assaring 21407	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
30	3	subrgid 20357	. . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝑉 ∈ (SubRing‘𝑊))
31	29, 30	syl 17	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ (SubRing‘𝑊))
32		assalmod 21406	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
33	3, 8	lss1 20541	. . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝐿)
34	32, 33	syl 17	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ 𝐿)
35	31, 34	elind 4193	. . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ ((SubRing‘𝑊) ∩ 𝐿))
36		sseq2 4007	. . . . . 6 ⊢ (𝑡 = 𝑉 → (𝑆 ⊆ 𝑡 ↔ 𝑆 ⊆ 𝑉))
37	36	rspcev 3612	. . . . 5 ⊢ ((𝑉 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∧ 𝑆 ⊆ 𝑉) → ∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡)
38	35, 37	sylan 580	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡)
39		intexrab 5339	. . . 4 ⊢ (∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡} ∈ V)
40	38, 39	sylib 217	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡} ∈ V)
41	22, 25, 28, 40	fvmptd3 7018	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
42	21, 41	eqtrd 2772	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 {crab 3432 Vcvv 3474 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 ∩ cint 4949 ↦ cmpt 5230 ‘cfv 6540 Basecbs 17140 Ringcrg 20049 SubRingcsubrg 20351 LModclmod 20463 LSubSpclss 20534 AssAlgcasa 21396 AlgSpancasp 21397

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-mgp 19982 df-ur 19999 df-ring 20051 df-subrg 20353 df-lmod 20465 df-lss 20535 df-assa 21399 df-asp 21400

This theorem is referenced by: asplss 21419 aspid 21420 aspsubrg 21421 aspss 21422 aspssid 21423 aspval2 21443

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