Theorem aspval 22088

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 6882	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		aspval.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
4	2, 3	eqtr4di 2815	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
5	4	pweqd 4577	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
6		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (SubRing‘𝑤) = (SubRing‘𝑊))
7		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8		aspval.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LSubSp‘𝑊)
9	7, 8	eqtr4di 2815	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
10	6, 9	ineq12d 4170	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) = ((SubRing‘𝑊) ∩ 𝐿))
11	10	rabeqdv 3429	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
12	11	inteqd 4915	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
13	5, 12	mpteq12dv 5196	. . . . . 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 22070	. . . . . 6 ⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
15	3	fvexi 6896	. . . . . . . 8 ⊢ 𝑉 ∈ V
16	15	pwex 5349	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
17	16	mptex 7225	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}) ∈ V
18	13, 14, 17	fvmpt 6990	. . . . 5 ⊢ (𝑊 ∈ AssAlg → (AlgSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}))
19	1, 18	eqtrid 2809	. . . 4 ⊢ (𝑊 ∈ AssAlg → 𝐴 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}))
20	19	fveq1d 6884	. . 3 ⊢ (𝑊 ∈ AssAlg → (𝐴‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆))
21	20	adantr 486	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆))
22		eqid 2762	. . 3 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
23		sseq1 3959	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑡 ↔ 𝑆 ⊆ 𝑡))
24	23	rabbidv 3421	. . . 4 ⊢ (𝑠 = 𝑆 → {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
25	24	inteqd 4915	. . 3 ⊢ (𝑠 = 𝑆 → ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
26	15	elpw2 5303	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 ⊆ 𝑉)
27	26	bilanri 512	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ∈ 𝒫 𝑉)
28		assaring 22077	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
29	3	subrgid 20736	. . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝑉 ∈ (SubRing‘𝑊))
30	28, 29	syl 18	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ (SubRing‘𝑊))
31		assalmod 22076	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
32	3, 8	lss1 21123	. . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝐿)
33	31, 32	syl 18	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ 𝐿)
34	30, 33	elind 4149	. . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ ((SubRing‘𝑊) ∩ 𝐿))
35		sseq2 3960	. . . . . 6 ⊢ (𝑡 = 𝑉 → (𝑆 ⊆ 𝑡 ↔ 𝑆 ⊆ 𝑉))
36	35	rspcev 3579	. . . . 5 ⊢ ((𝑉 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∧ 𝑆 ⊆ 𝑉) → ∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡)
37	34, 36	sylan 592	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡)
38		intexrab 5315	. . . 4 ⊢ (∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡} ∈ V)
39	37, 38	sylib 221	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡} ∈ V)
40	22, 25, 27, 39	fvmptd3 7014	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
41	21, 40	eqtrd 2797	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  {crab 3414  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560  ∩ cint 4910   ↦ cmpt 5190  ‘cfv 6537  Basecbs 17305  Ringcrg 20373  SubRingcsubrg 20732  LModclmod 21045  LSubSpclss 21116  AssAlgcasa 22066  AlgSpancasp 22067

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-0g 17530  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-grp 19061  df-mgp 20275  df-ur 20322  df-ring 20375  df-subrg 20733  df-lmod 21047  df-lss 21117  df-assa 22069  df-asp 22070

This theorem is used by:  asplss  22089  aspid  22090  aspsubrg  22091  aspss  22092  aspssid  22093  aspval2  22114