Theorem aspval 15015

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		df-asp 15000	. . . . . 6 ⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
3		fveq2 5695	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		aspval.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2289	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 3693	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7		fveq2 5695	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (SubRing‘𝑤) = (SubRing‘𝑊))
8		fveq2 5695	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
9		aspval.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LSubSp‘𝑊)
10	8, 9	eqtr4di 2289	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
11	7, 10	ineq12d 3433	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) = ((SubRing‘𝑊) ∩ 𝐿))
12	11	rabeqdv 2815	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
13	12	inteqd 3975	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
14	6, 13	mpteq12dv 4213	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}))
15		id 19	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ AssAlg)
16		basfn 13411	. . . . . . . . . 10 ⊢ Base Fn V
17		elex 2833	. . . . . . . . . 10 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ V)
18		funfvex 5712	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
19	18	funfni 5483	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
20	16, 17, 19	sylancr 418	. . . . . . . . 9 ⊢ (𝑊 ∈ AssAlg → (Base‘𝑊) ∈ V)
21	4, 20	eqeltrid 2325	. . . . . . . 8 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ V)
22	21	pwexd 4318	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝒫 𝑉 ∈ V)
23	22	mptexd 5944	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}) ∈ V)
24	2, 14, 15, 23	fvmptd3 5799	. . . . 5 ⊢ (𝑊 ∈ AssAlg → (AlgSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}))
25	1, 24	eqtrid 2283	. . . 4 ⊢ (𝑊 ∈ AssAlg → 𝐴 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}))
26	25	fveq1d 5697	. . 3 ⊢ (𝑊 ∈ AssAlg → (𝐴‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆))
27	26	adantr 276	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆))
28		eqid 2238	. . 3 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
29		sseq1 3271	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑡 ↔ 𝑆 ⊆ 𝑡))
30	29	rabbidv 2810	. . . 4 ⊢ (𝑠 = 𝑆 → {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
31	30	inteqd 3975	. . 3 ⊢ (𝑠 = 𝑆 → ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
32		elpw2g 4292	. . . . 5 ⊢ (𝑉 ∈ V → (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 ⊆ 𝑉))
33	21, 32	syl 14	. . . 4 ⊢ (𝑊 ∈ AssAlg → (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 ⊆ 𝑉))
34	33	biimpar 297	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ∈ 𝒫 𝑉)
35		assaring 15007	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
36	4	subrgid 14531	. . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝑉 ∈ (SubRing‘𝑊))
37	35, 36	syl 14	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ (SubRing‘𝑊))
38		assalmod 15006	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
39	4, 9	lss1 14699	. . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝐿)
40	38, 39	syl 14	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ 𝐿)
41	37, 40	elind 3414	. . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ ((SubRing‘𝑊) ∩ 𝐿))
42		sseq2 3272	. . . . . 6 ⊢ (𝑡 = 𝑉 → (𝑆 ⊆ 𝑡 ↔ 𝑆 ⊆ 𝑉))
43	42	rspcev 2929	. . . . 5 ⊢ ((𝑉 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∧ 𝑆 ⊆ 𝑉) → ∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡)
44	41, 43	sylan 283	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡)
45		intexrabim 4289	. . . 4 ⊢ (∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡 → ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡} ∈ V)
46	44, 45	syl 14	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡} ∈ V)
47	28, 31, 34, 46	fvmptd3 5799	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
48	27, 47	eqtrd 2271	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  {crab 2532  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220  𝒫 cpw 3688  ∩ cint 3970   ↦ cmpt 4192   Fn wfn 5372  ‘cfv 5377  Basecbs 13352  Ringcrg 14300  SubRingcsubrg 14525  LModclmod 14623  LSubSpclss 14689  AssAlgcasa 14996  AlgSpancasp 14997

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-sca 13447  df-vsca 13448  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-mgp 14218  df-ur 14263  df-ring 14302  df-subrg 14527  df-lmod 14625  df-lssm 14690  df-assa 14999  df-asp 15000

This theorem is used by:  asplss  15016  aspid  15017  aspsubrg  15018  aspss  15019  aspssid  15020