Theorem aspval 22001

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 6881	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3		aspval.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
4	2, 3	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
5	4	pweqd 4578	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
6		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (SubRing‘𝑤) = (SubRing‘𝑊))
7		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8		aspval.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LSubSp‘𝑊)
9	7, 8	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
10	6, 9	ineq12d 4173	. . . . . . . . 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 4916	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
13	5, 12	mpteq12dv 5197	. . . . . 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 21983	. . . . . 6 ⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
15	3	fvexi 6895	. . . . . . . 8 ⊢ 𝑉 ∈ V
16	15	pwex 5351	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
17	16	mptex 7221	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}) ∈ V
18	13, 14, 17	fvmpt 6989	. . . . 5 ⊢ (𝑊 ∈ AssAlg → (AlgSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}))
19	1, 18	eqtrid 2808	. . . 4 ⊢ (𝑊 ∈ AssAlg → 𝐴 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}))
20	19	fveq1d 6883	. . 3 ⊢ (𝑊 ∈ AssAlg → (𝐴‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆))
21	20	adantr 485	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})‘𝑆))
22		eqid 2761	. . 3 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡})
23		sseq1 3961	. . . . 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 4916	. . 3 ⊢ (𝑠 = 𝑆 → ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})
26	15	elpw2 5304	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 ⊆ 𝑉)
27	26	bilanri 511	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ∈ 𝒫 𝑉)
28		assaring 21990	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
29	3	subrgid 20657	. . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝑉 ∈ (SubRing‘𝑊))
30	28, 29	syl 18	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ (SubRing‘𝑊))
31		assalmod 21989	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
32	3, 8	lss1 21038	. . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝐿)
33	31, 32	syl 18	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ 𝐿)
34	30, 33	elind 4152	. . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑉 ∈ ((SubRing‘𝑊) ∩ 𝐿))
35		sseq2 3962	. . . . . 6 ⊢ (𝑡 = 𝑉 → (𝑆 ⊆ 𝑡 ↔ 𝑆 ⊆ 𝑉))
36	35	rspcev 3580	. . . . 5 ⊢ ((𝑉 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∧ 𝑆 ⊆ 𝑉) → ∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡)
37	34, 36	sylan 591	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ∃𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿)𝑆 ⊆ 𝑡)
38		intexrab 5317	. . . 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 7013	. 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 2796	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∃wrex 3087  {crab 3414  Vcvv 3453   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4561  ∩ cint 4911   ↦ cmpt 5191  ‘cfv 6536  Basecbs 17268  Ringcrg 20314  SubRingcsubrg 20653  LModclmod 20960  LSubSpclss 21031  AssAlgcasa 21979  AlgSpancasp 21980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-0g 17493  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-grp 19002  df-mgp 20216  df-ur 20263  df-ring 20316  df-subrg 20654  df-lmod 20962  df-lss 21032  df-assa 21982  df-asp 21983

This theorem is referenced by:  asplss  22002  aspid  22003  aspsubrg  22004  aspss  22005  aspssid  22006  aspval2  22027