Theorem lspval 21022

Description: The span of a set of vectors (in a left module). (spanval 31482 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
lspval.v	⊢ 𝑉 = (Base‘𝑊)
lspval.s	⊢ 𝑆 = (LSubSp‘𝑊)
lspval.n	⊢ 𝑁 = (LSpan‘𝑊)

Assertion

Ref	Expression
lspval	⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})

Distinct variable groups:   𝑡,𝑆   𝑡,𝑈   𝑡,𝑉

Allowed substitution hints:   𝑁(𝑡)   𝑊(𝑡)

Proof of Theorem lspval

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspval.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
2		lspval.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lspval.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
4	1, 2, 3	lspfval 21020	. . . 4 ⊢ (𝑊 ∈ LMod → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
5	4	fveq1d 6865	. . 3 ⊢ (𝑊 ∈ LMod → (𝑁‘𝑈) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈))
6	5	adantr 484	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈))
7		eqid 2761	. . 3 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})
8		sseq1 3961	. . . . 5 ⊢ (𝑠 = 𝑈 → (𝑠 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑡))
9	8	rabbidv 3420	. . . 4 ⊢ (𝑠 = 𝑈 → {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
10	9	inteqd 4909	. . 3 ⊢ (𝑠 = 𝑈 → ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
11	1	fvexi 6877	. . . . 5 ⊢ 𝑉 ∈ V
12	11	elpw2 5289	. . . 4 ⊢ (𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉)
13	12	bilanri 510	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ∈ 𝒫 𝑉)
14	1, 2	lss1 20985	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆)
15		sseq2 3962	. . . . . 6 ⊢ (𝑡 = 𝑉 → (𝑈 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑉))
16	15	rspcev 3581	. . . . 5 ⊢ ((𝑉 ∈ 𝑆 ∧ 𝑈 ⊆ 𝑉) → ∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡)
17	14, 16	sylan 589	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡)
18		intexrab 5302	. . . 4 ⊢ (∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} ∈ V)
19	17, 18	sylib 220	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} ∈ V)
20	7, 10, 13, 19	fvmptd3 6995	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
21	6, 20	eqtrd 2796	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  {crab 3413  Vcvv 3453   ⊆ wss 3904  𝒫 cpw 4554  ∩ cint 4904   ↦ cmpt 5180  ‘cfv 6517  Basecbs 17228  LModclmod 20907  LSubSpclss 20978  LSpanclspn 21018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961  df-lmod 20909  df-lss 20979  df-lsp 21019

This theorem is referenced by:  lspid  21029  lspss  21031  lspssid  21032  dochspss  41966  lcosslsp  49024