Theorem lcvfbr 39649

Description: The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)

Hypotheses

Ref	Expression
lcvfbr.s	⊢ 𝑆 = (LSubSp‘𝑊)
lcvfbr.c	⊢ 𝐶 = ( ⋖L ‘𝑊)
lcvfbr.w	⊢ (𝜑 → 𝑊 ∈ 𝑋)

Assertion

Ref	Expression
lcvfbr	⊢ (𝜑 → 𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})

Distinct variable groups:   𝑡,𝑠,𝑢,𝑆   𝑊,𝑠,𝑡,𝑢

Allowed substitution hints:   𝜑(𝑢,𝑡,𝑠)   𝐶(𝑢,𝑡,𝑠)   𝑋(𝑢,𝑡,𝑠)

Proof of Theorem lcvfbr

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcvfbr.c	. 2 ⊢ 𝐶 = ( ⋖L ‘𝑊)
2		lcvfbr.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
3		elex 3477	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
4		fveq2 6869	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
5		lcvfbr.s	. . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊)
6	4, 5	eqtr4di 2817	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
7	6	eleq2d 2850	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑡 ∈ (LSubSp‘𝑤) ↔ 𝑡 ∈ 𝑆))
8	6	eleq2d 2850	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑢 ∈ (LSubSp‘𝑤) ↔ 𝑢 ∈ 𝑆))
9	7, 8	anbi12d 641	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ↔ (𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)))
10	6	rexeqdv 3323	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))
11	10	notbid 320	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))
12	11	anbi2d 639	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)) ↔ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢))))
13	9, 12	anbi12d 641	. . . . 5 ⊢ (𝑤 = 𝑊 → (((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢))) ↔ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))))
14	13	opabbidv 5168	. . . 4 ⊢ (𝑤 = 𝑊 → {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))} = {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
15		df-lcv 39648	. . . 4 ⊢ ⋖L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
16	5	fvexi 6883	. . . . . 6 ⊢ 𝑆 ∈ V
17	16, 16	xpex 7738	. . . . 5 ⊢ (𝑆 × 𝑆) ∈ V
18		opabssxp 5741	. . . . 5 ⊢ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))} ⊆ (𝑆 × 𝑆)
19	17, 18	ssexi 5280	. . . 4 ⊢ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))} ∈ V
20	14, 15, 19	fvmpt 6977	. . 3 ⊢ (𝑊 ∈ V → ( ⋖L ‘𝑊) = {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
21	2, 3, 20	3syl 18	. 2 ⊢ (𝜑 → ( ⋖L ‘𝑊) = {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
22	1, 21	eqtrid 2811	1 ⊢ (𝜑 → 𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∃wrex 3088  Vcvv 3456   ⊊ wpss 3907  {copab 5164   × cxp 5647  ‘cfv 6523  LSubSpclss 21000   ⋖_L clcv 39647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-iota 6479  df-fun 6525  df-fv 6531  df-lcv 39648

This theorem is referenced by:  lcvbr  39650