Theorem lcvfbr 36316

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 3459	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
4		fveq2 6645	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
5		lcvfbr.s	. . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊)
6	4, 5	eqtr4di 2851	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
7	6	eleq2d 2875	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑡 ∈ (LSubSp‘𝑤) ↔ 𝑡 ∈ 𝑆))
8	6	eleq2d 2875	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑢 ∈ (LSubSp‘𝑤) ↔ 𝑢 ∈ 𝑆))
9	7, 8	anbi12d 633	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ↔ (𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)))
10	6	rexeqdv 3365	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))
11	10	notbid 321	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))
12	11	anbi2d 631	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)) ↔ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢))))
13	9, 12	anbi12d 633	. . . . 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 5096	. . . 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 36315	. . . 4 ⊢ ⋖L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
16	5	fvexi 6659	. . . . . 6 ⊢ 𝑆 ∈ V
17	16, 16	xpex 7456	. . . . 5 ⊢ (𝑆 × 𝑆) ∈ V
18		opabssxp 5607	. . . . 5 ⊢ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))} ⊆ (𝑆 × 𝑆)
19	17, 18	ssexi 5190	. . . 4 ⊢ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))} ∈ V
20	14, 15, 19	fvmpt 6745	. . 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	syl5eq 2845	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 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441   ⊊ wpss 3882  {copab 5092   × cxp 5517  ‘cfv 6324  LSubSpclss 19696   ⋖_L clcv 36314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-lcv 36315

This theorem is referenced by:  lcvbr  36317