Theorem lcvfbr 38976

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 3509	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
4		fveq2 6920	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
5		lcvfbr.s	. . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊)
6	4, 5	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
7	6	eleq2d 2830	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑡 ∈ (LSubSp‘𝑤) ↔ 𝑡 ∈ 𝑆))
8	6	eleq2d 2830	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑢 ∈ (LSubSp‘𝑤) ↔ 𝑢 ∈ 𝑆))
9	7, 8	anbi12d 631	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ↔ (𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)))
10	6	rexeqdv 3335	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))
11	10	notbid 318	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))
12	11	anbi2d 629	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)) ↔ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢))))
13	9, 12	anbi12d 631	. . . . 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 5232	. . . 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 38975	. . . 4 ⊢ ⋖L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
16	5	fvexi 6934	. . . . . 6 ⊢ 𝑆 ∈ V
17	16, 16	xpex 7788	. . . . 5 ⊢ (𝑆 × 𝑆) ∈ V
18		opabssxp 5792	. . . . 5 ⊢ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))} ⊆ (𝑆 × 𝑆)
19	17, 18	ssexi 5340	. . . 4 ⊢ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))} ∈ V
20	14, 15, 19	fvmpt 7029	. . 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 2792	1 ⊢ (𝜑 → 𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ⊊ wpss 3977  {copab 5228   × cxp 5698  ‘cfv 6573  LSubSpclss 20952   ⋖_L clcv 38974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-lcv 38975

This theorem is referenced by:  lcvbr  38977