Theorem lcvbr 34308

Description: The covers relation for a left vector space (or a left module). (cvbr 29141 analog.) (Contributed by NM, 9-Jan-2015.)

Hypotheses

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

Assertion

Ref	Expression
lcvbr	⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))))

Distinct variable groups:   𝑆,𝑠   𝑊,𝑠   𝑇,𝑠   𝑈,𝑠

Allowed substitution hints:   𝜑(𝑠)   𝐶(𝑠)   𝑋(𝑠)

Proof of Theorem lcvbr

Dummy variables 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcvfbr.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
2		lcvfbr.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
3		eleq1 2689	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡 ∈ 𝑆 ↔ 𝑇 ∈ 𝑆))
4	3	anbi1d 741	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ↔ (𝑇 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)))
5		psseq1 3694	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡 ⊊ 𝑢 ↔ 𝑇 ⊊ 𝑢))
6		psseq1 3694	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑡 ⊊ 𝑠 ↔ 𝑇 ⊊ 𝑠))
7	6	anbi1d 741	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))
8	7	rexbidv 3052	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))
9	8	notbid 308	. . . . . 6 ⊢ (𝑡 = 𝑇 → (¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))
10	5, 9	anbi12d 747	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)) ↔ (𝑇 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢))))
11	4, 10	anbi12d 747	. . . 4 ⊢ (𝑡 = 𝑇 → (((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢))) ↔ ((𝑇 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑇 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))))
12		eleq1 2689	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ 𝑆 ↔ 𝑈 ∈ 𝑆))
13	12	anbi2d 740	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑇 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ↔ (𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆)))
14		psseq2 3695	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑇 ⊊ 𝑢 ↔ 𝑇 ⊊ 𝑈))
15		psseq2 3695	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (𝑠 ⊊ 𝑢 ↔ 𝑠 ⊊ 𝑈))
16	15	anbi2d 740	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))
17	16	rexbidv 3052	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))
18	17	notbid 308	. . . . . 6 ⊢ (𝑢 = 𝑈 → (¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢) ↔ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))
19	14, 18	anbi12d 747	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑇 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)) ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))))
20	13, 19	anbi12d 747	. . . 4 ⊢ (𝑢 = 𝑈 → (((𝑇 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑇 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢))) ↔ ((𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) ∧ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))))
21		eqid 2622	. . . 4 ⊢ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))} = {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))}
22	11, 20, 21	brabg 4994	. . 3 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))}𝑈 ↔ ((𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) ∧ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))))
23	1, 2, 22	syl2anc 693	. 2 ⊢ (𝜑 → (𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))}𝑈 ↔ ((𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) ∧ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))))
24		lcvfbr.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑊)
25		lcvfbr.c	. . . 4 ⊢ 𝐶 = ( ⋖L ‘𝑊)
26		lcvfbr.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
27	24, 25, 26	lcvfbr 34307	. . 3 ⊢ (𝜑 → 𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
28	27	breqd 4664	. 2 ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ 𝑇{⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))}𝑈))
29	1, 2	jca 554	. . 3 ⊢ (𝜑 → (𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆))
30	29	biantrurd 529	. 2 ⊢ (𝜑 → ((𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)) ↔ ((𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) ∧ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))))
31	23, 28, 30	3bitr4d 300	1 ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ⊊ wpss 3575   class class class wbr 4653  {copab 4712  ‘cfv 5888  LSubSpclss 18932   ⋖_L clcv 34305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-lcv 34306

This theorem is referenced by:  lcvbr2  34309  lcvbr3  34310  lcvpss  34311  lcvnbtwn  34312  lsatcv0  34318  mapdcv  36949