Theorem lcvbr2 40345

Description: The covers relation for a left vector space (or a left module). (cvbr2 32942 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
lcvbr2	⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 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 lcvbr2

Step	Hyp	Ref	Expression
1		lcvfbr.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
2		lcvfbr.c	. . 3 ⊢ 𝐶 = ( ⋖L ‘𝑊)
3		lcvfbr.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
4		lcvfbr.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
5		lcvfbr.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
6	1, 2, 3, 4, 5	lcvbr 40344	. 2 ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))))
7		iman 407	. . . . . 6 ⊢ (((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈) ↔ ¬ ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) ∧ ¬ 𝑠 = 𝑈))
8		anass 474	. . . . . . 7 ⊢ (((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇 ⊊ 𝑠 ∧ (𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈)))
9		dfpss2 4018	. . . . . . . 8 ⊢ (𝑠 ⊊ 𝑈 ↔ (𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈))
10	9	anbi2i 635	. . . . . . 7 ⊢ ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈) ↔ (𝑇 ⊊ 𝑠 ∧ (𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈)))
11	8, 10	bitr4i 281	. . . . . 6 ⊢ (((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) ∧ ¬ 𝑠 = 𝑈) ↔ (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))
12	7, 11	xchbinx 337	. . . . 5 ⊢ (((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈) ↔ ¬ (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))
13	12	ralbii 3112	. . . 4 ⊢ (∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈) ↔ ∀𝑠 ∈ 𝑆 ¬ (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))
14		ralnex 3092	. . . 4 ⊢ (∀𝑠 ∈ 𝑆 ¬ (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈) ↔ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))
15	13, 14	bitri 278	. . 3 ⊢ (∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈) ↔ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))
16	15	anbi2i 635	. 2 ⊢ ((𝑇 ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈)) ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))
17	6, 16	bitr4di 292	1 ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ⊊ wpss 4016   class class class wbr 5108  ‘cfv 6566  LSubSpclss 21257   ⋖_L clcv 40341

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-lcv 40342

This theorem is used by:  lsmcv2  40352  lsat0cv  40356