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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvntr | Structured version Visualization version GIF version |
Description: The covers relation is not transitive. (cvntr 32115 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lcvnbtwn.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lcvnbtwn.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lcvnbtwn.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) |
lcvnbtwn.r | ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
lcvnbtwn.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
lcvnbtwn.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lcvnbtwn.d | ⊢ (𝜑 → 𝑅𝐶𝑇) |
lcvntr.p | ⊢ (𝜑 → 𝑇𝐶𝑈) |
Ref | Expression |
---|---|
lcvntr | ⊢ (𝜑 → ¬ 𝑅𝐶𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcvnbtwn.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
2 | lcvnbtwn.c | . . . 4 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
3 | lcvnbtwn.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
4 | lcvnbtwn.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) | |
5 | lcvnbtwn.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
6 | lcvnbtwn.d | . . . 4 ⊢ (𝜑 → 𝑅𝐶𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | lcvpss 38496 | . . 3 ⊢ (𝜑 → 𝑅 ⊊ 𝑇) |
8 | lcvnbtwn.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
9 | lcvntr.p | . . . 4 ⊢ (𝜑 → 𝑇𝐶𝑈) | |
10 | 1, 2, 3, 5, 8, 9 | lcvpss 38496 | . . 3 ⊢ (𝜑 → 𝑇 ⊊ 𝑈) |
11 | 7, 10 | jca 511 | . 2 ⊢ (𝜑 → (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈)) |
12 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑊 ∈ 𝑋) |
13 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅 ∈ 𝑆) |
14 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑈 ∈ 𝑆) |
15 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑇 ∈ 𝑆) |
16 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅𝐶𝑈) | |
17 | 1, 2, 12, 13, 14, 15, 16 | lcvnbtwn 38497 | . . 3 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈)) |
18 | 17 | ex 412 | . 2 ⊢ (𝜑 → (𝑅𝐶𝑈 → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈))) |
19 | 11, 18 | mt2d 136 | 1 ⊢ (𝜑 → ¬ 𝑅𝐶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊊ wpss 3948 class class class wbr 5148 ‘cfv 6548 LSubSpclss 20815 ⋖L clcv 38490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-lcv 38491 |
This theorem is referenced by: lsatcv0eq 38519 |
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