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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 31276 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 37515 | . . 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 37515 | . . 3 ⊢ (𝜑 → 𝑇 ⊊ 𝑈) |
11 | 7, 10 | jca 513 | . 2 ⊢ (𝜑 → (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈)) |
12 | 3 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑊 ∈ 𝑋) |
13 | 4 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅 ∈ 𝑆) |
14 | 8 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑈 ∈ 𝑆) |
15 | 5 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑇 ∈ 𝑆) |
16 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅𝐶𝑈) | |
17 | 1, 2, 12, 13, 14, 15, 16 | lcvnbtwn 37516 | . . 3 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈)) |
18 | 17 | ex 414 | . 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 397 = wceq 1542 ∈ wcel 2107 ⊊ wpss 3916 class class class wbr 5110 ‘cfv 6501 LSubSpclss 20408 ⋖L clcv 37509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-lcv 37510 |
This theorem is referenced by: lsatcv0eq 37538 |
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