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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 30071 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 36162 | . . 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 36162 | . . 3 ⊢ (𝜑 → 𝑇 ⊊ 𝑈) |
11 | 7, 10 | jca 514 | . 2 ⊢ (𝜑 → (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈)) |
12 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑊 ∈ 𝑋) |
13 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅 ∈ 𝑆) |
14 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑈 ∈ 𝑆) |
15 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑇 ∈ 𝑆) |
16 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅𝐶𝑈) | |
17 | 1, 2, 12, 13, 14, 15, 16 | lcvnbtwn 36163 | . . 3 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈)) |
18 | 17 | ex 415 | . 2 ⊢ (𝜑 → (𝑅𝐶𝑈 → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈))) |
19 | 11, 18 | mt2d 138 | 1 ⊢ (𝜑 → ¬ 𝑅𝐶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊊ wpss 3939 class class class wbr 5068 ‘cfv 6357 LSubSpclss 19705 ⋖L clcv 36156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-lcv 36157 |
This theorem is referenced by: lsatcv0eq 36185 |
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