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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg5 | Structured version Visualization version GIF version |
Description: TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 37173? TODO: The ∨ hypothesis is unused. FIX COMMENT. (Contributed by NM, 26-Apr-2013.) |
Ref | Expression |
---|---|
cdlemg5.l | ⊢ ≤ = (le‘𝐾) |
cdlemg5.j | ⊢ ∨ = (join‘𝐾) |
cdlemg5.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg5.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
cdlemg5 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg5.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | cdlemg5.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | cdlemg5.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 1, 2, 3 | lhpexle 37173 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑊) |
5 | 4 | adantr 483 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑊) |
6 | simpll 765 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | simpr 487 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊)) → (𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊)) | |
8 | simplr 767 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
9 | cdlemg5.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
10 | 1, 9, 2, 3 | cdlemf1 37729 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑟 ≤ (𝑃 ∨ 𝑞))) |
11 | 6, 7, 8, 10 | syl3anc 1367 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑟 ≤ (𝑃 ∨ 𝑞))) |
12 | 3simpa 1144 | . . . 4 ⊢ ((𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑟 ≤ (𝑃 ∨ 𝑞)) → (𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊)) | |
13 | 12 | reximi 3243 | . . 3 ⊢ (∃𝑞 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑟 ≤ (𝑃 ∨ 𝑞)) → ∃𝑞 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊)) |
14 | 11, 13 | syl 17 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑟 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊)) |
15 | 5, 14 | rexlimddv 3291 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑞 ≤ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 class class class wbr 5052 ‘cfv 6341 (class class class)co 7142 lecple 16555 joincjn 17537 Atomscatm 36431 HLchlt 36518 LHypclh 37152 |
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 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-proset 17521 df-poset 17539 df-plt 17551 df-lub 17567 df-glb 17568 df-join 17569 df-meet 17570 df-p0 17632 df-p1 17633 df-lat 17639 df-clat 17701 df-oposet 36344 df-ol 36346 df-oml 36347 df-covers 36434 df-ats 36435 df-atl 36466 df-cvlat 36490 df-hlat 36519 df-lhyp 37156 |
This theorem is referenced by: cdlemb3 37774 |
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