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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme35sn3a | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.) |
Ref | Expression |
---|---|
cdleme32s.b | ⊢ 𝐵 = (Base‘𝐾) |
cdleme32s.l | ⊢ ≤ = (le‘𝐾) |
cdleme32s.j | ⊢ ∨ = (join‘𝐾) |
cdleme32s.m | ⊢ ∧ = (meet‘𝐾) |
cdleme32s.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme32s.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme32s.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdleme32s.d | ⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) |
cdleme32s.n | ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) |
Ref | Expression |
---|---|
cdleme35sn3a | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ ⦋𝑅 / 𝑠⦌𝑁 ≤ (𝑃 ∨ 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme32s.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | cdleme32s.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | cdleme32s.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | cdleme32s.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdleme32s.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdleme32s.u | . . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
7 | eqid 2733 | . . 3 ⊢ ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | cdleme35fnpq 40393 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) ≤ (𝑃 ∨ 𝑄)) |
9 | simp2rl 1240 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑅 ∈ 𝐴) | |
10 | simp3 1136 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) | |
11 | cdleme32s.d | . . . . 5 ⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) | |
12 | cdleme32s.n | . . . . 5 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) | |
13 | 11, 12, 7 | cdleme31sn2 40333 | . . . 4 ⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))) |
14 | 9, 10, 13 | syl2anc 583 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))) |
15 | 14 | breq1d 5159 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (⦋𝑅 / 𝑠⦌𝑁 ≤ (𝑃 ∨ 𝑄) ↔ ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) ≤ (𝑃 ∨ 𝑄))) |
16 | 8, 15 | mtbird 325 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ ⦋𝑅 / 𝑠⦌𝑁 ≤ (𝑃 ∨ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1535 ∈ wcel 2104 ≠ wne 2936 ⦋csb 3908 ifcif 4530 class class class wbr 5149 ‘cfv 6558 (class class class)co 7425 Basecbs 17234 lecple 17294 joincjn 18357 meetcmee 18358 Atomscatm 39206 HLchlt 39293 LHypclh 39928 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-1st 8007 df-2nd 8008 df-proset 18341 df-poset 18359 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-oposet 39119 df-ol 39121 df-oml 39122 df-covers 39209 df-ats 39210 df-atl 39241 df-cvlat 39265 df-hlat 39294 df-lines 39445 df-psubsp 39447 df-pmap 39448 df-padd 39740 df-lhyp 39932 |
This theorem is referenced by: cdleme41sn3aw 40418 |
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