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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme6 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. This expresses (r ∨ fs(r)) ∧ w = u at the top of p. 114. (Contributed by NM, 7-Jun-2012.) |
Ref | Expression |
---|---|
cdleme4.l | ⊢ ≤ = (le‘𝐾) |
cdleme4.j | ⊢ ∨ = (join‘𝐾) |
cdleme4.m | ⊢ ∧ = (meet‘𝐾) |
cdleme4.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme4.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme4.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdleme4.f | ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) |
cdleme4.g | ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) |
Ref | Expression |
---|---|
cdleme6 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝐺) ∧ 𝑊) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme4.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | cdleme4.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | cdleme4.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
4 | cdleme4.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdleme4.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdleme4.u | . . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
7 | cdleme4.f | . . . 4 ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) | |
8 | cdleme4.g | . . . 4 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | cdleme5 40222 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝐺) = (𝑃 ∨ 𝑄)) |
10 | 9 | oveq1d 7445 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝐺) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊)) |
11 | 10, 6 | eqtr4di 2792 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝐺) ∧ 𝑊) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 lecple 17304 joincjn 18368 meetcmee 18369 Atomscatm 39244 HLchlt 39331 LHypclh 39966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 |
This theorem is referenced by: cdleme7 40231 |
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