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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme8tN | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. 𝑋 represents t1. In their notation, we prove p ∨ t1 = p ∨ t. (Contributed by NM, 8-Oct-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdleme8t.l | ⊢ ≤ = (le‘𝐾) |
cdleme8t.j | ⊢ ∨ = (join‘𝐾) |
cdleme8t.m | ⊢ ∧ = (meet‘𝐾) |
cdleme8t.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme8t.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme8t.x | ⊢ 𝑋 = ((𝑃 ∨ 𝑇) ∧ 𝑊) |
Ref | Expression |
---|---|
cdleme8tN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) → (𝑃 ∨ 𝑋) = (𝑃 ∨ 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme8t.l | . 2 ⊢ ≤ = (le‘𝐾) | |
2 | cdleme8t.j | . 2 ⊢ ∨ = (join‘𝐾) | |
3 | cdleme8t.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
4 | cdleme8t.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdleme8t.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdleme8t.x | . 2 ⊢ 𝑋 = ((𝑃 ∨ 𝑇) ∧ 𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | cdleme8 36324 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) → (𝑃 ∨ 𝑋) = (𝑃 ∨ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 lecple 16319 joincjn 17304 meetcmee 17305 Atomscatm 35337 HLchlt 35424 LHypclh 36058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-oposet 35250 df-ol 35252 df-oml 35253 df-covers 35340 df-ats 35341 df-atl 35372 df-cvlat 35396 df-hlat 35425 df-psubsp 35577 df-pmap 35578 df-padd 35870 df-lhyp 36062 |
This theorem is referenced by: (None) |
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