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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme43aN | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 115 penultimate line: g(f(r)) = (p v q) ^ (g(s) v v1). (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdleme43.b | ⊢ 𝐵 = (Base‘𝐾) |
cdleme43.l | ⊢ ≤ = (le‘𝐾) |
cdleme43.j | ⊢ ∨ = (join‘𝐾) |
cdleme43.m | ⊢ ∧ = (meet‘𝐾) |
cdleme43.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme43.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme43.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdleme43.x | ⊢ 𝑋 = ((𝑄 ∨ 𝑃) ∧ 𝑊) |
cdleme43.c | ⊢ 𝐶 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) |
cdleme43.f | ⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝐶 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) |
cdleme43.d | ⊢ 𝐷 = ((𝑆 ∨ 𝑋) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ 𝑊))) |
cdleme43.g | ⊢ 𝐺 = ((𝑄 ∨ 𝑃) ∧ (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊))) |
cdleme43.e | ⊢ 𝐸 = ((𝐷 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝐷) ∧ 𝑊))) |
cdleme43.v | ⊢ 𝑉 = ((𝑍 ∨ 𝑆) ∧ 𝑊) |
cdleme43.y | ⊢ 𝑌 = ((𝑅 ∨ 𝐷) ∧ 𝑊) |
Ref | Expression |
---|---|
cdleme43aN | ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ 𝑉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme43.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
2 | cdleme43.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | hlatjcom 35381 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
4 | cdleme43.v | . . . . 5 ⊢ 𝑉 = ((𝑍 ∨ 𝑆) ∧ 𝑊) | |
5 | 4 | oveq2i 6887 | . . . 4 ⊢ (𝐷 ∨ 𝑉) = (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊)) |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝐷 ∨ 𝑉) = (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊))) |
7 | 3, 6 | oveq12d 6894 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ 𝑉)) = ((𝑄 ∨ 𝑃) ∧ (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊)))) |
8 | cdleme43.g | . 2 ⊢ 𝐺 = ((𝑄 ∨ 𝑃) ∧ (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊))) | |
9 | 7, 8 | syl6reqr 2850 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ‘cfv 6099 (class class class)co 6876 Basecbs 16181 lecple 16271 joincjn 17256 meetcmee 17257 Atomscatm 35276 HLchlt 35363 LHypclh 35997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-lub 17286 df-join 17288 df-lat 17358 df-ats 35280 df-atl 35311 df-cvlat 35335 df-hlat 35364 |
This theorem is referenced by: (None) |
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