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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 33672 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
4 | cdleme43.v | . . . . 5 ⊢ 𝑉 = ((𝑍 ∨ 𝑆) ∧ 𝑊) | |
5 | 4 | oveq2i 6560 | . . . 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 6567 | . 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 2663 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 meetcmee 16768 Atomscatm 33568 HLchlt 33655 LHypclh 34288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-lub 16797 df-join 16799 df-lat 16869 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 |
This theorem is referenced by: (None) |
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