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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.g | . 2 ⊢ 𝐺 = ((𝑄 ∨ 𝑃) ∧ (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊))) | |
2 | cdleme43.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | cdleme43.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | hlatjcom 37309 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
5 | cdleme43.v | . . . . 5 ⊢ 𝑉 = ((𝑍 ∨ 𝑆) ∧ 𝑊) | |
6 | 5 | oveq2i 7266 | . . . 4 ⊢ (𝐷 ∨ 𝑉) = (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊)) |
7 | 6 | a1i 11 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝐷 ∨ 𝑉) = (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊))) |
8 | 4, 7 | oveq12d 7273 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ 𝑉)) = ((𝑄 ∨ 𝑃) ∧ (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊)))) |
9 | 1, 8 | eqtr4id 2798 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 lecple 16895 joincjn 17944 meetcmee 17945 Atomscatm 37204 HLchlt 37291 LHypclh 37925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-lub 17979 df-join 17981 df-lat 18065 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 |
This theorem is referenced by: (None) |
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