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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 39369 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
| 5 | cdleme43.v | . . . . 5 ⊢ 𝑉 = ((𝑍 ∨ 𝑆) ∧ 𝑊) | |
| 6 | 5 | oveq2i 7442 | . . . 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 7449 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ 𝑉)) = ((𝑄 ∨ 𝑃) ∧ (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊)))) |
| 9 | 1, 8 | eqtr4id 2796 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ 𝑉))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 joincjn 18357 meetcmee 18358 Atomscatm 39264 HLchlt 39351 LHypclh 39986 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-lub 18391 df-join 18393 df-lat 18477 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 |
| This theorem is referenced by: (None) |
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