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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemefs29pre00N | Structured version Visualization version GIF version |
Description: FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 37199. (Contributed by NM, 27-Mar-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdlemefs29.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemefs29.l | ⊢ ≤ = (le‘𝐾) |
cdlemefs29.j | ⊢ ∨ = (join‘𝐾) |
cdlemefs29.m | ⊢ ∧ = (meet‘𝐾) |
cdlemefs29.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemefs29.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
cdlemefs29pre00N | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑠 ∈ 𝐴) → (((¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemefs29.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemefs29.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemefs29.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemefs29.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
5 | cdlemefs29.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemefs29.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | breq1 5062 | . 2 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑃 ∨ 𝑄))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | cdlemefrs29pre00 37564 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑠 ∈ 𝐴) → (((¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 lecple 16567 joincjn 17549 meetcmee 17550 Atomscatm 36432 HLchlt 36519 LHypclh 37153 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-proset 17533 df-poset 17551 df-plt 17563 df-lub 17579 df-glb 17580 df-join 17581 df-meet 17582 df-p0 17644 df-lat 17651 df-oposet 36345 df-ol 36347 df-oml 36348 df-covers 36435 df-ats 36436 df-atl 36467 df-cvlat 36491 df-hlat 36520 df-lhyp 37157 |
This theorem is referenced by: (None) |
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