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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atexlemv | Structured version Visualization version GIF version |
Description: Lemma for 4atexlem7 36742. (Contributed by NM, 23-Nov-2012.) |
Ref | Expression |
---|---|
4thatlem.ph | ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) |
4thatlem0.l | ⊢ ≤ = (le‘𝐾) |
4thatlem0.j | ⊢ ∨ = (join‘𝐾) |
4thatlem0.m | ⊢ ∧ = (meet‘𝐾) |
4thatlem0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
4thatlem0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
4thatlem0.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
4thatlem0.v | ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊) |
Ref | Expression |
---|---|
4atexlemv | ⊢ (𝜑 → 𝑉 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4thatlem.ph | . . 3 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) | |
2 | 1 | 4atexlemk 36714 | . 2 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 1 | 4atexlemw 36715 | . 2 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
4 | 1 | 4atexlempw 36716 | . 2 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
5 | 1 | 4atexlems 36719 | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
6 | 4thatlem0.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
7 | 4thatlem0.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
8 | 4thatlem0.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 1, 6, 7, 8 | 4atexlempns 36729 | . 2 ⊢ (𝜑 → 𝑃 ≠ 𝑆) |
10 | 4thatlem0.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
11 | 4thatlem0.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
12 | 4thatlem0.v | . . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊) | |
13 | 6, 7, 10, 8, 11, 12 | lhpat2 36712 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆)) → 𝑉 ∈ 𝐴) |
14 | 2, 3, 4, 5, 9, 13 | syl212anc 1373 | 1 ⊢ (𝜑 → 𝑉 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 class class class wbr 4962 ‘cfv 6225 (class class class)co 7016 lecple 16401 joincjn 17383 meetcmee 17384 Atomscatm 35930 HLchlt 36017 LHypclh 36651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-proset 17367 df-poset 17385 df-plt 17397 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-p0 17478 df-p1 17479 df-lat 17485 df-clat 17547 df-oposet 35843 df-ol 35845 df-oml 35846 df-covers 35933 df-ats 35934 df-atl 35965 df-cvlat 35989 df-hlat 36018 df-lhyp 36655 |
This theorem is referenced by: 4atexlemunv 36733 4atexlemtlw 36734 4atexlemntlpq 36735 4atexlemc 36736 4atexlemnclw 36737 |
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