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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atexlempsb | Structured version Visualization version GIF version |
Description: Lemma for 4atexlem7 37852. (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)))) |
4thatlempqb.j | ⊢ ∨ = (join‘𝐾) |
4thatlempqb.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
4atexlempsb | ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
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 37824 | . 2 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 1 | 4atexlemp 37827 | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
4 | 1 | 4atexlems 37829 | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
5 | eqid 2738 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 4thatlempqb.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
7 | 4thatlempqb.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 5, 6, 7 | hlatjcl 37144 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
9 | 2, 3, 4, 8 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ≠ wne 2941 class class class wbr 5067 ‘cfv 6397 (class class class)co 7231 Basecbs 16784 joincjn 17842 Atomscatm 37040 HLchlt 37127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-lub 17876 df-glb 17877 df-join 17878 df-meet 17879 df-lat 17962 df-ats 37044 df-atl 37075 df-cvlat 37099 df-hlat 37128 |
This theorem is referenced by: 4atexlemunv 37843 4atexlemtlw 37844 4atexlemc 37846 4atexlemnclw 37847 4atexlemex2 37848 4atexlemcnd 37849 |
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