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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atlem4c | Structured version Visualization version GIF version |
Description: Lemma for 4at 36280. Frequently used associative law. (Contributed by NM, 9-Jul-2012.) |
Ref | Expression |
---|---|
4at.l | ⊢ ≤ = (le‘𝐾) |
4at.j | ⊢ ∨ = (join‘𝐾) |
4at.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
4atlem4c | ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = (𝑅 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1184 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ HL) | |
2 | 1 | hllatd 36031 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ Lat) |
3 | eqid 2795 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | 4at.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
5 | 4at.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 3, 4, 5 | hlatjcl 36034 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
7 | 6 | adantr 481 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
8 | 3, 5 | atbase 35956 | . . 3 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
9 | 8 | ad2antrl 724 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑅 ∈ (Base‘𝐾)) |
10 | 3, 5 | atbase 35956 | . . 3 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾)) |
11 | 10 | ad2antll 725 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑆 ∈ (Base‘𝐾)) |
12 | 3, 4 | latj12 17535 | . 2 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = (𝑅 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑆))) |
13 | 2, 7, 9, 11, 12 | syl13anc 1365 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = (𝑅 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 lecple 16401 joincjn 17383 Latclat 17484 Atomscatm 35930 HLchlt 36017 |
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-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-lat 17485 df-ats 35934 df-atl 35965 df-cvlat 35989 df-hlat 36018 |
This theorem is referenced by: 4atlem10a 36271 |
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