latj31 - Metamath Proof Explorer

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Theorem latj31 18545

Description: Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 23-Jun-2012.)

**Hypotheses**
Ref	Expression
latjass.b	⊢ 𝐵 = (Base‘𝐾)
latjass.j	⊢ ∨ = (join‘𝐾)

**Assertion**
Ref	Expression
latj31	⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑍 ∨ 𝑌) ∨ 𝑋))

**Proof of Theorem latj31**
Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat)
2		simpr3 1195	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
3		simpr1 1193	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
4		simpr2 1194	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
5		latjass.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6		latjass.j	. . . 4 ⊢ ∨ = (join‘𝐾)
7	5, 6	latj12 18542	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑍 ∨ (𝑋 ∨ 𝑌)) = (𝑋 ∨ (𝑍 ∨ 𝑌)))
8	1, 2, 3, 4, 7	syl13anc 1371	. 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 ∨ (𝑋 ∨ 𝑌)) = (𝑋 ∨ (𝑍 ∨ 𝑌)))
9	5, 6	latjcl 18497	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵)
10	9	3adant3r3 1183	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ 𝑌) ∈ 𝐵)
11	5, 6	latjcom 18505	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = (𝑍 ∨ (𝑋 ∨ 𝑌)))
12	1, 10, 2, 11	syl3anc 1370	. 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = (𝑍 ∨ (𝑋 ∨ 𝑌)))
13	5, 6	latjcl 18497	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍 ∨ 𝑌) ∈ 𝐵)
14	1, 2, 4, 13	syl3anc 1370	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 ∨ 𝑌) ∈ 𝐵)
15	5, 6	latjcom 18505	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑍 ∨ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑍 ∨ 𝑌) ∨ 𝑋) = (𝑋 ∨ (𝑍 ∨ 𝑌)))
16	1, 14, 3, 15	syl3anc 1370	. 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 ∨ 𝑌) ∨ 𝑋) = (𝑋 ∨ (𝑍 ∨ 𝑌)))
17	8, 12, 16	3eqtr4d 2785	1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑍 ∨ 𝑌) ∨ 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 joincjn 18369 Latclat 18489

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18352 df-poset 18371 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-lat 18490

This theorem is referenced by: latjrot 18546 4noncolr3 39436 3atlem5 39470 lplnexllnN 39547 dalawlem11 39864 cdleme20bN 40293

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