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Theorem latj31 17300

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 474	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat)
2		simpr3 1238	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
3		simpr1 1234	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
4		simpr2 1236	. . 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 17297	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑍 ∨ (𝑋 ∨ 𝑌)) = (𝑋 ∨ (𝑍 ∨ 𝑌)))
8	1, 2, 3, 4, 7	syl13anc 1479	. 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 ∨ (𝑋 ∨ 𝑌)) = (𝑋 ∨ (𝑍 ∨ 𝑌)))
9	5, 6	latjcl 17252	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵)
10	9	3adant3r3 1200	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ 𝑌) ∈ 𝐵)
11	5, 6	latjcom 17260	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = (𝑍 ∨ (𝑋 ∨ 𝑌)))
12	1, 10, 2, 11	syl3anc 1477	. 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = (𝑍 ∨ (𝑋 ∨ 𝑌)))
13	5, 6	latjcl 17252	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍 ∨ 𝑌) ∈ 𝐵)
14	1, 2, 4, 13	syl3anc 1477	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 ∨ 𝑌) ∈ 𝐵)
15	5, 6	latjcom 17260	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑍 ∨ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑍 ∨ 𝑌) ∨ 𝑋) = (𝑋 ∨ (𝑍 ∨ 𝑌)))
16	1, 14, 3, 15	syl3anc 1477	. 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 ∨ 𝑌) ∨ 𝑋) = (𝑋 ∨ (𝑍 ∨ 𝑌)))
17	8, 12, 16	3eqtr4d 2804	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 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 joincjn 17145 Latclat 17246

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-preset 17129 df-poset 17147 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-lat 17247

This theorem is referenced by: latjrot 17301 4noncolr3 35242 3atlem5 35276 lplnexllnN 35353 dalawlem11 35670 cdleme20bN 36100

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