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Theorem latmidm 18107

Description: Lattice meet is idempotent. Analogue of inidm 4149. (Contributed by NM, 8-Nov-2011.)

**Hypotheses**
Ref	Expression
latmidm.b	⊢ 𝐵 = (Base‘𝐾)
latmidm.m	⊢ ∧ = (meet‘𝐾)

**Assertion**
Ref	Expression
latmidm	⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋)

**Proof of Theorem latmidm**
Step	Hyp	Ref	Expression
1		latmidm.b	. 2 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2738	. 2 ⊢ (le‘𝐾) = (le‘𝐾)
3		simpl 482	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat)
4		latmidm.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
5	1, 4	latmcl 18073	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) ∈ 𝐵)
6	5	3anidm23 1419	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) ∈ 𝐵)
7		simpr 484	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
8	1, 2, 4	latmle1 18097	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋)(le‘𝐾)𝑋)
9	8	3anidm23 1419	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋)(le‘𝐾)𝑋)
10	1, 2	latref 18074	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋)
11	1, 2, 4	latlem12 18099	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋(le‘𝐾)𝑋 ∧ 𝑋(le‘𝐾)𝑋) ↔ 𝑋(le‘𝐾)(𝑋 ∧ 𝑋)))
12	3, 7, 7, 7, 11	syl13anc 1370	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → ((𝑋(le‘𝐾)𝑋 ∧ 𝑋(le‘𝐾)𝑋) ↔ 𝑋(le‘𝐾)(𝑋 ∧ 𝑋)))
13	10, 10, 12	mpbi2and 708	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∧ 𝑋))
14	1, 2, 3, 6, 7, 9, 13	latasymd 18078	1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 lecple 16895 meetcmee 17945 Latclat 18064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-proset 17928 df-poset 17946 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-lat 18065

This theorem is referenced by: latmmdiN 37175 latmmdir 37176 2llnm3N 37510

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