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Theorem latmidm 17080
Description: Lattice join is idempotent. (inidm 3820 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
latmidm.b 𝐵 = (Base‘𝐾)
latmidm.m = (meet‘𝐾)
Assertion
Ref Expression
latmidm ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)

Proof of Theorem latmidm
StepHypRef Expression
1 latmidm.b . 2 𝐵 = (Base‘𝐾)
2 eqid 2621 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 473 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → 𝐾 ∈ Lat)
4 latmidm.m . . . 4 = (meet‘𝐾)
51, 4latmcl 17046 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑋𝐵) → (𝑋 𝑋) ∈ 𝐵)
653anidm23 1384 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) ∈ 𝐵)
7 simpr 477 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → 𝑋𝐵)
81, 2, 4latmle1 17070 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑋𝐵) → (𝑋 𝑋)(le‘𝐾)𝑋)
983anidm23 1384 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋)(le‘𝐾)𝑋)
101, 2latref 17047 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → 𝑋(le‘𝐾)𝑋)
111, 2, 4latlem12 17072 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑋𝐵𝑋𝐵)) → ((𝑋(le‘𝐾)𝑋𝑋(le‘𝐾)𝑋) ↔ 𝑋(le‘𝐾)(𝑋 𝑋)))
123, 7, 7, 7, 11syl13anc 1327 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → ((𝑋(le‘𝐾)𝑋𝑋(le‘𝐾)𝑋) ↔ 𝑋(le‘𝐾)(𝑋 𝑋)))
1310, 10, 12mpbi2and 956 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → 𝑋(le‘𝐾)(𝑋 𝑋))
141, 2, 3, 6, 7, 9, 13latasymd 17051 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989   class class class wbr 4651  cfv 5886  (class class class)co 6647  Basecbs 15851  lecple 15942  meetcmee 16939  Latclat 17039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-preset 16922  df-poset 16940  df-lub 16968  df-glb 16969  df-join 16970  df-meet 16971  df-lat 17040
This theorem is referenced by:  latmmdiN  34347  latmmdir  34348  2llnm3N  34681
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