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Theorem meetcomALT 17707
 Description: The meet of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 17-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
meetcom.b 𝐵 = (Base‘𝐾)
meetcom.m = (meet‘𝐾)
Assertion
Ref Expression
meetcomALT ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))

Proof of Theorem meetcomALT
StepHypRef Expression
1 prcom 4625 . . . 4 {𝑌, 𝑋} = {𝑋, 𝑌}
21fveq2i 6661 . . 3 ((glb‘𝐾)‘{𝑌, 𝑋}) = ((glb‘𝐾)‘{𝑋, 𝑌})
32a1i 11 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → ((glb‘𝐾)‘{𝑌, 𝑋}) = ((glb‘𝐾)‘{𝑋, 𝑌}))
4 eqid 2758 . . 3 (glb‘𝐾) = (glb‘𝐾)
5 meetcom.m . . 3 = (meet‘𝐾)
6 simp1 1133 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝐾𝑉)
7 simp3 1135 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝑌𝐵)
8 simp2 1134 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝑋𝐵)
94, 5, 6, 7, 8meetval 17695 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = ((glb‘𝐾)‘{𝑌, 𝑋}))
104, 5, 6, 8, 7meetval 17695 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
113, 9, 103eqtr4rd 2804 1 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cpr 4524  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  glbcglb 17619  meetcmee 17621 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-glb 17651  df-meet 17653 This theorem is referenced by:  meetcom  17708
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