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Theorem meetcomALT 17252
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 4411 . . . 4 {𝑌, 𝑋} = {𝑋, 𝑌}
21fveq2i 6356 . . 3 ((glb‘𝐾)‘{𝑌, 𝑋}) = ((glb‘𝐾)‘{𝑋, 𝑌})
32a1i 11 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → ((glb‘𝐾)‘{𝑌, 𝑋}) = ((glb‘𝐾)‘{𝑋, 𝑌}))
4 eqid 2760 . . 3 (glb‘𝐾) = (glb‘𝐾)
5 meetcom.m . . 3 = (meet‘𝐾)
6 simp1 1131 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝐾𝑉)
7 simp3 1133 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝑌𝐵)
8 simp2 1132 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝑋𝐵)
94, 5, 6, 7, 8meetval 17240 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = ((glb‘𝐾)‘{𝑌, 𝑋}))
104, 5, 6, 8, 7meetval 17240 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
113, 9, 103eqtr4rd 2805 1 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139  {cpr 4323  cfv 6049  (class class class)co 6814  Basecbs 16079  glbcglb 17164  meetcmee 17166
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 7115
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 6775  df-ov 6817  df-oprab 6818  df-glb 17196  df-meet 17198
This theorem is referenced by:  meetcom  17253
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