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Theorem joincomALT 18392
Description: The join of a poset is commutative. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 16-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
joincom.b 𝐡 = (Baseβ€˜πΎ)
joincom.j ∨ = (joinβ€˜πΎ)
Assertion
Ref Expression
joincomALT ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) = (π‘Œ ∨ 𝑋))

Proof of Theorem joincomALT
StepHypRef Expression
1 prcom 4732 . . . 4 {π‘Œ, 𝑋} = {𝑋, π‘Œ}
21fveq2i 6895 . . 3 ((lubβ€˜πΎ)β€˜{π‘Œ, 𝑋}) = ((lubβ€˜πΎ)β€˜{𝑋, π‘Œ})
32a1i 11 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((lubβ€˜πΎ)β€˜{π‘Œ, 𝑋}) = ((lubβ€˜πΎ)β€˜{𝑋, π‘Œ}))
4 eqid 2725 . . 3 (lubβ€˜πΎ) = (lubβ€˜πΎ)
5 joincom.j . . 3 ∨ = (joinβ€˜πΎ)
6 simp1 1133 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ 𝑉)
7 simp3 1135 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
8 simp2 1134 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
94, 5, 6, 7, 8joinval 18368 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ ∨ 𝑋) = ((lubβ€˜πΎ)β€˜{π‘Œ, 𝑋}))
104, 5, 6, 8, 7joinval 18368 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) = ((lubβ€˜πΎ)β€˜{𝑋, π‘Œ}))
113, 9, 103eqtr4rd 2776 1 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) = (π‘Œ ∨ 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cpr 4626  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179  lubclub 18300  joincjn 18302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-lub 18337  df-join 18339
This theorem is referenced by:  joincom  18393
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