Theorem joincomALT 18459

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	⊢ 𝐵 = (Base‘𝐾)
joincom.j	⊢ ∨ = (join‘𝐾)

Assertion

Ref	Expression
joincomALT	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋))

Proof of Theorem joincomALT

Step	Hyp	Ref	Expression
1		prcom 4737	. . . 4 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌}
2	1	fveq2i 6910	. . 3 ⊢ ((lub‘𝐾)‘{𝑌, 𝑋}) = ((lub‘𝐾)‘{𝑋, 𝑌})
3	2	a1i 11	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((lub‘𝐾)‘{𝑌, 𝑋}) = ((lub‘𝐾)‘{𝑋, 𝑌}))
4		eqid 2735	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
5		joincom.j	. . 3 ⊢ ∨ = (join‘𝐾)
6		simp1 1135	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ 𝑉)
7		simp3 1137	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
8		simp2 1136	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
9	4, 5, 6, 7, 8	joinval 18435	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ 𝑋) = ((lub‘𝐾)‘{𝑌, 𝑋}))
10	4, 5, 6, 8, 7	joinval 18435	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) = ((lub‘𝐾)‘{𝑋, 𝑌}))
11	3, 9, 10	3eqtr4rd 2786	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {cpr 4633  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  lubclub 18367  joincjn 18369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-lub 18404  df-join 18406

This theorem is referenced by:  joincom  18460