Theorem joincomALT 18443

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 4694	. . . 4 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌}
2	1	fveq2i 6874	. . 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 2765	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
5		joincom.j	. . 3 ⊢ ∨ = (join‘𝐾)
6		simp1 1152	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ 𝑉)
7		simp3 1154	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
8		simp2 1153	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
9	4, 5, 6, 7, 8	joinval 18419	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ 𝑋) = ((lub‘𝐾)‘{𝑌, 𝑋}))
10	4, 5, 6, 8, 7	joinval 18419	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) = ((lub‘𝐾)‘{𝑋, 𝑌}))
11	3, 9, 10	3eqtr4rd 2811	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  {cpr 4587  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  lubclub 18353  joincjn 18355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-lub 18388  df-join 18390

This theorem is referenced by:  joincom  18444