Theorem meetcomALT 18366

Description: The meet of a poset is commutative. (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	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))

Proof of Theorem meetcomALT

Step	Hyp	Ref	Expression
1		prcom 4736	. . . 4 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌}
2	1	fveq2i 6894	. . 3 ⊢ ((glb‘𝐾)‘{𝑌, 𝑋}) = ((glb‘𝐾)‘{𝑋, 𝑌})
3	2	a1i 11	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((glb‘𝐾)‘{𝑌, 𝑋}) = ((glb‘𝐾)‘{𝑋, 𝑌}))
4		eqid 2731	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
5		meetcom.m	. . 3 ⊢ ∧ = (meet‘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	meetval 18354	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∧ 𝑋) = ((glb‘𝐾)‘{𝑌, 𝑋}))
10	4, 5, 6, 8, 7	meetval 18354	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
11	3, 9, 10	3eqtr4rd 2782	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {cpr 4630  ‘cfv 6543  (class class class)co 7412  Basecbs 17151  glbcglb 18273  meetcmee 18275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7368  df-ov 7415  df-oprab 7416  df-glb 18310  df-meet 18312

This theorem is referenced by:  meetcom  18367