Theorem meetcomALT 18416

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 4690	. . . 4 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌}
2	1	fveq2i 6866	. . 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 2761	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
5		meetcom.m	. . 3 ⊢ ∧ = (meet‘𝐾)
6		simp1 1148	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ 𝑉)
7		simp3 1150	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
8		simp2 1149	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
9	4, 5, 6, 7, 8	meetval 18404	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∧ 𝑋) = ((glb‘𝐾)‘{𝑌, 𝑋}))
10	4, 5, 6, 8, 7	meetval 18404	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
11	3, 9, 10	3eqtr4rd 2807	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))

Syntax hints:   → wi 4   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {cpr 4583  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  glbcglb 18325  meetcmee 18327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-glb 18360  df-meet 18362

This theorem is referenced by:  meetcom  18417