Metamath Proof Explorer

Theorem cmtvalN

Description: Equivalence for commutes relation. Definition of commutes in Kalmbach p. 20. ( cmbr analog.) (Contributed by NM, 6-Nov-2011) (New usage is discouraged.)

Ref Expression
Hypotheses cmtfval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cmtfval.j ⊢ ∨ = ( join ‘ 𝐾 )
cmtfval.m ⊢ ∧ = ( meet ‘ 𝐾 )
cmtfval.o ⊢ ⊥ = ( oc ‘ 𝐾 )
cmtfval.c ⊢ 𝐶 = ( cm ‘ 𝐾 )
Assertion cmtvalN ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ 𝑋 = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑌 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 cmtfval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cmtfval.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cmtfval.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cmtfval.o ⊢ ⊥ = ( oc ‘ 𝐾 )
5 cmtfval.c ⊢ 𝐶 = ( cm ‘ 𝐾 )
6 1 2 3 4 5 cmtfvalN ⊢ ( 𝐾 ∈ 𝐴 → 𝐶 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) } )
7 df-3an ⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) )
8 7 opabbii ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) }
9 6 8 eqtrdi ⊢ ( 𝐾 ∈ 𝐴 → 𝐶 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) } )
10 9 breqd ⊢ ( 𝐾 ∈ 𝐴 → ( 𝑋 𝐶 𝑌 ↔ 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) } 𝑌 ) )
11 10 3ad2ant1 ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) } 𝑌 ) )
12 df-br ⊢ ( 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) } 𝑌 ↔ ⟨ 𝑋 , 𝑌 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) } )
13 id ⊢ ( 𝑥 = 𝑋 → 𝑥 = 𝑋 )
14 oveq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 ∧ 𝑦 ) = ( 𝑋 ∧ 𝑦 ) )
15 oveq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) = ( 𝑋 ∧ ( ⊥ ‘ 𝑦 ) ) )
16 14 15 oveq12d ⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) = ( ( 𝑋 ∧ 𝑦 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑦 ) ) ) )
17 13 16 eqeq12d ⊢ ( 𝑥 = 𝑋 → ( 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ↔ 𝑋 = ( ( 𝑋 ∧ 𝑦 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) )
18 oveq2 ⊢ ( 𝑦 = 𝑌 → ( 𝑋 ∧ 𝑦 ) = ( 𝑋 ∧ 𝑌 ) )
19 fveq2 ⊢ ( 𝑦 = 𝑌 → ( ⊥ ‘ 𝑦 ) = ( ⊥ ‘ 𝑌 ) )
20 19 oveq2d ⊢ ( 𝑦 = 𝑌 → ( 𝑋 ∧ ( ⊥ ‘ 𝑦 ) ) = ( 𝑋 ∧ ( ⊥ ‘ 𝑌 ) ) )
21 18 20 oveq12d ⊢ ( 𝑦 = 𝑌 → ( ( 𝑋 ∧ 𝑦 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑦 ) ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑌 ) ) ) )
22 21 eqeq2d ⊢ ( 𝑦 = 𝑌 → ( 𝑋 = ( ( 𝑋 ∧ 𝑦 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑦 ) ) ) ↔ 𝑋 = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑌 ) ) ) ) )
23 17 22 opelopab2 ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) } ↔ 𝑋 = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑌 ) ) ) ) )
24 12 23 syl5bb ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) } 𝑌 ↔ 𝑋 = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑌 ) ) ) ) )
25 24 3adant1 ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ ( ⊥ ‘ 𝑦 ) ) ) ) } 𝑌 ↔ 𝑋 = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑌 ) ) ) ) )
26 11 25 bitrd ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ 𝑋 = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑋 ∧ ( ⊥ ‘ 𝑌 ) ) ) ) )