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 ( ( 𝐾𝐴𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝐶 𝑌𝑋 = ( ( 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 ( 𝐾𝐴𝐶 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐵𝑦𝐵𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) } )
7 df-3an ( ( 𝑥𝐵𝑦𝐵𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) ↔ ( ( 𝑥𝐵𝑦𝐵 ) ∧ 𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) )
8 7 opabbii { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐵𝑦𝐵𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐵𝑦𝐵 ) ∧ 𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) }
9 6 8 eqtrdi ( 𝐾𝐴𝐶 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐵𝑦𝐵 ) ∧ 𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) } )
10 9 breqd ( 𝐾𝐴 → ( 𝑋 𝐶 𝑌𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐵𝑦𝐵 ) ∧ 𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) } 𝑌 ) )
11 10 3ad2ant1 ( ( 𝐾𝐴𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝐶 𝑌𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐵𝑦𝐵 ) ∧ 𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) } 𝑌 ) )
12 df-br ( 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐵𝑦𝐵 ) ∧ 𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) } 𝑌 ↔ ⟨ 𝑋 , 𝑌 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐵𝑦𝐵 ) ∧ 𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) } )
13 id ( 𝑥 = 𝑋𝑥 = 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 ( ( 𝑋𝐵𝑌𝐵 ) → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐵𝑦𝐵 ) ∧ 𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) } ↔ 𝑋 = ( ( 𝑋 𝑌 ) ( 𝑋 ( 𝑌 ) ) ) ) )
24 12 23 syl5bb ( ( 𝑋𝐵𝑌𝐵 ) → ( 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐵𝑦𝐵 ) ∧ 𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) } 𝑌𝑋 = ( ( 𝑋 𝑌 ) ( 𝑋 ( 𝑌 ) ) ) ) )
25 24 3adant1 ( ( 𝐾𝐴𝑋𝐵𝑌𝐵 ) → ( 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐵𝑦𝐵 ) ∧ 𝑥 = ( ( 𝑥 𝑦 ) ( 𝑥 ( 𝑦 ) ) ) ) } 𝑌𝑋 = ( ( 𝑋 𝑌 ) ( 𝑋 ( 𝑌 ) ) ) ) )
26 11 25 bitrd ( ( 𝐾𝐴𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝐶 𝑌𝑋 = ( ( 𝑋 𝑌 ) ( 𝑋 ( 𝑌 ) ) ) ) )