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Metamath Proof Explorer

Theorem pjoml3

Description: Variation of orthomodular law. (Contributed by NM, 24-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion pjoml3 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 sseq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
2 id ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) )
3 fveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ⊥ ‘ 𝐴 ) = ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
4 3 oveq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) )
5 2 4 ineq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) )
6 5 eqeq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) = 𝐵 ) )
7 1 6 imbi12d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ) ↔ ( 𝐵 ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) = 𝐵 ) ) )
8 sseq1 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( 𝐵 ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ↔ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
9 oveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
10 9 ineq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) )
11 id ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) )
12 10 11 eqeq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) = 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
13 8 12 imbi12d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( 𝐵 ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) = 𝐵 ) ↔ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) )
14 ifchhv ⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
15 ifchhv ⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
16 14 15 pjoml3i ⊢ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) )
17 7 13 16 dedth2h ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ) )