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Metamath Proof Explorer

Theorem pjoml2

Description: Variation of orthomodular law. Definition in Kalmbach p. 22. (Contributed by NM, 13-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion pjoml2 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = 𝐵 )

Proof

Step Hyp Ref Expression
1 sseq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( 𝐴 ⊆ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 ) )
2 id ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) )
3 fveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ⊥ ‘ 𝐴 ) = ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) )
4 3 ineq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ 𝐵 ) )
5 2 4 oveq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ 𝐵 ) ) )
6 5 eqeq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ 𝐵 ) ) = 𝐵 ) )
7 1 6 imbi12d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = 𝐵 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ 𝐵 ) ) = 𝐵 ) ) )
8 sseq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) )
9 ineq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ 𝐵 ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) )
10 9 oveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ 𝐵 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) ) )
11 id ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) )
12 10 11 eqeq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ 𝐵 ) ) = 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) )
13 8 12 imbi12d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ 𝐵 → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ 𝐵 ) ) = 𝐵 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) ) )
14 h0elch ⊢ 0ℋ ∈ Cℋ
15 14 elimel ⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∈ Cℋ
16 14 elimel ⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ∈ Cℋ
17 15 16 pjoml2i ⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∨ℋ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , 0ℋ ) )
18 7 13 17 dedth2h ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = 𝐵 ) )
19 18 3impia ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = 𝐵 )