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Metamath Proof Explorer

Theorem chsscon3

Description: Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chsscon3 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 sseq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ⊆ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ 𝐵 ) )
2 fveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ⊥ ‘ 𝐴 ) = ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
3 2 sseq2d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) )
4 1 3 bibi12d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) ) )
5 sseq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
6 fveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ⊥ ‘ 𝐵 ) = ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
7 6 sseq1d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ↔ ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ⊆ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) )
8 5 7 bibi12d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ↔ ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ⊆ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) ) )
9 ifchhv ⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
10 ifchhv ⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
11 9 10 chsscon3i ⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ↔ ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ⊆ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
12 4 8 11 dedth2h ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ) )