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Metamath Proof Explorer

Theorem dihcnvord

Description: Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014)

Ref Expression
Hypotheses dihcnvord.l ⊢ ≤ = ( le ‘ 𝐾 )
dihcnvord.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihcnvord.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihcnvord.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dihcnvord.x ⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
dihcnvord.y ⊢ ( 𝜑 → 𝑌 ∈ ran 𝐼 )
Assertion dihcnvord ( 𝜑 → ( ( ◡ 𝐼 ‘ 𝑋 ) ≤ ( ◡ 𝐼 ‘ 𝑌 ) ↔ 𝑋 ⊆ 𝑌 ) )

Proof

Step Hyp Ref Expression
1 dihcnvord.l ⊢ ≤ = ( le ‘ 𝐾 )
2 dihcnvord.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 dihcnvord.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4 dihcnvord.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
5 dihcnvord.x ⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
6 dihcnvord.y ⊢ ( 𝜑 → 𝑌 ∈ ran 𝐼 )
7 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8 7 2 3 dihcnvcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) )
9 4 5 8 syl2anc ⊢ ( 𝜑 → ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) )
10 7 2 3 dihcnvcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) )
11 4 6 10 syl2anc ⊢ ( 𝜑 → ( ◡ 𝐼 ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) )
12 7 1 2 3 dihord ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ◡ 𝐼 ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) ⊆ ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) ↔ ( ◡ 𝐼 ‘ 𝑋 ) ≤ ( ◡ 𝐼 ‘ 𝑌 ) ) )
13 4 9 11 12 syl3anc ⊢ ( 𝜑 → ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) ⊆ ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) ↔ ( ◡ 𝐼 ‘ 𝑋 ) ≤ ( ◡ 𝐼 ‘ 𝑌 ) ) )
14 2 3 dihcnvid2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 )
15 4 5 14 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 )
16 2 3 dihcnvid2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) = 𝑌 )
17 4 6 16 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) = 𝑌 )
18 15 17 sseq12d ⊢ ( 𝜑 → ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) ⊆ ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) ↔ 𝑋 ⊆ 𝑌 ) )
19 13 18 bitr3d ⊢ ( 𝜑 → ( ( ◡ 𝐼 ‘ 𝑋 ) ≤ ( ◡ 𝐼 ‘ 𝑌 ) ↔ 𝑋 ⊆ 𝑌 ) )