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Metamath Proof Explorer

Theorem dih11

Description: The isomorphism H is one-to-one. Part of proof after Lemma N of Crawley p. 122 line 6. (Contributed by NM, 7-Mar-2014)

Ref Expression
Hypotheses dih11.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dih11.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dih11.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dih11 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) = ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )

Proof

Step Hyp Ref Expression
1 dih11.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dih11.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 dih11.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4 eqss ⊢ ( ( 𝐼 ‘ 𝑋 ) = ( 𝐼 ‘ 𝑌 ) ↔ ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ 𝑋 ) ) )
5 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6 1 5 2 3 dihord ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ( le ‘ 𝐾 ) 𝑌 ) )
7 1 5 2 3 dihord ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ 𝑋 ) ↔ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) )
8 7 3com23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ 𝑋 ) ↔ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) )
9 6 8 anbi12d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ 𝑋 ) ) ↔ ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ) )
10 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ HL )
11 10 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Lat )
12 1 5 latasymb ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ↔ 𝑋 = 𝑌 ) )
13 11 12 syld3an1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ↔ 𝑋 = 𝑌 ) )
14 9 13 bitrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ 𝑋 ) ) ↔ 𝑋 = 𝑌 ) )
15 4 14 bitrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) = ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )