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	syl5bb	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) = ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )