Metamath Proof Explorer

Theorem dihord

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

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

Proof

Step Hyp Ref Expression
1 dihord.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihord.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihord.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihord.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
7 simprl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → 𝑋 ≤ 𝑊 )
8 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → 𝑌 ∈ 𝐵 )
9 simprr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → 𝑌 ≤ 𝑊 )
10 1 2 3 4 dihord3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )
11 5 6 7 8 9 10 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )
12 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
13 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
14 simprl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → 𝑋 ≤ 𝑊 )
15 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → 𝑌 ∈ 𝐵 )
16 simprr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ¬ 𝑌 ≤ 𝑊 )
17 1 2 3 4 dihord5a ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → 𝑋 ≤ 𝑌 )
18 1 2 3 4 dihord5b ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) )
19 17 18 impbida ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )
20 12 13 14 15 16 19 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )
21 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
22 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
23 simprl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → ¬ 𝑋 ≤ 𝑊 )
24 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → 𝑌 ∈ 𝐵 )
25 simprr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → 𝑌 ≤ 𝑊 )
26 1 2 3 4 dihord6a ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → 𝑋 ≤ 𝑌 )
27 1 2 3 4 dihord6b ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) )
28 26 27 impbida ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )
29 21 22 23 24 25 28 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )
30 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
31 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
32 simprl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ¬ 𝑋 ≤ 𝑊 )
33 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → 𝑌 ∈ 𝐵 )
34 simprr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ¬ 𝑌 ≤ 𝑊 )
35 1 2 3 4 dihord4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )
36 30 31 32 33 34 35 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )
37 11 20 29 36 4casesdan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )