This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dian0

Description: The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014)

Ref Expression
Hypotheses dian0.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dian0.l ⊢ ≤ = ( le ‘ 𝐾 )
dian0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dian0.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion dian0 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ≠ ∅ )

Proof

Step Hyp Ref Expression
1 dian0.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dian0.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dian0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dian0.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 1 3 5 idltrn ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝐵 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
7 6 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( I ↾ 𝐵 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
8 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
9 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
10 1 8 3 9 trlid0 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( I ↾ 𝐵 ) ) = ( 0. ‘ 𝐾 ) )
11 10 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( I ↾ 𝐵 ) ) = ( 0. ‘ 𝐾 ) )
12 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
13 12 adantr ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ AtLat )
14 simpl ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) → 𝑋 ∈ 𝐵 )
15 1 2 8 atl0le ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ) → ( 0. ‘ 𝐾 ) ≤ 𝑋 )
16 13 14 15 syl2an ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 0. ‘ 𝐾 ) ≤ 𝑋 )
17 11 16 eqbrtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( I ↾ 𝐵 ) ) ≤ 𝑋 )
18 1 2 3 5 9 4 diaelval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( I ↾ 𝐵 ) ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( ( I ↾ 𝐵 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( I ↾ 𝐵 ) ) ≤ 𝑋 ) ) )
19 7 17 18 mpbir2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( I ↾ 𝐵 ) ∈ ( 𝐼 ‘ 𝑋 ) )
20 19 ne0d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ≠ ∅ )