Metamath Proof Explorer

Theorem cdlemk19w

Description: Use a fixed element to eliminate P in cdlemk19u . (Contributed by NM, 1-Aug-2013)

Ref Expression
Hypotheses cdlemk6.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemk6.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemk6.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemk6.o ⊢ ⊥ = ( oc ‘ 𝐾 )
cdlemk6.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemk6.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemk6.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemk6.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemk6.p ⊢ 𝑃 = ( ⊥ ‘ 𝑊 )
cdlemk6.z ⊢ 𝑍 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) )
cdlemk6.y ⊢ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) )
cdlemk6.x ⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑃 ) = 𝑌 ) )
cdlemk6.u ⊢ 𝑈 = ( 𝑔 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑔 , 𝑋 ) )
Assertion cdlemk19w ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( 𝑈 ‘ 𝐹 ) = 𝑁 )

Proof

Step Hyp Ref Expression
1 cdlemk6.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemk6.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemk6.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdlemk6.o ⊢ ⊥ = ( oc ‘ 𝐾 )
5 cdlemk6.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemk6.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemk6.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemk6.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemk6.p ⊢ 𝑃 = ( ⊥ ‘ 𝑊 )
10 cdlemk6.z ⊢ 𝑍 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) )
11 cdlemk6.y ⊢ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) )
12 cdlemk6.x ⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑃 ) = 𝑌 ) )
13 cdlemk6.u ⊢ 𝑈 = ( 𝑔 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑔 , 𝑋 ) )
14 3simpb ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) )
15 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) )
16 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
17 16 4 5 6 lhpocnel ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ⊥ ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ⊥ ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) )
18 17 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( ( ⊥ ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ⊥ ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) )
19 9 eleq1i ⊢ ( 𝑃 ∈ 𝐴 ↔ ( ⊥ ‘ 𝑊 ) ∈ 𝐴 )
20 9 breq1i ⊢ ( 𝑃 ( le ‘ 𝐾 ) 𝑊 ↔ ( ⊥ ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 )
21 20 notbii ⊢ ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ↔ ¬ ( ⊥ ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 )
22 19 21 anbi12i ⊢ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) ↔ ( ( ⊥ ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ⊥ ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) )
23 18 22 sylibr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) )
24 1 16 2 3 5 6 7 8 10 11 12 13 cdlemk19u ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑈 ‘ 𝐹 ) = 𝑁 )
25 14 15 23 24 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( 𝑈 ‘ 𝐹 ) = 𝑁 )