Metamath Proof Explorer

Theorem cdlemg4b2

Description: TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013)

Ref Expression
Hypotheses cdlemg4.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg4.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemg4.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg4b.v ⊢ 𝑉 = ( 𝑅 ‘ 𝐺 )
Assertion cdlemg4b2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝐺 ‘ 𝑃 ) ∨ 𝑉 ) = ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg4.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 cdlemg4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 cdlemg4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 cdlemg4.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 cdlemg4.j ⊢ ∨ = ( join ‘ 𝐾 )
7 cdlemg4b.v ⊢ 𝑉 = ( 𝑅 ‘ 𝐺 )
8 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
9 1 6 8 2 3 4 5 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
10 9 3com23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
11 7 10 eqtrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → 𝑉 = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
12 11 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝐺 ‘ 𝑃 ) ∨ 𝑉 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
13 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → 𝑃 ∈ 𝐴 )
15 1 2 3 4 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
16 15 3com23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
17 eqid ⊢ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 )
18 1 6 8 2 3 17 cdleme0cq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
19 13 14 16 18 syl12anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
20 12 19 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝐺 ‘ 𝑃 ) ∨ 𝑉 ) = ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )