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Metamath Proof Explorer

Theorem cdlemg8d

Description: TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013)

Ref Expression
Hypotheses cdlemg8.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg8.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg8.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemg8.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg8.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg8.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg8d ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≠ 𝑃 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑊 ) )

Proof

Step Hyp Ref Expression
1 cdlemg8.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg8.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemg8.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdlemg8.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemg8.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemg8.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 1 2 3 4 5 6 cdlemg8b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≠ 𝑃 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( 𝑃 ∨ 𝑄 ) )
8 1 2 3 4 5 6 cdlemg8c ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≠ 𝑃 ) ) → ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( 𝑃 ∨ 𝑄 ) )
9 7 8 eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≠ 𝑃 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) )
10 9 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≠ 𝑃 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑊 ) )