Metamath Proof Explorer


Theorem cdlemg25zz

Description: cdlemg16zz restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013)

Ref Expression
Hypotheses cdlemg12.l = ( le ‘ 𝐾 )
cdlemg12.j = ( join ‘ 𝐾 )
cdlemg12.m = ( meet ‘ 𝐾 )
cdlemg12.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg12.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg12.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg12b.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg25zz ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑧𝐴 ∧ ¬ 𝑧 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑧 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑧 ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑧 ( 𝐹 ‘ ( 𝐺𝑧 ) ) ) 𝑊 ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l = ( le ‘ 𝐾 )
2 cdlemg12.j = ( join ‘ 𝐾 )
3 cdlemg12.m = ( meet ‘ 𝐾 )
4 cdlemg12.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemg12.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemg12.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemg12b.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 1 2 3 4 5 6 7 cdlemg16zz ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑧𝐴 ∧ ¬ 𝑧 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑧 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑧 ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑧 ( 𝐹 ‘ ( 𝐺𝑧 ) ) ) 𝑊 ) )