Metamath Proof Explorer


Theorem cdlemg17j

Description: TODO: fix comment. (Contributed by NM, 11-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 cdlemg17j ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺 ‘ ( 𝐹𝑃 ) ) = ( 𝐹 ‘ ( 𝐺𝑃 ) ) )

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 cdlemg17i ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺 ‘ ( 𝐹𝑃 ) ) = ( 𝐹𝑄 ) )
9 1 2 3 4 5 6 7 cdlemg17ir ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐹 ‘ ( 𝐺𝑃 ) ) = ( 𝐹𝑄 ) )
10 8 9 eqtr4d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺 ‘ ( 𝐹𝑃 ) ) = ( 𝐹 ‘ ( 𝐺𝑃 ) ) )