Metamath Proof Explorer


Theorem cdlemg17bq

Description: cdlemg17b with P and Q swapped. Antecedent F e. ( TW ) is redundant for easier use. TODO: should we have redundant antecedent for cdlemg17b also? (Contributed by NM, 13-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 cdlemg17bq ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺𝑄 ) = 𝑃 )

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 cdlemg17pq ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) )
9 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
11 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
12 simp22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) → 𝐺𝑇 )
13 simp23 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) → 𝑄𝑃 )
14 simp3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) → ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) )
15 1 2 3 4 5 6 7 cdlemg17b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) → ( 𝐺𝑄 ) = 𝑃 )
16 9 10 11 12 13 14 15 syl321anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) → ( 𝐺𝑄 ) = 𝑃 )
17 8 16 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺𝑄 ) = 𝑃 )