Metamath Proof Explorer


Theorem cdlemkfid2N

Description: Lemma for cdlemkfid3N . (Contributed by NM, 29-Jul-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemk5.b 𝐵 = ( Base ‘ 𝐾 )
cdlemk5.l = ( le ‘ 𝐾 )
cdlemk5.j = ( join ‘ 𝐾 )
cdlemk5.m = ( meet ‘ 𝐾 )
cdlemk5.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemk5.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemk5.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemk5.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemk5.z 𝑍 = ( ( 𝑃 ( 𝑅𝑏 ) ) ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) )
Assertion cdlemkfid2N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹 = 𝑁 ) ∧ ( 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝑏𝑇 ) ∧ ( ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → 𝑍 = ( 𝑏𝑃 ) )

Proof

Step Hyp Ref Expression
1 cdlemk5.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemk5.l = ( le ‘ 𝐾 )
3 cdlemk5.j = ( join ‘ 𝐾 )
4 cdlemk5.m = ( meet ‘ 𝐾 )
5 cdlemk5.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemk5.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemk5.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemk5.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemk5.z 𝑍 = ( ( 𝑃 ( 𝑅𝑏 ) ) ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) )
10 simp1r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹 = 𝑁 ) ∧ ( 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝑏𝑇 ) ∧ ( ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → 𝐹 = 𝑁 )
11 10 fveq1d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹 = 𝑁 ) ∧ ( 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝑏𝑇 ) ∧ ( ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → ( 𝐹𝑃 ) = ( 𝑁𝑃 ) )
12 11 oveq1d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹 = 𝑁 ) ∧ ( 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝑏𝑇 ) ∧ ( ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → ( ( 𝐹𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) = ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) )
13 12 oveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹 = 𝑁 ) ∧ ( 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝑏𝑇 ) ∧ ( ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → ( ( 𝑃 ( 𝑅𝑏 ) ) ( ( 𝐹𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) ) = ( ( 𝑃 ( 𝑅𝑏 ) ) ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) ) )
14 1 2 3 4 5 6 7 8 cdlemkfid1N ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝑏𝑇 ) ∧ ( ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → ( ( 𝑃 ( 𝑅𝑏 ) ) ( ( 𝐹𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) ) = ( 𝑏𝑃 ) )
15 14 3adant1r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹 = 𝑁 ) ∧ ( 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝑏𝑇 ) ∧ ( ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → ( ( 𝑃 ( 𝑅𝑏 ) ) ( ( 𝐹𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) ) = ( 𝑏𝑃 ) )
16 13 15 eqtr3d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹 = 𝑁 ) ∧ ( 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝑏𝑇 ) ∧ ( ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → ( ( 𝑃 ( 𝑅𝑏 ) ) ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) ) = ( 𝑏𝑃 ) )
17 9 16 syl5eq ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹 = 𝑁 ) ∧ ( 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝑏𝑇 ) ∧ ( ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ) → 𝑍 = ( 𝑏𝑃 ) )