Metamath Proof Explorer


Theorem cdlemk56w

Description: Use a fixed element to eliminate P in cdlemk56 . (Contributed by NM, 1-Aug-2013)

Ref Expression
Hypotheses cdlemk6.b 𝐵 = ( Base ‘ 𝐾 )
cdlemk6.j = ( join ‘ 𝐾 )
cdlemk6.m = ( meet ‘ 𝐾 )
cdlemk6.o = ( oc ‘ 𝐾 )
cdlemk6.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemk6.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemk6.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemk6.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemk6.p 𝑃 = ( 𝑊 )
cdlemk6.z 𝑍 = ( ( 𝑃 ( 𝑅𝑏 ) ) ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) )
cdlemk6.y 𝑌 = ( ( 𝑃 ( 𝑅𝑔 ) ) ( 𝑍 ( 𝑅 ‘ ( 𝑔 𝑏 ) ) ) )
cdlemk6.x 𝑋 = ( 𝑧𝑇𝑏𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅𝑔 ) ) → ( 𝑧𝑃 ) = 𝑌 ) )
cdlemk6.u 𝑈 = ( 𝑔𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑔 , 𝑋 ) )
cdlemk6.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemk56w ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) → ( 𝑈𝐸 ∧ ( 𝑈𝐹 ) = 𝑁 ) )

Proof

Step Hyp Ref Expression
1 cdlemk6.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemk6.j = ( join ‘ 𝐾 )
3 cdlemk6.m = ( meet ‘ 𝐾 )
4 cdlemk6.o = ( oc ‘ 𝐾 )
5 cdlemk6.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemk6.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemk6.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemk6.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemk6.p 𝑃 = ( 𝑊 )
10 cdlemk6.z 𝑍 = ( ( 𝑃 ( 𝑅𝑏 ) ) ( ( 𝑁𝑃 ) ( 𝑅 ‘ ( 𝑏 𝐹 ) ) ) )
11 cdlemk6.y 𝑌 = ( ( 𝑃 ( 𝑅𝑔 ) ) ( 𝑍 ( 𝑅 ‘ ( 𝑔 𝑏 ) ) ) )
12 cdlemk6.x 𝑋 = ( 𝑧𝑇𝑏𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅𝑔 ) ) → ( 𝑧𝑃 ) = 𝑌 ) )
13 cdlemk6.u 𝑈 = ( 𝑔𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑔 , 𝑋 ) )
14 cdlemk6.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
15 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
16 simp2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) → 𝐹𝑇 )
17 simp2r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) → 𝑁𝑇 )
18 simp3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) → ( 𝑅𝐹 ) = ( 𝑅𝑁 ) )
19 eqid ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
20 4 fveq1i ( 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
21 9 20 eqtri 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
22 19 5 6 21 lhpocnel2 ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) )
23 22 3ad2ant1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) )
24 1 19 2 3 5 6 7 8 10 11 12 13 14 cdlemk56 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑈𝐸 )
25 15 16 17 18 23 24 syl311anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) → 𝑈𝐸 )
26 1 2 3 4 5 6 7 8 9 10 11 12 13 cdlemk19w ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) → ( 𝑈𝐹 ) = 𝑁 )
27 25 26 jca ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) → ( 𝑈𝐸 ∧ ( 𝑈𝐹 ) = 𝑁 ) )