Metamath Proof Explorer


Theorem cdlemk19u

Description: Part of Lemma K of Crawley p. 118. Line 12, p. 120, "f (exponent) tau = k". We represent f, k, tau with F , N , U . (Contributed by NM, 31-Jul-2013)

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 ) ) ) )
cdlemk5.y 𝑌 = ( ( 𝑃 ( 𝑅𝑔 ) ) ( 𝑍 ( 𝑅 ‘ ( 𝑔 𝑏 ) ) ) )
cdlemk5.x 𝑋 = ( 𝑧𝑇𝑏𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅𝑔 ) ) → ( 𝑧𝑃 ) = 𝑌 ) )
cdlemk5.u 𝑈 = ( 𝑔𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑔 , 𝑋 ) )
Assertion cdlemk19u ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑈𝐹 ) = 𝑁 )

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 cdlemk5.y 𝑌 = ( ( 𝑃 ( 𝑅𝑔 ) ) ( 𝑍 ( 𝑅 ‘ ( 𝑔 𝑏 ) ) ) )
11 cdlemk5.x 𝑋 = ( 𝑧𝑇𝑏𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅𝐹 ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅𝑔 ) ) → ( 𝑧𝑃 ) = 𝑌 ) )
12 cdlemk5.u 𝑈 = ( 𝑔𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑔 , 𝑋 ) )
13 simp1l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
14 simp1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) )
15 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐹𝑇 )
16 simp2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑁𝑇 )
17 simp3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
18 1 2 3 4 5 6 7 8 9 10 11 12 cdlemk35u ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇𝐹𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑈𝐹 ) ∈ 𝑇 )
19 14 15 16 15 17 18 syl131anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑈𝐹 ) ∈ 𝑇 )
20 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → 𝐹 = 𝑁 )
21 simpl2l ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → 𝐹𝑇 )
22 11 12 cdlemk40t ( ( 𝐹 = 𝑁𝐹𝑇 ) → ( 𝑈𝐹 ) = 𝐹 )
23 20 21 22 syl2anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → ( 𝑈𝐹 ) = 𝐹 )
24 23 fveq1d ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → ( ( 𝑈𝐹 ) ‘ 𝑃 ) = ( 𝐹𝑃 ) )
25 fveq1 ( 𝐹 = 𝑁 → ( 𝐹𝑃 ) = ( 𝑁𝑃 ) )
26 25 adantl ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → ( 𝐹𝑃 ) = ( 𝑁𝑃 ) )
27 24 26 eqtrd ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → ( ( 𝑈𝐹 ) ‘ 𝑃 ) = ( 𝑁𝑃 ) )
28 simpl1 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹𝑁 ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) )
29 simpl2l ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹𝑁 ) → 𝐹𝑇 )
30 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹𝑁 ) → 𝐹𝑁 )
31 simpl2r ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹𝑁 ) → 𝑁𝑇 )
32 simpl3 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹𝑁 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
33 1 2 3 4 5 6 7 8 9 10 11 12 cdlemk19u1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝐹𝑁𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝑈𝐹 ) ‘ 𝑃 ) = ( 𝑁𝑃 ) )
34 28 29 30 31 32 33 syl131anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ 𝐹𝑁 ) → ( ( 𝑈𝐹 ) ‘ 𝑃 ) = ( 𝑁𝑃 ) )
35 27 34 pm2.61dane ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝑈𝐹 ) ‘ 𝑃 ) = ( 𝑁𝑃 ) )
36 2 5 6 7 cdlemd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑈𝐹 ) ∈ 𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( ( 𝑈𝐹 ) ‘ 𝑃 ) = ( 𝑁𝑃 ) ) → ( 𝑈𝐹 ) = 𝑁 )
37 13 19 16 17 35 36 syl311anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝑁 ) ) ∧ ( 𝐹𝑇𝑁𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑈𝐹 ) = 𝑁 )