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Metamath Proof Explorer

Theorem cdlemk55u

Description: Part of proof of Lemma K of Crawley p. 118. Line 11, p. 120. G , I stand for g, h. X represents tau. (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 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) )
cdlemk5.y ⊢ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) )
cdlemk5.x ⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑃 ) = 𝑌 ) )
cdlemk5.u ⊢ 𝑈 = ( 𝑔 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑔 , 𝑋 ) )
Assertion cdlemk55u ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑈 ‘ ( 𝐺 ∘ 𝐼 ) ) = ( ( 𝑈 ‘ 𝐺 ) ∘ ( 𝑈 ‘ 𝐼 ) ) )

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 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) )
10 cdlemk5.y ⊢ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) )
11 cdlemk5.x ⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑃 ) = 𝑌 ) )
12 cdlemk5.u ⊢ 𝑈 = ( 𝑔 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑔 , 𝑋 ) )
13 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → 𝐹 = 𝑁 )
14 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15 simp22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐺 ∈ 𝑇 )
16 simp23 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐼 ∈ 𝑇 )
17 6 7 ltrnco ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) → ( 𝐺 ∘ 𝐼 ) ∈ 𝑇 )
18 14 15 16 17 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐺 ∘ 𝐼 ) ∈ 𝑇 )
19 18 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → ( 𝐺 ∘ 𝐼 ) ∈ 𝑇 )
20 11 12 cdlemk40t ⊢ ( ( 𝐹 = 𝑁 ∧ ( 𝐺 ∘ 𝐼 ) ∈ 𝑇 ) → ( 𝑈 ‘ ( 𝐺 ∘ 𝐼 ) ) = ( 𝐺 ∘ 𝐼 ) )
21 13 19 20 syl2anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → ( 𝑈 ‘ ( 𝐺 ∘ 𝐼 ) ) = ( 𝐺 ∘ 𝐼 ) )
22 simpl22 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → 𝐺 ∈ 𝑇 )
23 11 12 cdlemk40t ⊢ ( ( 𝐹 = 𝑁 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐺 ) = 𝐺 )
24 13 22 23 syl2anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → ( 𝑈 ‘ 𝐺 ) = 𝐺 )
25 simpl23 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → 𝐼 ∈ 𝑇 )
26 11 12 cdlemk40t ⊢ ( ( 𝐹 = 𝑁 ∧ 𝐼 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐼 ) = 𝐼 )
27 13 25 26 syl2anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → ( 𝑈 ‘ 𝐼 ) = 𝐼 )
28 24 27 coeq12d ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → ( ( 𝑈 ‘ 𝐺 ) ∘ ( 𝑈 ‘ 𝐼 ) ) = ( 𝐺 ∘ 𝐼 ) )
29 21 28 eqtr4d ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝑁 ) → ( 𝑈 ‘ ( 𝐺 ∘ 𝐼 ) ) = ( ( 𝑈 ‘ 𝐺 ) ∘ ( 𝑈 ‘ 𝐼 ) ) )
30 simpl1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ 𝑁 ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) )
31 simpl21 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ 𝑁 ) → ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) )
32 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ 𝑁 ) → 𝐹 ≠ 𝑁 )
33 31 32 jca ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ 𝑁 ) → ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐹 ≠ 𝑁 ) )
34 simpl22 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ 𝑁 ) → 𝐺 ∈ 𝑇 )
35 simpl23 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ 𝑁 ) → 𝐼 ∈ 𝑇 )
36 simpl3 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ 𝑁 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
37 1 2 3 4 5 6 7 8 9 10 11 12 cdlemk55u1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐹 ≠ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑈 ‘ ( 𝐺 ∘ 𝐼 ) ) = ( ( 𝑈 ‘ 𝐺 ) ∘ ( 𝑈 ‘ 𝐼 ) ) )
38 30 33 34 35 36 37 syl131anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ 𝑁 ) → ( 𝑈 ‘ ( 𝐺 ∘ 𝐼 ) ) = ( ( 𝑈 ‘ 𝐺 ) ∘ ( 𝑈 ‘ 𝐼 ) ) )
39 29 38 pm2.61dane ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑈 ‘ ( 𝐺 ∘ 𝐼 ) ) = ( ( 𝑈 ‘ 𝐺 ) ∘ ( 𝑈 ‘ 𝐼 ) ) )