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Metamath Proof Explorer

Theorem cdlemk

Description: Lemma K of Crawley p. 118. Final result, lines 11 and 12 on p. 120: given two translations f and k with the same trace, there exists a trace-preserving endomorphism tau whose value at f is k. We use F , N , and u to represent f, k, and tau. (Contributed by NM, 1-Aug-2013)

Ref Expression
Hypotheses cdlemk7.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemk7.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemk7.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemk7.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemk ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ∃ 𝑢 ∈ 𝐸 ( 𝑢 ‘ 𝐹 ) = 𝑁 )

Proof

Step Hyp Ref Expression
1 cdlemk7.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 cdlemk7.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 cdlemk7.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
4 cdlemk7.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
6 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
7 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
8 eqid ⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
9 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
10 eqid ⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
11 eqid ⊢ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) )
12 eqid ⊢ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) )
13 eqid ⊢ ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) )
14 eqid ⊢ ( 𝑓 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑓 , ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) = ( 𝑓 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑓 , ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) ) )
15 5 6 7 8 9 1 2 3 10 11 12 13 14 4 cdlemk56w ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( ( 𝑓 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑓 , ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) ∈ 𝐸 ∧ ( ( 𝑓 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑓 , ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) ‘ 𝐹 ) = 𝑁 ) )
16 fveq1 ⊢ ( 𝑢 = ( 𝑓 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑓 , ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) → ( 𝑢 ‘ 𝐹 ) = ( ( 𝑓 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑓 , ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) ‘ 𝐹 ) )
17 16 eqeq1d ⊢ ( 𝑢 = ( 𝑓 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑓 , ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) → ( ( 𝑢 ‘ 𝐹 ) = 𝑁 ↔ ( ( 𝑓 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑓 , ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) ‘ 𝐹 ) = 𝑁 ) )
18 17 rspcev ⊢ ( ( ( 𝑓 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑓 , ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) ∈ 𝐸 ∧ ( ( 𝑓 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑓 , ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑓 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑓 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝑅 ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑁 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) ‘ 𝐹 ) = 𝑁 ) → ∃ 𝑢 ∈ 𝐸 ( 𝑢 ‘ 𝐹 ) = 𝑁 )
19 15 18 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ∃ 𝑢 ∈ 𝐸 ( 𝑢 ‘ 𝐹 ) = 𝑁 )