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Metamath Proof Explorer

Theorem cdlemftr1

Description: Part of proof of Lemma G of Crawley p. 116, sixth line of third paragraph on p. 117: there is "a translation h, different from the identity, such that tr h =/= tr f." (Contributed by NM, 25-Jul-2013)

Ref Expression
Hypotheses cdlemftr.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemftr.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemftr.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemftr.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemftr1 ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑓 ∈ 𝑇 ( 𝑓 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑓 ) ≠ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 cdlemftr.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemftr.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 cdlemftr.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 cdlemftr.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
5 1 2 3 4 cdlemftr2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑓 ∈ 𝑇 ( 𝑓 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑓 ) ≠ 𝑋 ∧ ( 𝑅 ‘ 𝑓 ) ≠ 𝑋 ) )
6 3simpa ⊢ ( ( 𝑓 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑓 ) ≠ 𝑋 ∧ ( 𝑅 ‘ 𝑓 ) ≠ 𝑋 ) → ( 𝑓 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑓 ) ≠ 𝑋 ) )
7 6 reximi ⊢ ( ∃ 𝑓 ∈ 𝑇 ( 𝑓 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑓 ) ≠ 𝑋 ∧ ( 𝑅 ‘ 𝑓 ) ≠ 𝑋 ) → ∃ 𝑓 ∈ 𝑇 ( 𝑓 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑓 ) ≠ 𝑋 ) )
8 5 7 syl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑓 ∈ 𝑇 ( 𝑓 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑓 ) ≠ 𝑋 ) )