Metamath Proof Explorer


Theorem trlnid

Description: Different translations with the same trace cannot be the identity. (Contributed by NM, 26-Jul-2013)

Ref Expression
Hypotheses trlnid.b 𝐵 = ( Base ‘ 𝐾 )
trlnid.h 𝐻 = ( LHyp ‘ 𝐾 )
trlnid.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlnid.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlnid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → 𝐹 ≠ ( I ↾ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 trlnid.b 𝐵 = ( Base ‘ 𝐾 )
2 trlnid.h 𝐻 = ( LHyp ‘ 𝐾 )
3 trlnid.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 trlnid.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
5 simp3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → 𝐹𝐺 )
6 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
7 simp2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → 𝐹𝑇 )
8 eqid ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
9 1 8 2 3 4 trlid0b ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) ) )
10 6 7 9 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) ) )
11 10 biimpar ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) ∧ ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) ) → 𝐹 = ( I ↾ 𝐵 ) )
12 simp3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → ( 𝑅𝐹 ) = ( 𝑅𝐺 ) )
13 12 eqeq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → ( ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) ↔ ( 𝑅𝐺 ) = ( 0. ‘ 𝐾 ) ) )
14 13 biimpa ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) ∧ ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) ) → ( 𝑅𝐺 ) = ( 0. ‘ 𝐾 ) )
15 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) ∧ ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
16 simpl2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) ∧ ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) ) → 𝐺𝑇 )
17 1 8 2 3 4 trlid0b ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇 ) → ( 𝐺 = ( I ↾ 𝐵 ) ↔ ( 𝑅𝐺 ) = ( 0. ‘ 𝐾 ) ) )
18 15 16 17 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) ∧ ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) ) → ( 𝐺 = ( I ↾ 𝐵 ) ↔ ( 𝑅𝐺 ) = ( 0. ‘ 𝐾 ) ) )
19 14 18 mpbird ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) ∧ ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) ) → 𝐺 = ( I ↾ 𝐵 ) )
20 11 19 eqtr4d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) ∧ ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) ) → 𝐹 = 𝐺 )
21 20 ex ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → ( ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) → 𝐹 = 𝐺 ) )
22 10 21 sylbid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → ( 𝐹 = ( I ↾ 𝐵 ) → 𝐹 = 𝐺 ) )
23 22 necon3d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → ( 𝐹𝐺𝐹 ≠ ( I ↾ 𝐵 ) ) )
24 5 23 mpd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝐹𝐺 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) ) → 𝐹 ≠ ( I ↾ 𝐵 ) )