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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → 𝐹 ≠ 𝐺 )
6 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → 𝐹 ∈ 𝑇 )
8 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
9 1 8 2 3 4 trlid0b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) ) )
10 6 7 9 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) ) )
11 10 biimpar ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) ) → 𝐹 = ( I ↾ 𝐵 ) )
12 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) )
13 12 eqeq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → ( ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) ↔ ( 𝑅 ‘ 𝐺 ) = ( 0. ‘ 𝐾 ) ) )
14 13 biimpa ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) ) → ( 𝑅 ‘ 𝐺 ) = ( 0. ‘ 𝐾 ) )
15 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
16 simpl2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) ) → 𝐺 ∈ 𝑇 )
17 1 8 2 3 4 trlid0b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ( 𝐺 = ( I ↾ 𝐵 ) ↔ ( 𝑅 ‘ 𝐺 ) = ( 0. ‘ 𝐾 ) ) )
18 15 16 17 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) ) → ( 𝐺 = ( I ↾ 𝐵 ) ↔ ( 𝑅 ‘ 𝐺 ) = ( 0. ‘ 𝐾 ) ) )
19 14 18 mpbird ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) ) → 𝐺 = ( I ↾ 𝐵 ) )
20 11 19 eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) ) → 𝐹 = 𝐺 )
21 20 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → ( ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) → 𝐹 = 𝐺 ) )
22 10 21 sylbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐹 = ( I ↾ 𝐵 ) → 𝐹 = 𝐺 ) )
23 22 necon3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐹 ≠ 𝐺 → 𝐹 ≠ ( I ↾ 𝐵 ) ) )
24 5 23 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ 𝐺 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) ) → 𝐹 ≠ ( I ↾ 𝐵 ) )