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 ) ) |