Step |
Hyp |
Ref |
Expression |
1 |
|
tendoid0.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
tendoid0.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
tendoid0.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
tendoid0.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
tendoid0.o |
⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
6 |
1 2 3
|
cdlemftr0 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑔 ∈ 𝑇 𝑔 ≠ ( I ↾ 𝐵 ) ) |
7 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵 ) ) → 𝑔 ≠ ( I ↾ 𝐵 ) ) |
8 |
|
fveq1 |
⊢ ( ( I ↾ 𝑇 ) = 𝑂 → ( ( I ↾ 𝑇 ) ‘ 𝑔 ) = ( 𝑂 ‘ 𝑔 ) ) |
9 |
8
|
adantl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵 ) ) ∧ ( I ↾ 𝑇 ) = 𝑂 ) → ( ( I ↾ 𝑇 ) ‘ 𝑔 ) = ( 𝑂 ‘ 𝑔 ) ) |
10 |
|
simpl2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵 ) ) ∧ ( I ↾ 𝑇 ) = 𝑂 ) → 𝑔 ∈ 𝑇 ) |
11 |
|
fvresi |
⊢ ( 𝑔 ∈ 𝑇 → ( ( I ↾ 𝑇 ) ‘ 𝑔 ) = 𝑔 ) |
12 |
10 11
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵 ) ) ∧ ( I ↾ 𝑇 ) = 𝑂 ) → ( ( I ↾ 𝑇 ) ‘ 𝑔 ) = 𝑔 ) |
13 |
5 1
|
tendo02 |
⊢ ( 𝑔 ∈ 𝑇 → ( 𝑂 ‘ 𝑔 ) = ( I ↾ 𝐵 ) ) |
14 |
10 13
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵 ) ) ∧ ( I ↾ 𝑇 ) = 𝑂 ) → ( 𝑂 ‘ 𝑔 ) = ( I ↾ 𝐵 ) ) |
15 |
9 12 14
|
3eqtr3d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵 ) ) ∧ ( I ↾ 𝑇 ) = 𝑂 ) → 𝑔 = ( I ↾ 𝐵 ) ) |
16 |
15
|
ex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵 ) ) → ( ( I ↾ 𝑇 ) = 𝑂 → 𝑔 = ( I ↾ 𝐵 ) ) ) |
17 |
16
|
necon3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵 ) ) → ( 𝑔 ≠ ( I ↾ 𝐵 ) → ( I ↾ 𝑇 ) ≠ 𝑂 ) ) |
18 |
7 17
|
mpd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵 ) ) → ( I ↾ 𝑇 ) ≠ 𝑂 ) |
19 |
18
|
rexlimdv3a |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ∃ 𝑔 ∈ 𝑇 𝑔 ≠ ( I ↾ 𝐵 ) → ( I ↾ 𝑇 ) ≠ 𝑂 ) ) |
20 |
6 19
|
mpd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝑇 ) ≠ 𝑂 ) |