Step |
Hyp |
Ref |
Expression |
1 |
|
tendotr.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
tendotr.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
tendotr.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
tendotr.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
tendotr.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
tendotr.o |
⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
7 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
8 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → 𝑈 ∈ 𝐸 ) |
9 |
1 2 5
|
tendoid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ) → ( 𝑈 ‘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) ) |
10 |
7 8 9
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝑈 ‘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) ) |
11 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → 𝐹 = ( I ↾ 𝐵 ) ) |
12 |
11
|
fveq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝑈 ‘ 𝐹 ) = ( 𝑈 ‘ ( I ↾ 𝐵 ) ) ) |
13 |
10 12 11
|
3eqtr4d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝑈 ‘ 𝐹 ) = 𝐹 ) |
14 |
13
|
fveq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) = ( 𝑅 ‘ 𝐹 ) ) |
15 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
16 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → 𝑈 ∈ 𝐸 ) |
17 |
|
simpl3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → 𝐹 ∈ 𝑇 ) |
18 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
19 |
18 2 3 4 5
|
tendotp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( le ‘ 𝐾 ) ( 𝑅 ‘ 𝐹 ) ) |
20 |
15 16 17 19
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( le ‘ 𝐾 ) ( 𝑅 ‘ 𝐹 ) ) |
21 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → 𝐾 ∈ HL ) |
22 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
23 |
21 22
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → 𝐾 ∈ AtLat ) |
24 |
2 3 5
|
tendocl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 ) |
25 |
15 16 17 24
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 ) |
26 |
|
simpl2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → 𝑈 ≠ 𝑂 ) |
27 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → 𝐹 ≠ ( I ↾ 𝐵 ) ) |
28 |
1 2 3 5 6
|
tendoid0 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → ( ( 𝑈 ‘ 𝐹 ) = ( I ↾ 𝐵 ) ↔ 𝑈 = 𝑂 ) ) |
29 |
15 16 17 27 28
|
syl112anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑈 ‘ 𝐹 ) = ( I ↾ 𝐵 ) ↔ 𝑈 = 𝑂 ) ) |
30 |
29
|
necon3bid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑈 ‘ 𝐹 ) ≠ ( I ↾ 𝐵 ) ↔ 𝑈 ≠ 𝑂 ) ) |
31 |
26 30
|
mpbird |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝑈 ‘ 𝐹 ) ≠ ( I ↾ 𝐵 ) ) |
32 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
33 |
1 32 2 3 4
|
trlnidat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 ∧ ( 𝑈 ‘ 𝐹 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ∈ ( Atoms ‘ 𝐾 ) ) |
34 |
15 25 31 33
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ∈ ( Atoms ‘ 𝐾 ) ) |
35 |
1 32 2 3 4
|
trlnidat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Atoms ‘ 𝐾 ) ) |
36 |
15 17 27 35
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Atoms ‘ 𝐾 ) ) |
37 |
18 32
|
atcmp |
⊢ ( ( 𝐾 ∈ AtLat ∧ ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ∈ ( Atoms ‘ 𝐾 ) ∧ ( 𝑅 ‘ 𝐹 ) ∈ ( Atoms ‘ 𝐾 ) ) → ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( le ‘ 𝐾 ) ( 𝑅 ‘ 𝐹 ) ↔ ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) = ( 𝑅 ‘ 𝐹 ) ) ) |
38 |
23 34 36 37
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( le ‘ 𝐾 ) ( 𝑅 ‘ 𝐹 ) ↔ ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) = ( 𝑅 ‘ 𝐹 ) ) ) |
39 |
20 38
|
mpbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) = ( 𝑅 ‘ 𝐹 ) ) |
40 |
14 39
|
pm2.61dane |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑈 ≠ 𝑂 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) = ( 𝑅 ‘ 𝐹 ) ) |