| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dihatlat.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
| 2 |
|
dihatlat.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
| 3 |
|
dihatlat.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
| 4 |
|
dihatlat.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
| 5 |
|
dihatlat.l |
⊢ 𝐿 = ( LSAtoms ‘ 𝑈 ) |
| 6 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
| 7 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
| 8 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
| 9 |
|
eqid |
⊢ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) |
| 10 |
|
eqid |
⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 ) |
| 11 |
6 7 1 2 8 9 3 4 10
|
dih1dimb2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ∃ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) |
| 12 |
11
|
anassrs |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ∃ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) |
| 13 |
|
simp3rr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) |
| 14 |
|
simp1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
| 15 |
2 3 14
|
dvhlmod |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → 𝑈 ∈ LMod ) |
| 16 |
|
simp3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
| 17 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
| 18 |
6 2 8 17 9
|
tendo0cl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
| 19 |
14 18
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
| 20 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
| 21 |
2 8 17 3 20
|
dvhelvbasei |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ∈ ( Base ‘ 𝑈 ) ) |
| 22 |
14 16 19 21
|
syl12anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ∈ ( Base ‘ 𝑈 ) ) |
| 23 |
|
simp3rl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) |
| 24 |
23
|
neneqd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ¬ 𝑔 = ( I ↾ ( Base ‘ 𝐾 ) ) ) |
| 25 |
24
|
intnanrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ¬ ( 𝑔 = ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) ) |
| 26 |
|
vex |
⊢ 𝑔 ∈ V |
| 27 |
|
fvex |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∈ V |
| 28 |
27
|
mptex |
⊢ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∈ V |
| 29 |
26 28
|
opth |
⊢ ( ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ↔ ( 𝑔 = ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) ) |
| 30 |
29
|
necon3abii |
⊢ ( ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ≠ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ↔ ¬ ( 𝑔 = ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) ) |
| 31 |
25 30
|
sylibr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ≠ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ) |
| 32 |
|
eqid |
⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 ) |
| 33 |
6 2 8 3 32 9
|
dvh0g |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 0g ‘ 𝑈 ) = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ) |
| 34 |
14 33
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ( 0g ‘ 𝑈 ) = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ) |
| 35 |
31 34
|
neeqtrrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ≠ ( 0g ‘ 𝑈 ) ) |
| 36 |
20 10 32 5
|
lsatlspsn2 |
⊢ ( ( 𝑈 ∈ LMod ∧ ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ∈ ( Base ‘ 𝑈 ) ∧ ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ≠ ( 0g ‘ 𝑈 ) ) → ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ∈ 𝐿 ) |
| 37 |
15 22 35 36
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ∈ 𝐿 ) |
| 38 |
13 37
|
eqeltrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ( 𝐼 ‘ 𝑄 ) ∈ 𝐿 ) |
| 39 |
38
|
3expa |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ∧ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑔 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ 𝑔 , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ } ) ) ) ) → ( 𝐼 ‘ 𝑄 ) ∈ 𝐿 ) |
| 40 |
12 39
|
rexlimddv |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ 𝑄 ) ∈ 𝐿 ) |
| 41 |
|
eqid |
⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
| 42 |
|
eqid |
⊢ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) |
| 43 |
7 1 2 41 8 4 3 10 42
|
dih1dimc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ } ) ) |
| 44 |
43
|
anassrs |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ 𝑄 ) = ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ } ) ) |
| 45 |
|
simpll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
| 46 |
2 3 45
|
dvhlmod |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → 𝑈 ∈ LMod ) |
| 47 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
| 48 |
7 47 1 2
|
lhpocnel |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) |
| 49 |
48
|
ad2antrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) |
| 50 |
|
simplr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → 𝑄 ∈ 𝐴 ) |
| 51 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) |
| 52 |
7 1 2 8 42
|
ltrniotacl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
| 53 |
45 49 50 51 52
|
syl112anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
| 54 |
2 8 17
|
tendoidcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
| 55 |
54
|
ad2antrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
| 56 |
2 8 17 3 20
|
dvhelvbasei |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ∈ ( Base ‘ 𝑈 ) ) |
| 57 |
45 53 55 56
|
syl12anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ∈ ( Base ‘ 𝑈 ) ) |
| 58 |
6 2 8 17 9
|
tendo1ne0 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ≠ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) |
| 59 |
58
|
ad2antrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ≠ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) |
| 60 |
59
|
neneqd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ¬ ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) |
| 61 |
60
|
intnand |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ¬ ( ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) ) |
| 62 |
|
riotaex |
⊢ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ∈ V |
| 63 |
|
resiexg |
⊢ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∈ V → ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ V ) |
| 64 |
27 63
|
ax-mp |
⊢ ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ V |
| 65 |
62 64
|
opth |
⊢ ( ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ↔ ( ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) ) |
| 66 |
65
|
necon3abii |
⊢ ( ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ≠ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ↔ ¬ ( ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) ) |
| 67 |
61 66
|
sylibr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ≠ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ) |
| 68 |
33
|
ad2antrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ( 0g ‘ 𝑈 ) = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ⟩ ) |
| 69 |
67 68
|
neeqtrrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ≠ ( 0g ‘ 𝑈 ) ) |
| 70 |
20 10 32 5
|
lsatlspsn2 |
⊢ ( ( 𝑈 ∈ LMod ∧ ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ∈ ( Base ‘ 𝑈 ) ∧ ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ ≠ ( 0g ‘ 𝑈 ) ) → ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ } ) ∈ 𝐿 ) |
| 71 |
46 57 69 70
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ( ( LSpan ‘ 𝑈 ) ‘ { ⟨ ( ℩ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ } ) ∈ 𝐿 ) |
| 72 |
44 71
|
eqeltrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ 𝑄 ) ∈ 𝐿 ) |
| 73 |
40 72
|
pm2.61dan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐴 ) → ( 𝐼 ‘ 𝑄 ) ∈ 𝐿 ) |