Step |
Hyp |
Ref |
Expression |
1 |
|
ttgval.n |
⊢ 𝐺 = ( toTG ‘ 𝐻 ) |
2 |
|
ttgds.1 |
⊢ 𝐷 = ( dist ‘ 𝐻 ) |
3 |
|
dsid |
⊢ dist = Slot ( dist ‘ ndx ) |
4 |
|
slotslnbpsd |
⊢ ( ( ( LineG ‘ ndx ) ≠ ( Base ‘ ndx ) ∧ ( LineG ‘ ndx ) ≠ ( +g ‘ ndx ) ) ∧ ( ( LineG ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( LineG ‘ ndx ) ≠ ( dist ‘ ndx ) ) ) |
5 |
|
simprr |
⊢ ( ( ( ( LineG ‘ ndx ) ≠ ( Base ‘ ndx ) ∧ ( LineG ‘ ndx ) ≠ ( +g ‘ ndx ) ) ∧ ( ( LineG ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( LineG ‘ ndx ) ≠ ( dist ‘ ndx ) ) ) → ( LineG ‘ ndx ) ≠ ( dist ‘ ndx ) ) |
6 |
4 5
|
ax-mp |
⊢ ( LineG ‘ ndx ) ≠ ( dist ‘ ndx ) |
7 |
6
|
necomi |
⊢ ( dist ‘ ndx ) ≠ ( LineG ‘ ndx ) |
8 |
|
slotsinbpsd |
⊢ ( ( ( Itv ‘ ndx ) ≠ ( Base ‘ ndx ) ∧ ( Itv ‘ ndx ) ≠ ( +g ‘ ndx ) ) ∧ ( ( Itv ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( Itv ‘ ndx ) ≠ ( dist ‘ ndx ) ) ) |
9 |
|
simprr |
⊢ ( ( ( ( Itv ‘ ndx ) ≠ ( Base ‘ ndx ) ∧ ( Itv ‘ ndx ) ≠ ( +g ‘ ndx ) ) ∧ ( ( Itv ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( Itv ‘ ndx ) ≠ ( dist ‘ ndx ) ) ) → ( Itv ‘ ndx ) ≠ ( dist ‘ ndx ) ) |
10 |
8 9
|
ax-mp |
⊢ ( Itv ‘ ndx ) ≠ ( dist ‘ ndx ) |
11 |
10
|
necomi |
⊢ ( dist ‘ ndx ) ≠ ( Itv ‘ ndx ) |
12 |
1 3 7 11
|
ttglem |
⊢ ( dist ‘ 𝐻 ) = ( dist ‘ 𝐺 ) |
13 |
2 12
|
eqtri |
⊢ 𝐷 = ( dist ‘ 𝐺 ) |