Step |
Hyp |
Ref |
Expression |
1 |
|
ttgval.n |
⊢ 𝐺 = ( toTG ‘ 𝐻 ) |
2 |
|
ttgval.b |
⊢ 𝐵 = ( Base ‘ 𝐻 ) |
3 |
|
ttgval.m |
⊢ − = ( -g ‘ 𝐻 ) |
4 |
|
ttgval.s |
⊢ · = ( ·𝑠 ‘ 𝐻 ) |
5 |
|
ttgval.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
6 |
1
|
a1i |
⊢ ( 𝐻 ∈ 𝑉 → 𝐺 = ( toTG ‘ 𝐻 ) ) |
7 |
|
elex |
⊢ ( 𝐻 ∈ 𝑉 → 𝐻 ∈ V ) |
8 |
|
fveq2 |
⊢ ( 𝑤 = 𝐻 → ( Base ‘ 𝑤 ) = ( Base ‘ 𝐻 ) ) |
9 |
8 2
|
eqtr4di |
⊢ ( 𝑤 = 𝐻 → ( Base ‘ 𝑤 ) = 𝐵 ) |
10 |
|
fveq2 |
⊢ ( 𝑤 = 𝐻 → ( -g ‘ 𝑤 ) = ( -g ‘ 𝐻 ) ) |
11 |
10 3
|
eqtr4di |
⊢ ( 𝑤 = 𝐻 → ( -g ‘ 𝑤 ) = − ) |
12 |
11
|
oveqd |
⊢ ( 𝑤 = 𝐻 → ( 𝑧 ( -g ‘ 𝑤 ) 𝑥 ) = ( 𝑧 − 𝑥 ) ) |
13 |
|
fveq2 |
⊢ ( 𝑤 = 𝐻 → ( ·𝑠 ‘ 𝑤 ) = ( ·𝑠 ‘ 𝐻 ) ) |
14 |
13 4
|
eqtr4di |
⊢ ( 𝑤 = 𝐻 → ( ·𝑠 ‘ 𝑤 ) = · ) |
15 |
|
eqidd |
⊢ ( 𝑤 = 𝐻 → 𝑘 = 𝑘 ) |
16 |
11
|
oveqd |
⊢ ( 𝑤 = 𝐻 → ( 𝑦 ( -g ‘ 𝑤 ) 𝑥 ) = ( 𝑦 − 𝑥 ) ) |
17 |
14 15 16
|
oveq123d |
⊢ ( 𝑤 = 𝐻 → ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑦 ( -g ‘ 𝑤 ) 𝑥 ) ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) ) |
18 |
12 17
|
eqeq12d |
⊢ ( 𝑤 = 𝐻 → ( ( 𝑧 ( -g ‘ 𝑤 ) 𝑥 ) = ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑦 ( -g ‘ 𝑤 ) 𝑥 ) ) ↔ ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) ) ) |
19 |
18
|
rexbidv |
⊢ ( 𝑤 = 𝐻 → ( ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 ( -g ‘ 𝑤 ) 𝑥 ) = ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑦 ( -g ‘ 𝑤 ) 𝑥 ) ) ↔ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) ) ) |
20 |
9 19
|
rabeqbidv |
⊢ ( 𝑤 = 𝐻 → { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 ( -g ‘ 𝑤 ) 𝑥 ) = ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑦 ( -g ‘ 𝑤 ) 𝑥 ) ) } = { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) |
21 |
9 9 20
|
mpoeq123dv |
⊢ ( 𝑤 = 𝐻 → ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 ( -g ‘ 𝑤 ) 𝑥 ) = ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑦 ( -g ‘ 𝑤 ) 𝑥 ) ) } ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ) |
22 |
21
|
csbeq1d |
⊢ ( 𝑤 = 𝐻 → ⦋ ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 ( -g ‘ 𝑤 ) 𝑥 ) = ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑦 ( -g ‘ 𝑤 ) 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝑤 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) = ⦋ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝑤 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) ) |
23 |
|
oveq1 |
⊢ ( 𝑤 = 𝐻 → ( 𝑤 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) = ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) ) |
24 |
9
|
rabeqdv |
⊢ ( 𝑤 = 𝐻 → { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } = { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) |
25 |
9 9 24
|
mpoeq123dv |
⊢ ( 𝑤 = 𝐻 → ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) ) |
26 |
25
|
opeq2d |
⊢ ( 𝑤 = 𝐻 → 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 = 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) |
27 |
23 26
|
oveq12d |
⊢ ( 𝑤 = 𝐻 → ( ( 𝑤 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) = ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) ) |
28 |
27
|
csbeq2dv |
⊢ ( 𝑤 = 𝐻 → ⦋ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝑤 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) = ⦋ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) ) |
29 |
22 28
|
eqtrd |
⊢ ( 𝑤 = 𝐻 → ⦋ ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 ( -g ‘ 𝑤 ) 𝑥 ) = ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑦 ( -g ‘ 𝑤 ) 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝑤 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) = ⦋ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) ) |
30 |
|
df-ttg |
⊢ toTG = ( 𝑤 ∈ V ↦ ⦋ ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 ( -g ‘ 𝑤 ) 𝑥 ) = ( 𝑘 ( ·𝑠 ‘ 𝑤 ) ( 𝑦 ( -g ‘ 𝑤 ) 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝑤 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑤 ) , 𝑦 ∈ ( Base ‘ 𝑤 ) ↦ { 𝑧 ∈ ( Base ‘ 𝑤 ) ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) ) |
31 |
|
ovex |
⊢ ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) ∈ V |
32 |
31
|
csbex |
⊢ ⦋ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) ∈ V |
33 |
29 30 32
|
fvmpt |
⊢ ( 𝐻 ∈ V → ( toTG ‘ 𝐻 ) = ⦋ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) ) |
34 |
7 33
|
syl |
⊢ ( 𝐻 ∈ 𝑉 → ( toTG ‘ 𝐻 ) = ⦋ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) ) |
35 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
36 |
35 35
|
mpoex |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ∈ V |
37 |
36
|
a1i |
⊢ ( 𝐻 ∈ 𝑉 → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ∈ V ) |
38 |
|
simpr |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ) → 𝑖 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ) |
39 |
|
oveq2 |
⊢ ( 𝑎 = 𝑥 → ( 𝑐 − 𝑎 ) = ( 𝑐 − 𝑥 ) ) |
40 |
|
oveq2 |
⊢ ( 𝑎 = 𝑥 → ( 𝑏 − 𝑎 ) = ( 𝑏 − 𝑥 ) ) |
41 |
40
|
oveq2d |
⊢ ( 𝑎 = 𝑥 → ( 𝑘 · ( 𝑏 − 𝑎 ) ) = ( 𝑘 · ( 𝑏 − 𝑥 ) ) ) |
42 |
39 41
|
eqeq12d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) ↔ ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑏 − 𝑥 ) ) ) ) |
43 |
42
|
rexbidv |
⊢ ( 𝑎 = 𝑥 → ( ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) ↔ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑏 − 𝑥 ) ) ) ) |
44 |
43
|
rabbidv |
⊢ ( 𝑎 = 𝑥 → { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } = { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑏 − 𝑥 ) ) } ) |
45 |
|
oveq1 |
⊢ ( 𝑏 = 𝑦 → ( 𝑏 − 𝑥 ) = ( 𝑦 − 𝑥 ) ) |
46 |
45
|
oveq2d |
⊢ ( 𝑏 = 𝑦 → ( 𝑘 · ( 𝑏 − 𝑥 ) ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) ) |
47 |
46
|
eqeq2d |
⊢ ( 𝑏 = 𝑦 → ( ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑏 − 𝑥 ) ) ↔ ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) ) ) |
48 |
47
|
rexbidv |
⊢ ( 𝑏 = 𝑦 → ( ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑏 − 𝑥 ) ) ↔ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) ) ) |
49 |
48
|
rabbidv |
⊢ ( 𝑏 = 𝑦 → { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑏 − 𝑥 ) ) } = { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) |
50 |
|
oveq1 |
⊢ ( 𝑐 = 𝑧 → ( 𝑐 − 𝑥 ) = ( 𝑧 − 𝑥 ) ) |
51 |
50
|
eqeq1d |
⊢ ( 𝑐 = 𝑧 → ( ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) ↔ ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) ) ) |
52 |
51
|
rexbidv |
⊢ ( 𝑐 = 𝑧 → ( ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) ↔ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) ) ) |
53 |
52
|
cbvrabv |
⊢ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } = { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } |
54 |
49 53
|
eqtrdi |
⊢ ( 𝑏 = 𝑦 → { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑥 ) = ( 𝑘 · ( 𝑏 − 𝑥 ) ) } = { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) |
55 |
44 54
|
cbvmpov |
⊢ ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) |
56 |
38 55
|
eqtr4di |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ) → 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) |
57 |
|
simpr |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) |
58 |
57 55
|
eqtrdi |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → 𝑖 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ) |
59 |
58
|
opeq2d |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → 〈 ( Itv ‘ ndx ) , 𝑖 〉 = 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) |
60 |
59
|
oveq2d |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) = ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) ) |
61 |
58
|
oveqd |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → ( 𝑥 𝑖 𝑦 ) = ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ) |
62 |
61
|
eleq2d |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ↔ 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ) ) |
63 |
58
|
oveqd |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → ( 𝑧 𝑖 𝑦 ) = ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ) |
64 |
63
|
eleq2d |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → ( 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ↔ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ) ) |
65 |
58
|
oveqd |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → ( 𝑥 𝑖 𝑧 ) = ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) |
66 |
65
|
eleq2d |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → ( 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ↔ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) ) |
67 |
62 64 66
|
3orbi123d |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → ( ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) ↔ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) ) ) |
68 |
67
|
rabbidv |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } = { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) |
69 |
68
|
mpoeq3dv |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) ) |
70 |
69
|
opeq2d |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 = 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) 〉 ) |
71 |
60 70
|
oveq12d |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑎 ∈ 𝐵 , 𝑏 ∈ 𝐵 ↦ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑐 − 𝑎 ) = ( 𝑘 · ( 𝑏 − 𝑎 ) ) } ) ) → ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) = ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) 〉 ) ) |
72 |
56 71
|
syldan |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ 𝑖 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ) → ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) = ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) 〉 ) ) |
73 |
37 72
|
csbied |
⊢ ( 𝐻 ∈ 𝑉 → ⦋ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) / 𝑖 ⦌ ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , 𝑖 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) 〉 ) = ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) 〉 ) ) |
74 |
6 34 73
|
3eqtrd |
⊢ ( 𝐻 ∈ 𝑉 → 𝐺 = ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) 〉 ) ) |
75 |
74
|
fveq2d |
⊢ ( 𝐻 ∈ 𝑉 → ( Itv ‘ 𝐺 ) = ( Itv ‘ ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) 〉 ) ) ) |
76 |
|
itvid |
⊢ Itv = Slot ( Itv ‘ ndx ) |
77 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
78 |
|
6nn |
⊢ 6 ∈ ℕ |
79 |
77 78
|
decnncl |
⊢ ; 1 6 ∈ ℕ |
80 |
79
|
nnrei |
⊢ ; 1 6 ∈ ℝ |
81 |
|
6nn0 |
⊢ 6 ∈ ℕ0 |
82 |
|
7nn |
⊢ 7 ∈ ℕ |
83 |
|
6lt7 |
⊢ 6 < 7 |
84 |
77 81 82 83
|
declt |
⊢ ; 1 6 < ; 1 7 |
85 |
80 84
|
ltneii |
⊢ ; 1 6 ≠ ; 1 7 |
86 |
|
itvndx |
⊢ ( Itv ‘ ndx ) = ; 1 6 |
87 |
|
lngndx |
⊢ ( LineG ‘ ndx ) = ; 1 7 |
88 |
86 87
|
neeq12i |
⊢ ( ( Itv ‘ ndx ) ≠ ( LineG ‘ ndx ) ↔ ; 1 6 ≠ ; 1 7 ) |
89 |
85 88
|
mpbir |
⊢ ( Itv ‘ ndx ) ≠ ( LineG ‘ ndx ) |
90 |
76 89
|
setsnid |
⊢ ( Itv ‘ ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) ) = ( Itv ‘ ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) 〉 ) ) |
91 |
75 90
|
eqtr4di |
⊢ ( 𝐻 ∈ 𝑉 → ( Itv ‘ 𝐺 ) = ( Itv ‘ ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) ) ) |
92 |
5
|
a1i |
⊢ ( 𝐻 ∈ 𝑉 → 𝐼 = ( Itv ‘ 𝐺 ) ) |
93 |
76
|
setsid |
⊢ ( ( 𝐻 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ∈ V ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) = ( Itv ‘ ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) ) ) |
94 |
36 93
|
mpan2 |
⊢ ( 𝐻 ∈ 𝑉 → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) = ( Itv ‘ ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) ) ) |
95 |
91 92 94
|
3eqtr4d |
⊢ ( 𝐻 ∈ 𝑉 → 𝐼 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ) |
96 |
95
|
oveqd |
⊢ ( 𝐻 ∈ 𝑉 → ( 𝑥 𝐼 𝑦 ) = ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ) |
97 |
96
|
eleq2d |
⊢ ( 𝐻 ∈ 𝑉 → ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ↔ 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ) ) |
98 |
95
|
oveqd |
⊢ ( 𝐻 ∈ 𝑉 → ( 𝑧 𝐼 𝑦 ) = ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ) |
99 |
98
|
eleq2d |
⊢ ( 𝐻 ∈ 𝑉 → ( 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ↔ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ) ) |
100 |
95
|
oveqd |
⊢ ( 𝐻 ∈ 𝑉 → ( 𝑥 𝐼 𝑧 ) = ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) |
101 |
100
|
eleq2d |
⊢ ( 𝐻 ∈ 𝑉 → ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ↔ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) ) |
102 |
97 99 101
|
3orbi123d |
⊢ ( 𝐻 ∈ 𝑉 → ( ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ↔ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) ) ) |
103 |
102
|
rabbidv |
⊢ ( 𝐻 ∈ 𝑉 → { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) } = { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) |
104 |
103
|
mpoeq3dv |
⊢ ( 𝐻 ∈ 𝑉 → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) } ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) ) |
105 |
104
|
opeq2d |
⊢ ( 𝐻 ∈ 𝑉 → 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) } ) 〉 = 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) 〉 ) |
106 |
105
|
oveq2d |
⊢ ( 𝐻 ∈ 𝑉 → ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) } ) 〉 ) = ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 𝑧 ) ) } ) 〉 ) ) |
107 |
74 106
|
eqtr4d |
⊢ ( 𝐻 ∈ 𝑉 → 𝐺 = ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) } ) 〉 ) ) |
108 |
107 95
|
jca |
⊢ ( 𝐻 ∈ 𝑉 → ( 𝐺 = ( ( 𝐻 sSet 〈 ( Itv ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) 〉 ) sSet 〈 ( LineG ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) } ) 〉 ) ∧ 𝐼 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑧 ∈ 𝐵 ∣ ∃ 𝑘 ∈ ( 0 [,] 1 ) ( 𝑧 − 𝑥 ) = ( 𝑘 · ( 𝑦 − 𝑥 ) ) } ) ) ) |