Step |
Hyp |
Ref |
Expression |
1 |
|
axcontlem7.1 |
⊢ 𝐷 = { 𝑝 ∈ ( 𝔼 ‘ 𝑁 ) ∣ ( 𝑈 Btwn 〈 𝑍 , 𝑝 〉 ∨ 𝑝 Btwn 〈 𝑍 , 𝑈 〉 ) } |
2 |
|
axcontlem7.2 |
⊢ 𝐹 = { 〈 𝑥 , 𝑡 〉 ∣ ( 𝑥 ∈ 𝐷 ∧ ( 𝑡 ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑥 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑈 ‘ 𝑖 ) ) ) ) ) } |
3 |
1
|
ssrab3 |
⊢ 𝐷 ⊆ ( 𝔼 ‘ 𝑁 ) |
4 |
3
|
sseli |
⊢ ( 𝑃 ∈ 𝐷 → 𝑃 ∈ ( 𝔼 ‘ 𝑁 ) ) |
5 |
4
|
ad2antrl |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ) ) → 𝑃 ∈ ( 𝔼 ‘ 𝑁 ) ) |
6 |
|
simpll2 |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ) ) → 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ) |
7 |
3
|
sseli |
⊢ ( 𝑄 ∈ 𝐷 → 𝑄 ∈ ( 𝔼 ‘ 𝑁 ) ) |
8 |
7
|
ad2antll |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ) ) → 𝑄 ∈ ( 𝔼 ‘ 𝑁 ) ) |
9 |
|
brbtwn |
⊢ ( ( 𝑃 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑄 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝑃 Btwn 〈 𝑍 , 𝑄 〉 ↔ ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ) ) |
10 |
5 6 8 9
|
syl3anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ) ) → ( 𝑃 Btwn 〈 𝑍 , 𝑄 〉 ↔ ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ) ) |
11 |
1 2
|
axcontlem6 |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ 𝑃 ∈ 𝐷 ) → ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) |
12 |
1 2
|
axcontlem6 |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ 𝑄 ∈ 𝐷 ) → ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) |
13 |
11 12
|
anim12dan |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ) ) → ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) |
14 |
|
an4 |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) |
15 |
|
r19.26 |
⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ↔ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) |
16 |
15
|
anbi2i |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) |
17 |
14 16
|
bitr4i |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) |
18 |
|
id |
⊢ ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) → ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) |
19 |
|
oveq2 |
⊢ ( ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) → ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) = ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) |
20 |
19
|
oveq2d |
⊢ ( ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) → ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) |
21 |
18 20
|
eqeqan12d |
⊢ ( ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) ) |
22 |
21
|
ralimi |
⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) ) |
23 |
|
ralbi |
⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) ) |
24 |
22 23
|
syl |
⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) ) |
25 |
24
|
rexbidv |
⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ↔ ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) ) |
26 |
|
simpll2 |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ) |
27 |
|
fveecn |
⊢ ( ( 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑍 ‘ 𝑖 ) ∈ ℂ ) |
28 |
26 27
|
sylan |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑍 ‘ 𝑖 ) ∈ ℂ ) |
29 |
|
simpll3 |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) |
30 |
|
fveecn |
⊢ ( ( 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) |
31 |
29 30
|
sylan |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) |
32 |
|
elicc01 |
⊢ ( 𝑡 ∈ ( 0 [,] 1 ) ↔ ( 𝑡 ∈ ℝ ∧ 0 ≤ 𝑡 ∧ 𝑡 ≤ 1 ) ) |
33 |
32
|
simp1bi |
⊢ ( 𝑡 ∈ ( 0 [,] 1 ) → 𝑡 ∈ ℝ ) |
34 |
33
|
recnd |
⊢ ( 𝑡 ∈ ( 0 [,] 1 ) → 𝑡 ∈ ℂ ) |
35 |
34
|
ad2antll |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → 𝑡 ∈ ℂ ) |
36 |
35
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → 𝑡 ∈ ℂ ) |
37 |
|
elrege0 |
⊢ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ↔ ( ( 𝐹 ‘ 𝑃 ) ∈ ℝ ∧ 0 ≤ ( 𝐹 ‘ 𝑃 ) ) ) |
38 |
37
|
simplbi |
⊢ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) → ( 𝐹 ‘ 𝑃 ) ∈ ℝ ) |
39 |
38
|
recnd |
⊢ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) |
40 |
39
|
adantr |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) |
41 |
40
|
ad2antrl |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) |
42 |
41
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) |
43 |
|
elrege0 |
⊢ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ↔ ( ( 𝐹 ‘ 𝑄 ) ∈ ℝ ∧ 0 ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
44 |
43
|
simplbi |
⊢ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) → ( 𝐹 ‘ 𝑄 ) ∈ ℝ ) |
45 |
44
|
recnd |
⊢ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) → ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) |
46 |
45
|
adantl |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) |
47 |
46
|
ad2antrl |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) |
48 |
47
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) |
49 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
50 |
|
simpr1 |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → 𝑡 ∈ ℂ ) |
51 |
|
simpr3 |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) |
52 |
50 51
|
mulcld |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ) |
53 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ) → ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ∈ ℂ ) |
54 |
49 52 53
|
sylancr |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ∈ ℂ ) |
55 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) → ( 1 − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ ) |
56 |
49 55
|
mpan |
⊢ ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ → ( 1 − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ ) |
57 |
56
|
3ad2ant2 |
⊢ ( ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( 1 − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ ) |
58 |
57
|
adantl |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 1 − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ ) |
59 |
|
simpll |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 𝑍 ‘ 𝑖 ) ∈ ℂ ) |
60 |
54 58 59
|
subdird |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) − ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) − ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) |
61 |
|
simpr2 |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) |
62 |
|
nnncan1 |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) → ( ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) − ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
63 |
49 52 61 62
|
mp3an2i |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) − ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
64 |
63
|
oveq1d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) − ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) ) |
65 |
|
subdi |
⊢ ( ( 𝑡 ∈ ℂ ∧ 1 ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) = ( ( 𝑡 · 1 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
66 |
49 65
|
mp3an2 |
⊢ ( ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) = ( ( 𝑡 · 1 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
67 |
|
mulid1 |
⊢ ( 𝑡 ∈ ℂ → ( 𝑡 · 1 ) = 𝑡 ) |
68 |
67
|
adantr |
⊢ ( ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( 𝑡 · 1 ) = 𝑡 ) |
69 |
68
|
oveq1d |
⊢ ( ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( ( 𝑡 · 1 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = ( 𝑡 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
70 |
66 69
|
eqtrd |
⊢ ( ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) = ( 𝑡 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
71 |
50 51 70
|
syl2anc |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) = ( 𝑡 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
72 |
71
|
oveq2d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 1 − 𝑡 ) + ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) ) = ( ( 1 − 𝑡 ) + ( 𝑡 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) ) |
73 |
|
npncan |
⊢ ( ( 1 ∈ ℂ ∧ 𝑡 ∈ ℂ ∧ ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ) → ( ( 1 − 𝑡 ) + ( 𝑡 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) = ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
74 |
49 50 52 73
|
mp3an2i |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 1 − 𝑡 ) + ( 𝑡 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) = ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
75 |
72 74
|
eqtr2d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = ( ( 1 − 𝑡 ) + ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) ) ) |
76 |
75
|
oveq1d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 1 − 𝑡 ) + ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) ) · ( 𝑍 ‘ 𝑖 ) ) ) |
77 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ 𝑡 ∈ ℂ ) → ( 1 − 𝑡 ) ∈ ℂ ) |
78 |
49 77
|
mpan |
⊢ ( 𝑡 ∈ ℂ → ( 1 − 𝑡 ) ∈ ℂ ) |
79 |
78
|
3ad2ant1 |
⊢ ( ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( 1 − 𝑡 ) ∈ ℂ ) |
80 |
79
|
adantl |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 1 − 𝑡 ) ∈ ℂ ) |
81 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( 1 − ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ) |
82 |
49 81
|
mpan |
⊢ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ → ( 1 − ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ) |
83 |
82
|
3ad2ant3 |
⊢ ( ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( 1 − ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ) |
84 |
83
|
adantl |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 1 − ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ) |
85 |
50 84
|
mulcld |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) ∈ ℂ ) |
86 |
80 85 59
|
adddird |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 1 − 𝑡 ) + ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) |
87 |
50 84 59
|
mulassd |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) |
88 |
87
|
oveq2d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝑡 · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) ) |
89 |
76 86 88
|
3eqtrd |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) ) |
90 |
89
|
oveq1d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 1 − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) − ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) = ( ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) − ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) |
91 |
60 64 90
|
3eqtr3d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) − ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) |
92 |
|
simplr |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) |
93 |
61 52 92
|
subdird |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) − ( ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) · ( 𝑈 ‘ 𝑖 ) ) ) ) |
94 |
50 51 92
|
mulassd |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) · ( 𝑈 ‘ 𝑖 ) ) = ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) |
95 |
94
|
oveq2d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) − ( ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) − ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) |
96 |
93 95
|
eqtrd |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) − ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) |
97 |
91 96
|
eqeq12d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) ↔ ( ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) − ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) = ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) − ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) |
98 |
58 59
|
mulcld |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ∈ ℂ ) |
99 |
61 92
|
mulcld |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ∈ ℂ ) |
100 |
80 59
|
mulcld |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) ∈ ℂ ) |
101 |
84 59
|
mulcld |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ∈ ℂ ) |
102 |
50 101
|
mulcld |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ∈ ℂ ) |
103 |
100 102
|
addcld |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) ∈ ℂ ) |
104 |
51 92
|
mulcld |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ∈ ℂ ) |
105 |
50 104
|
mulcld |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∈ ℂ ) |
106 |
98 99 103 105
|
addsubeq4d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ↔ ( ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) − ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) = ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) − ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) |
107 |
100 102 105
|
addassd |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) |
108 |
50 101 104
|
adddid |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) |
109 |
108
|
oveq2d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) |
110 |
107 109
|
eqtr4d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) |
111 |
110
|
eqeq2d |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) ) |
112 |
97 106 111
|
3bitr2rd |
⊢ ( ( ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ∧ ( 𝑡 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) ) → ( ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) ) ) |
113 |
28 31 36 42 48 112
|
syl23anc |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) ) ) |
114 |
113
|
ralbidva |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) ) ) |
115 |
36 48
|
mulcld |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ) |
116 |
42 115
|
subcld |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ∈ ℂ ) |
117 |
|
mulcan1g |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ∈ ℂ ∧ ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) → ( ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) ) |
118 |
116 28 31 117
|
syl3anc |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) ) |
119 |
118
|
ralbidva |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) ) |
120 |
|
r19.32v |
⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
121 |
|
simplr |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → 𝑍 ≠ 𝑈 ) |
122 |
121
|
neneqd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ¬ 𝑍 = 𝑈 ) |
123 |
|
biorf |
⊢ ( ¬ 𝑍 = 𝑈 → ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ↔ ( 𝑍 = 𝑈 ∨ ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ) ) ) |
124 |
|
orcom |
⊢ ( ( 𝑍 = 𝑈 ∨ ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ 𝑍 = 𝑈 ) ) |
125 |
123 124
|
bitrdi |
⊢ ( ¬ 𝑍 = 𝑈 → ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ↔ ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ 𝑍 = 𝑈 ) ) ) |
126 |
122 125
|
syl |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ↔ ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ 𝑍 = 𝑈 ) ) ) |
127 |
35 47
|
mulcld |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ) |
128 |
41 127
|
subeq0ad |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ↔ ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
129 |
|
eqeefv |
⊢ ( ( 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝑍 = 𝑈 ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
130 |
129
|
3adant1 |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝑍 = 𝑈 ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
131 |
130
|
adantr |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) → ( 𝑍 = 𝑈 ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
132 |
131
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( 𝑍 = 𝑈 ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
133 |
132
|
orbi2d |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ 𝑍 = 𝑈 ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) ) |
134 |
126 128 133
|
3bitr3rd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ↔ ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
135 |
120 134
|
syl5bb |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐹 ‘ 𝑃 ) − ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) = 0 ∨ ( 𝑍 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ↔ ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
136 |
114 119 135
|
3bitrd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
137 |
136
|
anassrs |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
138 |
137
|
rexbidva |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
139 |
33
|
adantl |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) → 𝑡 ∈ ℝ ) |
140 |
|
1red |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) → 1 ∈ ℝ ) |
141 |
43
|
biimpi |
⊢ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) → ( ( 𝐹 ‘ 𝑄 ) ∈ ℝ ∧ 0 ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
142 |
141
|
ad2antlr |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) → ( ( 𝐹 ‘ 𝑄 ) ∈ ℝ ∧ 0 ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
143 |
32
|
simp3bi |
⊢ ( 𝑡 ∈ ( 0 [,] 1 ) → 𝑡 ≤ 1 ) |
144 |
143
|
adantl |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) → 𝑡 ≤ 1 ) |
145 |
|
lemul1a |
⊢ ( ( ( 𝑡 ∈ ℝ ∧ 1 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℝ ∧ 0 ≤ ( 𝐹 ‘ 𝑄 ) ) ) ∧ 𝑡 ≤ 1 ) → ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ≤ ( 1 · ( 𝐹 ‘ 𝑄 ) ) ) |
146 |
139 140 142 144 145
|
syl31anc |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) → ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ≤ ( 1 · ( 𝐹 ‘ 𝑄 ) ) ) |
147 |
45
|
ad2antlr |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) |
148 |
147
|
mulid2d |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) → ( 1 · ( 𝐹 ‘ 𝑄 ) ) = ( 𝐹 ‘ 𝑄 ) ) |
149 |
146 148
|
breqtrd |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) → ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ≤ ( 𝐹 ‘ 𝑄 ) ) |
150 |
|
breq1 |
⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) → ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ↔ ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
151 |
149 150
|
syl5ibrcom |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ 𝑡 ∈ ( 0 [,] 1 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
152 |
151
|
rexlimdva |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
153 |
|
0elunit |
⊢ 0 ∈ ( 0 [,] 1 ) |
154 |
|
simpl |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) = 0 ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( 𝐹 ‘ 𝑃 ) = 0 ) |
155 |
45
|
mul02d |
⊢ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) → ( 0 · ( 𝐹 ‘ 𝑄 ) ) = 0 ) |
156 |
155
|
adantl |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) = 0 ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( 0 · ( 𝐹 ‘ 𝑄 ) ) = 0 ) |
157 |
154 156
|
eqtr4d |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) = 0 ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( 𝐹 ‘ 𝑃 ) = ( 0 · ( 𝐹 ‘ 𝑄 ) ) ) |
158 |
|
oveq1 |
⊢ ( 𝑡 = 0 → ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) = ( 0 · ( 𝐹 ‘ 𝑄 ) ) ) |
159 |
158
|
rspceeqv |
⊢ ( ( 0 ∈ ( 0 [,] 1 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 0 · ( 𝐹 ‘ 𝑄 ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) |
160 |
153 157 159
|
sylancr |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) = 0 ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) |
161 |
160
|
adantrl |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) = 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) |
162 |
161
|
a1d |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) = 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
163 |
162
|
ex |
⊢ ( ( 𝐹 ‘ 𝑃 ) = 0 → ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) ) |
164 |
|
simp3 |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) |
165 |
38
|
adantr |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℝ ) |
166 |
165
|
3ad2ant2 |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℝ ) |
167 |
37
|
simprbi |
⊢ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) → 0 ≤ ( 𝐹 ‘ 𝑃 ) ) |
168 |
167
|
adantr |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → 0 ≤ ( 𝐹 ‘ 𝑃 ) ) |
169 |
168
|
3ad2ant2 |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → 0 ≤ ( 𝐹 ‘ 𝑃 ) ) |
170 |
44
|
adantl |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℝ ) |
171 |
170
|
3ad2ant2 |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℝ ) |
172 |
|
0red |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → 0 ∈ ℝ ) |
173 |
|
simp1 |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) ≠ 0 ) |
174 |
166 169 173
|
ne0gt0d |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → 0 < ( 𝐹 ‘ 𝑃 ) ) |
175 |
172 166 171 174 164
|
ltletrd |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → 0 < ( 𝐹 ‘ 𝑄 ) ) |
176 |
|
divelunit |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℝ ∧ 0 ≤ ( 𝐹 ‘ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℝ ∧ 0 < ( 𝐹 ‘ 𝑄 ) ) ) → ( ( ( 𝐹 ‘ 𝑃 ) / ( 𝐹 ‘ 𝑄 ) ) ∈ ( 0 [,] 1 ) ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
177 |
166 169 171 175 176
|
syl22anc |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( ( ( 𝐹 ‘ 𝑃 ) / ( 𝐹 ‘ 𝑄 ) ) ∈ ( 0 [,] 1 ) ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
178 |
164 177
|
mpbird |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( ( 𝐹 ‘ 𝑃 ) / ( 𝐹 ‘ 𝑄 ) ) ∈ ( 0 [,] 1 ) ) |
179 |
40
|
3ad2ant2 |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) |
180 |
46
|
3ad2ant2 |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) |
181 |
175
|
gt0ne0d |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( 𝐹 ‘ 𝑄 ) ≠ 0 ) |
182 |
179 180 181
|
divcan1d |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( ( ( 𝐹 ‘ 𝑃 ) / ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑄 ) ) = ( 𝐹 ‘ 𝑃 ) ) |
183 |
182
|
eqcomd |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) = ( ( ( 𝐹 ‘ 𝑃 ) / ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑄 ) ) ) |
184 |
|
oveq1 |
⊢ ( 𝑡 = ( ( 𝐹 ‘ 𝑃 ) / ( 𝐹 ‘ 𝑄 ) ) → ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) / ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑄 ) ) ) |
185 |
184
|
rspceeqv |
⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) / ( 𝐹 ‘ 𝑄 ) ) ∈ ( 0 [,] 1 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( ( ( 𝐹 ‘ 𝑃 ) / ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑄 ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) |
186 |
178 183 185
|
syl2anc |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ 0 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) |
187 |
186
|
3exp |
⊢ ( ( 𝐹 ‘ 𝑃 ) ≠ 0 → ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) ) |
188 |
163 187
|
pm2.61ine |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ) ) |
189 |
152 188
|
impbid |
⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
190 |
189
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ( 𝐹 ‘ 𝑃 ) = ( 𝑡 · ( 𝐹 ‘ 𝑄 ) ) ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
191 |
138 190
|
bitrd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
192 |
25 191
|
sylan9bbr |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
193 |
192
|
anasss |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
194 |
17 193
|
sylan2b |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
195 |
13 194
|
syldan |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑄 ‘ 𝑖 ) ) ) ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |
196 |
10 195
|
bitrd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ) ) → ( 𝑃 Btwn 〈 𝑍 , 𝑄 〉 ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) ) |