Step |
Hyp |
Ref |
Expression |
1 |
|
iccpartiun.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
2 |
|
iccpartiun.p |
⊢ ( 𝜑 → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
3 |
|
iccelpart |
⊢ ( 𝑀 ∈ ℕ → ∀ 𝑝 ∈ ( RePart ‘ 𝑀 ) ( 𝑥 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
4 |
|
fveq1 |
⊢ ( 𝑝 = 𝑃 → ( 𝑝 ‘ 0 ) = ( 𝑃 ‘ 0 ) ) |
5 |
|
fveq1 |
⊢ ( 𝑝 = 𝑃 → ( 𝑝 ‘ 𝑀 ) = ( 𝑃 ‘ 𝑀 ) ) |
6 |
4 5
|
oveq12d |
⊢ ( 𝑝 = 𝑃 → ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) = ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) ) |
7 |
6
|
eleq2d |
⊢ ( 𝑝 = 𝑃 → ( 𝑥 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) ↔ 𝑥 ∈ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) ) ) |
8 |
|
fveq1 |
⊢ ( 𝑝 = 𝑃 → ( 𝑝 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) |
9 |
|
fveq1 |
⊢ ( 𝑝 = 𝑃 → ( 𝑝 ‘ ( 𝑖 + 1 ) ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) |
10 |
8 9
|
oveq12d |
⊢ ( 𝑝 = 𝑃 → ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) |
11 |
10
|
eleq2d |
⊢ ( 𝑝 = 𝑃 → ( 𝑥 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) ) |
12 |
11
|
rexbidv |
⊢ ( 𝑝 = 𝑃 → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) ) |
13 |
7 12
|
imbi12d |
⊢ ( 𝑝 = 𝑃 → ( ( 𝑥 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( 𝑥 ∈ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
14 |
13
|
rspcva |
⊢ ( ( 𝑃 ∈ ( RePart ‘ 𝑀 ) ∧ ∀ 𝑝 ∈ ( RePart ‘ 𝑀 ) ( 𝑥 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( 𝑥 ∈ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) ) |
15 |
14
|
expcom |
⊢ ( ∀ 𝑝 ∈ ( RePart ‘ 𝑀 ) ( 𝑥 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑃 ∈ ( RePart ‘ 𝑀 ) → ( 𝑥 ∈ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
16 |
1 3 15
|
3syl |
⊢ ( 𝜑 → ( 𝑃 ∈ ( RePart ‘ 𝑀 ) → ( 𝑥 ∈ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
17 |
2 16
|
mpd |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) ) |
18 |
|
nnnn0 |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℕ0 ) |
19 |
|
0elfz |
⊢ ( 𝑀 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑀 ) ) |
20 |
1 18 19
|
3syl |
⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑀 ) ) |
21 |
1 2 20
|
iccpartxr |
⊢ ( 𝜑 → ( 𝑃 ‘ 0 ) ∈ ℝ* ) |
22 |
|
nn0fz0 |
⊢ ( 𝑀 ∈ ℕ0 ↔ 𝑀 ∈ ( 0 ... 𝑀 ) ) |
23 |
22
|
biimpi |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ( 0 ... 𝑀 ) ) |
24 |
1 18 23
|
3syl |
⊢ ( 𝜑 → 𝑀 ∈ ( 0 ... 𝑀 ) ) |
25 |
1 2 24
|
iccpartxr |
⊢ ( 𝜑 → ( 𝑃 ‘ 𝑀 ) ∈ ℝ* ) |
26 |
21 25
|
jca |
⊢ ( 𝜑 → ( ( 𝑃 ‘ 0 ) ∈ ℝ* ∧ ( 𝑃 ‘ 𝑀 ) ∈ ℝ* ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑃 ‘ 0 ) ∈ ℝ* ∧ ( 𝑃 ‘ 𝑀 ) ∈ ℝ* ) ) |
28 |
|
elfzofz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
29 |
1 2
|
iccpartgel |
⊢ ( 𝜑 → ∀ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑃 ‘ 0 ) ≤ ( 𝑃 ‘ 𝑗 ) ) |
30 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 𝑖 ) ) |
31 |
30
|
breq2d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝑃 ‘ 0 ) ≤ ( 𝑃 ‘ 𝑗 ) ↔ ( 𝑃 ‘ 0 ) ≤ ( 𝑃 ‘ 𝑖 ) ) ) |
32 |
31
|
rspcva |
⊢ ( ( 𝑖 ∈ ( 0 ... 𝑀 ) ∧ ∀ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑃 ‘ 0 ) ≤ ( 𝑃 ‘ 𝑗 ) ) → ( 𝑃 ‘ 0 ) ≤ ( 𝑃 ‘ 𝑖 ) ) |
33 |
28 29 32
|
syl2anr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑃 ‘ 0 ) ≤ ( 𝑃 ‘ 𝑖 ) ) |
34 |
|
fzofzp1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
35 |
1 2
|
iccpartleu |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑃 ‘ 𝑘 ) ≤ ( 𝑃 ‘ 𝑀 ) ) |
36 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑖 + 1 ) → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) |
37 |
36
|
breq1d |
⊢ ( 𝑘 = ( 𝑖 + 1 ) → ( ( 𝑃 ‘ 𝑘 ) ≤ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑃 ‘ ( 𝑖 + 1 ) ) ≤ ( 𝑃 ‘ 𝑀 ) ) ) |
38 |
37
|
rspcva |
⊢ ( ( ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ∧ ∀ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑃 ‘ 𝑘 ) ≤ ( 𝑃 ‘ 𝑀 ) ) → ( 𝑃 ‘ ( 𝑖 + 1 ) ) ≤ ( 𝑃 ‘ 𝑀 ) ) |
39 |
34 35 38
|
syl2anr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑃 ‘ ( 𝑖 + 1 ) ) ≤ ( 𝑃 ‘ 𝑀 ) ) |
40 |
|
icossico |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ* ∧ ( 𝑃 ‘ 𝑀 ) ∈ ℝ* ) ∧ ( ( 𝑃 ‘ 0 ) ≤ ( 𝑃 ‘ 𝑖 ) ∧ ( 𝑃 ‘ ( 𝑖 + 1 ) ) ≤ ( 𝑃 ‘ 𝑀 ) ) ) → ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) ) |
41 |
27 33 39 40
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) ) |
42 |
41
|
sseld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) ) ) |
43 |
42
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) ) ) |
44 |
17 43
|
impbid |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) ) |
45 |
|
eliun |
⊢ ( 𝑥 ∈ ∪ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑥 ∈ ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) |
46 |
44 45
|
bitr4di |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) ↔ 𝑥 ∈ ∪ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) ) |
47 |
46
|
eqrdv |
⊢ ( 𝜑 → ( ( 𝑃 ‘ 0 ) [,) ( 𝑃 ‘ 𝑀 ) ) = ∪ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( ( 𝑃 ‘ 𝑖 ) [,) ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) |