Step |
Hyp |
Ref |
Expression |
1 |
|
iccpartgtprec.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
2 |
|
iccpartgtprec.p |
⊢ ( 𝜑 → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
3 |
|
nnnn0 |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℕ0 ) |
4 |
|
elnn0uz |
⊢ ( 𝑀 ∈ ℕ0 ↔ 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
5 |
3 4
|
sylib |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
6 |
1 5
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
7 |
|
fzisfzounsn |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 0 ) → ( 0 ... 𝑀 ) = ( ( 0 ..^ 𝑀 ) ∪ { 𝑀 } ) ) |
8 |
6 7
|
syl |
⊢ ( 𝜑 → ( 0 ... 𝑀 ) = ( ( 0 ..^ 𝑀 ) ∪ { 𝑀 } ) ) |
9 |
8
|
eleq2d |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ... 𝑀 ) ↔ 𝑖 ∈ ( ( 0 ..^ 𝑀 ) ∪ { 𝑀 } ) ) ) |
10 |
|
elun |
⊢ ( 𝑖 ∈ ( ( 0 ..^ 𝑀 ) ∪ { 𝑀 } ) ↔ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∨ 𝑖 ∈ { 𝑀 } ) ) |
11 |
10
|
a1i |
⊢ ( 𝜑 → ( 𝑖 ∈ ( ( 0 ..^ 𝑀 ) ∪ { 𝑀 } ) ↔ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∨ 𝑖 ∈ { 𝑀 } ) ) ) |
12 |
|
velsn |
⊢ ( 𝑖 ∈ { 𝑀 } ↔ 𝑖 = 𝑀 ) |
13 |
12
|
a1i |
⊢ ( 𝜑 → ( 𝑖 ∈ { 𝑀 } ↔ 𝑖 = 𝑀 ) ) |
14 |
13
|
orbi2d |
⊢ ( 𝜑 → ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∨ 𝑖 ∈ { 𝑀 } ) ↔ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∨ 𝑖 = 𝑀 ) ) ) |
15 |
9 11 14
|
3bitrd |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ... 𝑀 ) ↔ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∨ 𝑖 = 𝑀 ) ) ) |
16 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑀 ∈ ℕ ) |
17 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
18 |
|
fzossfz |
⊢ ( 0 ..^ 𝑀 ) ⊆ ( 0 ... 𝑀 ) |
19 |
18
|
a1i |
⊢ ( 𝜑 → ( 0 ..^ 𝑀 ) ⊆ ( 0 ... 𝑀 ) ) |
20 |
19
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
21 |
16 17 20
|
iccpartxr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) ∈ ℝ* ) |
22 |
|
nn0fz0 |
⊢ ( 𝑀 ∈ ℕ0 ↔ 𝑀 ∈ ( 0 ... 𝑀 ) ) |
23 |
3 22
|
sylib |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ( 0 ... 𝑀 ) ) |
24 |
1 23
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ( 0 ... 𝑀 ) ) |
25 |
1 2 24
|
iccpartxr |
⊢ ( 𝜑 → ( 𝑃 ‘ 𝑀 ) ∈ ℝ* ) |
26 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑀 ) ∈ ℝ* ) |
27 |
1 2
|
iccpartltu |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ 𝑀 ) ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ 𝑀 ) ) |
28 |
|
fveq2 |
⊢ ( 𝑘 = 𝑖 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 𝑖 ) ) |
29 |
28
|
breq1d |
⊢ ( 𝑘 = 𝑖 → ( ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
30 |
29
|
rspccv |
⊢ ( ∀ 𝑘 ∈ ( 0 ..^ 𝑀 ) ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ 𝑀 ) → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
31 |
27 30
|
syl |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
32 |
31
|
imp |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) |
33 |
21 26 32
|
xrltled |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) ≤ ( 𝑃 ‘ 𝑀 ) ) |
34 |
33
|
expcom |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝜑 → ( 𝑃 ‘ 𝑖 ) ≤ ( 𝑃 ‘ 𝑀 ) ) ) |
35 |
|
fveq2 |
⊢ ( 𝑖 = 𝑀 → ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑀 ) ) |
36 |
35
|
adantr |
⊢ ( ( 𝑖 = 𝑀 ∧ 𝜑 ) → ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑀 ) ) |
37 |
25
|
xrleidd |
⊢ ( 𝜑 → ( 𝑃 ‘ 𝑀 ) ≤ ( 𝑃 ‘ 𝑀 ) ) |
38 |
37
|
adantl |
⊢ ( ( 𝑖 = 𝑀 ∧ 𝜑 ) → ( 𝑃 ‘ 𝑀 ) ≤ ( 𝑃 ‘ 𝑀 ) ) |
39 |
36 38
|
eqbrtrd |
⊢ ( ( 𝑖 = 𝑀 ∧ 𝜑 ) → ( 𝑃 ‘ 𝑖 ) ≤ ( 𝑃 ‘ 𝑀 ) ) |
40 |
39
|
ex |
⊢ ( 𝑖 = 𝑀 → ( 𝜑 → ( 𝑃 ‘ 𝑖 ) ≤ ( 𝑃 ‘ 𝑀 ) ) ) |
41 |
34 40
|
jaoi |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∨ 𝑖 = 𝑀 ) → ( 𝜑 → ( 𝑃 ‘ 𝑖 ) ≤ ( 𝑃 ‘ 𝑀 ) ) ) |
42 |
41
|
com12 |
⊢ ( 𝜑 → ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∨ 𝑖 = 𝑀 ) → ( 𝑃 ‘ 𝑖 ) ≤ ( 𝑃 ‘ 𝑀 ) ) ) |
43 |
15 42
|
sylbid |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ... 𝑀 ) → ( 𝑃 ‘ 𝑖 ) ≤ ( 𝑃 ‘ 𝑀 ) ) ) |
44 |
43
|
ralrimiv |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ... 𝑀 ) ( 𝑃 ‘ 𝑖 ) ≤ ( 𝑃 ‘ 𝑀 ) ) |