Step |
Hyp |
Ref |
Expression |
1 |
|
iccpartgtprec.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
2 |
|
iccpartgtprec.p |
⊢ ( 𝜑 → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
3 |
|
0zd |
⊢ ( 𝑀 ∈ ℕ → 0 ∈ ℤ ) |
4 |
|
nnz |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℤ ) |
5 |
|
nngt0 |
⊢ ( 𝑀 ∈ ℕ → 0 < 𝑀 ) |
6 |
3 4 5
|
3jca |
⊢ ( 𝑀 ∈ ℕ → ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) ) |
7 |
1 6
|
syl |
⊢ ( 𝜑 → ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) ) |
8 |
|
fzopred |
⊢ ( ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) → ( 0 ..^ 𝑀 ) = ( { 0 } ∪ ( ( 0 + 1 ) ..^ 𝑀 ) ) ) |
9 |
7 8
|
syl |
⊢ ( 𝜑 → ( 0 ..^ 𝑀 ) = ( { 0 } ∪ ( ( 0 + 1 ) ..^ 𝑀 ) ) ) |
10 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
11 |
10
|
a1i |
⊢ ( 𝜑 → ( 0 + 1 ) = 1 ) |
12 |
11
|
oveq1d |
⊢ ( 𝜑 → ( ( 0 + 1 ) ..^ 𝑀 ) = ( 1 ..^ 𝑀 ) ) |
13 |
12
|
uneq2d |
⊢ ( 𝜑 → ( { 0 } ∪ ( ( 0 + 1 ) ..^ 𝑀 ) ) = ( { 0 } ∪ ( 1 ..^ 𝑀 ) ) ) |
14 |
9 13
|
eqtrd |
⊢ ( 𝜑 → ( 0 ..^ 𝑀 ) = ( { 0 } ∪ ( 1 ..^ 𝑀 ) ) ) |
15 |
14
|
eleq2d |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ 𝑖 ∈ ( { 0 } ∪ ( 1 ..^ 𝑀 ) ) ) ) |
16 |
|
elun |
⊢ ( 𝑖 ∈ ( { 0 } ∪ ( 1 ..^ 𝑀 ) ) ↔ ( 𝑖 ∈ { 0 } ∨ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) |
17 |
|
elsni |
⊢ ( 𝑖 ∈ { 0 } → 𝑖 = 0 ) |
18 |
|
fveq2 |
⊢ ( 𝑖 = 0 → ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ 0 ) ) |
19 |
18
|
adantr |
⊢ ( ( 𝑖 = 0 ∧ 𝜑 ) → ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ 0 ) ) |
20 |
1 2
|
iccpartlt |
⊢ ( 𝜑 → ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑀 ) ) |
21 |
20
|
adantl |
⊢ ( ( 𝑖 = 0 ∧ 𝜑 ) → ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑀 ) ) |
22 |
19 21
|
eqbrtrd |
⊢ ( ( 𝑖 = 0 ∧ 𝜑 ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) |
23 |
22
|
ex |
⊢ ( 𝑖 = 0 → ( 𝜑 → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
24 |
17 23
|
syl |
⊢ ( 𝑖 ∈ { 0 } → ( 𝜑 → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
25 |
|
fveq2 |
⊢ ( 𝑘 = 𝑖 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 𝑖 ) ) |
26 |
25
|
breq1d |
⊢ ( 𝑘 = 𝑖 → ( ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
27 |
26
|
rspccv |
⊢ ( ∀ 𝑘 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ 𝑀 ) → ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
28 |
1 2
|
iccpartiltu |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ 𝑀 ) ) |
29 |
27 28
|
syl11 |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝜑 → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
30 |
24 29
|
jaoi |
⊢ ( ( 𝑖 ∈ { 0 } ∨ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝜑 → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
31 |
30
|
com12 |
⊢ ( 𝜑 → ( ( 𝑖 ∈ { 0 } ∨ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
32 |
16 31
|
syl5bi |
⊢ ( 𝜑 → ( 𝑖 ∈ ( { 0 } ∪ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
33 |
15 32
|
sylbid |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
34 |
33
|
ralrimiv |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) |