Step |
Hyp |
Ref |
Expression |
1 |
|
uzp1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 − 1 ) ) → ( 𝑁 = ( 𝑀 − 1 ) ∨ 𝑁 ∈ ( ℤ≥ ‘ ( ( 𝑀 − 1 ) + 1 ) ) ) ) |
2 |
|
zcn |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℂ ) |
3 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
4 |
|
npcan |
⊢ ( ( 𝑀 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑀 − 1 ) + 1 ) = 𝑀 ) |
5 |
2 3 4
|
sylancl |
⊢ ( 𝑀 ∈ ℤ → ( ( 𝑀 − 1 ) + 1 ) = 𝑀 ) |
6 |
5
|
oveq2d |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 ... ( ( 𝑀 − 1 ) + 1 ) ) = ( 𝑀 ... 𝑀 ) ) |
7 |
|
uncom |
⊢ ( ∅ ∪ { 𝑀 } ) = ( { 𝑀 } ∪ ∅ ) |
8 |
|
un0 |
⊢ ( { 𝑀 } ∪ ∅ ) = { 𝑀 } |
9 |
7 8
|
eqtri |
⊢ ( ∅ ∪ { 𝑀 } ) = { 𝑀 } |
10 |
|
zre |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℝ ) |
11 |
10
|
ltm1d |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 − 1 ) < 𝑀 ) |
12 |
|
peano2zm |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 − 1 ) ∈ ℤ ) |
13 |
|
fzn |
⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑀 − 1 ) ∈ ℤ ) → ( ( 𝑀 − 1 ) < 𝑀 ↔ ( 𝑀 ... ( 𝑀 − 1 ) ) = ∅ ) ) |
14 |
12 13
|
mpdan |
⊢ ( 𝑀 ∈ ℤ → ( ( 𝑀 − 1 ) < 𝑀 ↔ ( 𝑀 ... ( 𝑀 − 1 ) ) = ∅ ) ) |
15 |
11 14
|
mpbid |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 ... ( 𝑀 − 1 ) ) = ∅ ) |
16 |
5
|
sneqd |
⊢ ( 𝑀 ∈ ℤ → { ( ( 𝑀 − 1 ) + 1 ) } = { 𝑀 } ) |
17 |
15 16
|
uneq12d |
⊢ ( 𝑀 ∈ ℤ → ( ( 𝑀 ... ( 𝑀 − 1 ) ) ∪ { ( ( 𝑀 − 1 ) + 1 ) } ) = ( ∅ ∪ { 𝑀 } ) ) |
18 |
|
fzsn |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 ... 𝑀 ) = { 𝑀 } ) |
19 |
9 17 18
|
3eqtr4a |
⊢ ( 𝑀 ∈ ℤ → ( ( 𝑀 ... ( 𝑀 − 1 ) ) ∪ { ( ( 𝑀 − 1 ) + 1 ) } ) = ( 𝑀 ... 𝑀 ) ) |
20 |
6 19
|
eqtr4d |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 ... ( ( 𝑀 − 1 ) + 1 ) ) = ( ( 𝑀 ... ( 𝑀 − 1 ) ) ∪ { ( ( 𝑀 − 1 ) + 1 ) } ) ) |
21 |
|
oveq1 |
⊢ ( 𝑁 = ( 𝑀 − 1 ) → ( 𝑁 + 1 ) = ( ( 𝑀 − 1 ) + 1 ) ) |
22 |
21
|
oveq2d |
⊢ ( 𝑁 = ( 𝑀 − 1 ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( 𝑀 ... ( ( 𝑀 − 1 ) + 1 ) ) ) |
23 |
|
oveq2 |
⊢ ( 𝑁 = ( 𝑀 − 1 ) → ( 𝑀 ... 𝑁 ) = ( 𝑀 ... ( 𝑀 − 1 ) ) ) |
24 |
21
|
sneqd |
⊢ ( 𝑁 = ( 𝑀 − 1 ) → { ( 𝑁 + 1 ) } = { ( ( 𝑀 − 1 ) + 1 ) } ) |
25 |
23 24
|
uneq12d |
⊢ ( 𝑁 = ( 𝑀 − 1 ) → ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) = ( ( 𝑀 ... ( 𝑀 − 1 ) ) ∪ { ( ( 𝑀 − 1 ) + 1 ) } ) ) |
26 |
22 25
|
eqeq12d |
⊢ ( 𝑁 = ( 𝑀 − 1 ) → ( ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ↔ ( 𝑀 ... ( ( 𝑀 − 1 ) + 1 ) ) = ( ( 𝑀 ... ( 𝑀 − 1 ) ) ∪ { ( ( 𝑀 − 1 ) + 1 ) } ) ) ) |
27 |
20 26
|
syl5ibrcom |
⊢ ( 𝑀 ∈ ℤ → ( 𝑁 = ( 𝑀 − 1 ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ) ) |
28 |
27
|
imp |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 = ( 𝑀 − 1 ) ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ) |
29 |
5
|
fveq2d |
⊢ ( 𝑀 ∈ ℤ → ( ℤ≥ ‘ ( ( 𝑀 − 1 ) + 1 ) ) = ( ℤ≥ ‘ 𝑀 ) ) |
30 |
29
|
eleq2d |
⊢ ( 𝑀 ∈ ℤ → ( 𝑁 ∈ ( ℤ≥ ‘ ( ( 𝑀 − 1 ) + 1 ) ) ↔ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ) |
31 |
30
|
biimpa |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ( ℤ≥ ‘ ( ( 𝑀 − 1 ) + 1 ) ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
32 |
|
fzsuc |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ) |
33 |
31 32
|
syl |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ( ℤ≥ ‘ ( ( 𝑀 − 1 ) + 1 ) ) ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ) |
34 |
28 33
|
jaodan |
⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑁 = ( 𝑀 − 1 ) ∨ 𝑁 ∈ ( ℤ≥ ‘ ( ( 𝑀 − 1 ) + 1 ) ) ) ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ) |
35 |
1 34
|
sylan2 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 − 1 ) ) ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ) |