Step |
Hyp |
Ref |
Expression |
1 |
|
gsumncl.k |
⊢ 𝐾 = ( Base ‘ 𝑀 ) |
2 |
|
gsumncl.w |
⊢ ( 𝜑 → 𝑀 ∈ Mnd ) |
3 |
|
gsumncl.p |
⊢ ( 𝜑 → 𝑃 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
4 |
|
gsumncl.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑁 ... 𝑃 ) ) → 𝐵 ∈ 𝐾 ) |
5 |
|
gsumnunsn.a |
⊢ + = ( +g ‘ 𝑀 ) |
6 |
|
gsumnunsn.l |
⊢ ( 𝜑 → 𝐶 ∈ 𝐾 ) |
7 |
|
gsumnunsn.c |
⊢ ( ( 𝜑 ∧ 𝑘 = ( 𝑃 + 1 ) ) → 𝐵 = 𝐶 ) |
8 |
|
seqp1 |
⊢ ( 𝑃 ∈ ( ℤ≥ ‘ 𝑁 ) → ( seq 𝑁 ( + , ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) ‘ ( 𝑃 + 1 ) ) = ( ( seq 𝑁 ( + , ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) ‘ 𝑃 ) + ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ‘ ( 𝑃 + 1 ) ) ) ) |
9 |
3 8
|
syl |
⊢ ( 𝜑 → ( seq 𝑁 ( + , ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) ‘ ( 𝑃 + 1 ) ) = ( ( seq 𝑁 ( + , ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) ‘ 𝑃 ) + ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ‘ ( 𝑃 + 1 ) ) ) ) |
10 |
|
peano2uz |
⊢ ( 𝑃 ∈ ( ℤ≥ ‘ 𝑁 ) → ( 𝑃 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
11 |
3 10
|
syl |
⊢ ( 𝜑 → ( 𝑃 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
12 |
4
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ) ∧ 𝑘 ∈ ( 𝑁 ... 𝑃 ) ) → 𝐵 ∈ 𝐾 ) |
13 |
7
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ) ∧ 𝑘 = ( 𝑃 + 1 ) ) → 𝐵 = 𝐶 ) |
14 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ) ∧ 𝑘 = ( 𝑃 + 1 ) ) → 𝐶 ∈ 𝐾 ) |
15 |
13 14
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ) ∧ 𝑘 = ( 𝑃 + 1 ) ) → 𝐵 ∈ 𝐾 ) |
16 |
|
elfzp1 |
⊢ ( 𝑃 ∈ ( ℤ≥ ‘ 𝑁 ) → ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↔ ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ∨ 𝑘 = ( 𝑃 + 1 ) ) ) ) |
17 |
3 16
|
syl |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↔ ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ∨ 𝑘 = ( 𝑃 + 1 ) ) ) ) |
18 |
17
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ) → ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ∨ 𝑘 = ( 𝑃 + 1 ) ) ) |
19 |
12 15 18
|
mpjaodan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ) → 𝐵 ∈ 𝐾 ) |
20 |
19
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) : ( 𝑁 ... ( 𝑃 + 1 ) ) ⟶ 𝐾 ) |
21 |
1 5 2 11 20
|
gsumval2 |
⊢ ( 𝜑 → ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) = ( seq 𝑁 ( + , ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) ‘ ( 𝑃 + 1 ) ) ) |
22 |
4
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) : ( 𝑁 ... 𝑃 ) ⟶ 𝐾 ) |
23 |
1 5 2 3 22
|
gsumval2 |
⊢ ( 𝜑 → ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) ) = ( seq 𝑁 ( + , ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) ) ‘ 𝑃 ) ) |
24 |
|
fvres |
⊢ ( 𝑖 ∈ ( 𝑁 ... 𝑃 ) → ( ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ↾ ( 𝑁 ... 𝑃 ) ) ‘ 𝑖 ) = ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ‘ 𝑖 ) ) |
25 |
24
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ... 𝑃 ) ) → ( ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ↾ ( 𝑁 ... 𝑃 ) ) ‘ 𝑖 ) = ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ‘ 𝑖 ) ) |
26 |
|
fzssp1 |
⊢ ( 𝑁 ... 𝑃 ) ⊆ ( 𝑁 ... ( 𝑃 + 1 ) ) |
27 |
|
resmpt |
⊢ ( ( 𝑁 ... 𝑃 ) ⊆ ( 𝑁 ... ( 𝑃 + 1 ) ) → ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ↾ ( 𝑁 ... 𝑃 ) ) = ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) ) |
28 |
26 27
|
ax-mp |
⊢ ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ↾ ( 𝑁 ... 𝑃 ) ) = ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) |
29 |
28
|
fveq1i |
⊢ ( ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ↾ ( 𝑁 ... 𝑃 ) ) ‘ 𝑖 ) = ( ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) ‘ 𝑖 ) |
30 |
25 29
|
eqtr3di |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ... 𝑃 ) ) → ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ‘ 𝑖 ) = ( ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) ‘ 𝑖 ) ) |
31 |
3 30
|
seqfveq |
⊢ ( 𝜑 → ( seq 𝑁 ( + , ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) ‘ 𝑃 ) = ( seq 𝑁 ( + , ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) ) ‘ 𝑃 ) ) |
32 |
23 31
|
eqtr4d |
⊢ ( 𝜑 → ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) ) = ( seq 𝑁 ( + , ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) ‘ 𝑃 ) ) |
33 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) = ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) |
34 |
|
eluzfz2 |
⊢ ( ( 𝑃 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) → ( 𝑃 + 1 ) ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ) |
35 |
11 34
|
syl |
⊢ ( 𝜑 → ( 𝑃 + 1 ) ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ) |
36 |
33 7 35 6
|
fvmptd |
⊢ ( 𝜑 → ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ‘ ( 𝑃 + 1 ) ) = 𝐶 ) |
37 |
36
|
eqcomd |
⊢ ( 𝜑 → 𝐶 = ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ‘ ( 𝑃 + 1 ) ) ) |
38 |
32 37
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) ) + 𝐶 ) = ( ( seq 𝑁 ( + , ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) ‘ 𝑃 ) + ( ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ‘ ( 𝑃 + 1 ) ) ) ) |
39 |
9 21 38
|
3eqtr4d |
⊢ ( 𝜑 → ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ... ( 𝑃 + 1 ) ) ↦ 𝐵 ) ) = ( ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ... 𝑃 ) ↦ 𝐵 ) ) + 𝐶 ) ) |