Step | Hyp | Ref
| Expression |
1 | | fsummulc2.2 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ ℂ) |
2 | 1 | mul01d 8263 |
. . 3
⊢ (𝜑 → (𝐶 · 0) = 0) |
3 | | sumeq1 11247 |
. . . . . 6
⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵) |
4 | | sum0 11280 |
. . . . . 6
⊢
Σ𝑘 ∈
∅ 𝐵 =
0 |
5 | 3, 4 | eqtrdi 2206 |
. . . . 5
⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
6 | 5 | oveq2d 5837 |
. . . 4
⊢ (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = (𝐶 · 0)) |
7 | | sumeq1 11247 |
. . . . 5
⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵)) |
8 | | sum0 11280 |
. . . . 5
⊢
Σ𝑘 ∈
∅ (𝐶 · 𝐵) = 0 |
9 | 7, 8 | eqtrdi 2206 |
. . . 4
⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = 0) |
10 | 6, 9 | eqeq12d 2172 |
. . 3
⊢ (𝐴 = ∅ → ((𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) ↔ (𝐶 · 0) = 0)) |
11 | 2, 10 | syl5ibrcom 156 |
. 2
⊢ (𝜑 → (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))) |
12 | | addcl 7852 |
. . . . . . . . 9
⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 + 𝑣) ∈ ℂ) |
13 | 12 | adantl 275 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 + 𝑣) ∈ ℂ) |
14 | 1 | ad2antrr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝐶 ∈ ℂ) |
15 | | simprl 521 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑢 ∈ ℂ) |
16 | | simprr 522 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑣 ∈ ℂ) |
17 | 14, 15, 16 | adddid 7897 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝐶 · (𝑢 + 𝑣)) = ((𝐶 · 𝑢) + (𝐶 · 𝑣))) |
18 | | simprl 521 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈
ℕ) |
19 | | nnuz 9469 |
. . . . . . . . 9
⊢ ℕ =
(ℤ≥‘1) |
20 | 18, 19 | eleqtrdi 2250 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈
(ℤ≥‘1)) |
21 | | elnnuz 9470 |
. . . . . . . . . . . 12
⊢ (𝑢 ∈ ℕ ↔ 𝑢 ∈
(ℤ≥‘1)) |
22 | 21 | biimpri 132 |
. . . . . . . . . . 11
⊢ (𝑢 ∈
(ℤ≥‘1) → 𝑢 ∈ ℕ) |
23 | 22 | adantl 275 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ 𝑢 ∈
ℕ) |
24 | | f1of 5413 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴) |
25 | 24 | ad2antll 483 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴) |
26 | 25 | ad2antrr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
𝑓:(1...(♯‘𝐴))⟶𝐴) |
27 | | 1zzd 9189 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) → 1
∈ ℤ) |
28 | 18 | ad2antrr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(♯‘𝐴) ∈
ℕ) |
29 | 28 | nnzd 9280 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(♯‘𝐴) ∈
ℤ) |
30 | | eluzelz 9443 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 ∈
(ℤ≥‘1) → 𝑢 ∈ ℤ) |
31 | 30 | ad2antlr 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
𝑢 ∈
ℤ) |
32 | 27, 29, 31 | 3jca 1162 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ)) |
33 | | eluzle 9446 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 ∈
(ℤ≥‘1) → 1 ≤ 𝑢) |
34 | 33 | ad2antlr 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) → 1
≤ 𝑢) |
35 | | simpr 109 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
𝑢 ≤ (♯‘𝐴)) |
36 | 34, 35 | jca 304 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(1 ≤ 𝑢 ∧ 𝑢 ≤ (♯‘𝐴))) |
37 | | elfz2 9914 |
. . . . . . . . . . . . . 14
⊢ (𝑢 ∈
(1...(♯‘𝐴))
↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ) ∧ (1 ≤ 𝑢 ∧ 𝑢 ≤ (♯‘𝐴)))) |
38 | 32, 36, 37 | sylanbrc 414 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
𝑢 ∈
(1...(♯‘𝐴))) |
39 | | fvco3 5538 |
. . . . . . . . . . . . 13
⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑢 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢))) |
40 | 26, 38, 39 | syl2anc 409 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢))) |
41 | 26, 38 | ffvelrnd 5602 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(𝑓‘𝑢) ∈ 𝐴) |
42 | | fsummulc2.3 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
43 | 42 | ralrimiva 2530 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
44 | 43 | ad3antrrr 484 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
45 | | nfcsb1v 3064 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘⦋(𝑓‘𝑢) / 𝑘⦌𝐵 |
46 | 45 | nfel1 2310 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ |
47 | | csbeq1a 3040 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = (𝑓‘𝑢) → 𝐵 = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵) |
48 | 47 | eleq1d 2226 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (𝑓‘𝑢) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ)) |
49 | 46, 48 | rspc 2810 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓‘𝑢) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ)) |
50 | 41, 44, 49 | sylc 62 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ) |
51 | | eqid 2157 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) |
52 | 51 | fvmpts 5545 |
. . . . . . . . . . . . . 14
⊢ (((𝑓‘𝑢) ∈ 𝐴 ∧ ⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢)) = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵) |
53 | 41, 50, 52 | syl2anc 409 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢)) = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵) |
54 | 53, 50 | eqeltrd 2234 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢)) ∈ ℂ) |
55 | 40, 54 | eqeltrd 2234 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢) ∈ ℂ) |
56 | | 0cnd 7866 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ ¬ 𝑢 ≤
(♯‘𝐴)) → 0
∈ ℂ) |
57 | 23 | nnzd 9280 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ 𝑢 ∈
ℤ) |
58 | 18 | adantr 274 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ (♯‘𝐴)
∈ ℕ) |
59 | 58 | nnzd 9280 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ (♯‘𝐴)
∈ ℤ) |
60 | | zdcle 9235 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ ℤ ∧
(♯‘𝐴) ∈
ℤ) → DECID 𝑢 ≤ (♯‘𝐴)) |
61 | 57, 59, 60 | syl2anc 409 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ DECID 𝑢 ≤ (♯‘𝐴)) |
62 | 55, 56, 61 | ifcldadc 3534 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0) ∈ ℂ) |
63 | | breq1 3968 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑢 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑢 ≤ (♯‘𝐴))) |
64 | | fveq2 5467 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑢 → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢)) |
65 | 63, 64 | ifbieq1d 3527 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)) |
66 | | eqid 2157 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)) |
67 | 65, 66 | fvmptg 5543 |
. . . . . . . . . 10
⊢ ((𝑢 ∈ ℕ ∧ if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)) |
68 | 23, 62, 67 | syl2anc 409 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ ((𝑛 ∈ ℕ
↦ if(𝑛 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)) |
69 | 68, 62 | eqeltrd 2234 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ ((𝑛 ∈ ℕ
↦ if(𝑛 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢) ∈ ℂ) |
70 | | csbov2g 5859 |
. . . . . . . . . . . 12
⊢ ((𝑓‘𝑢) ∈ 𝐴 → ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵) = (𝐶 · ⦋(𝑓‘𝑢) / 𝑘⦌𝐵)) |
71 | 41, 70 | syl 14 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵) = (𝐶 · ⦋(𝑓‘𝑢) / 𝑘⦌𝐵)) |
72 | 35 | iftrued 3512 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢)) |
73 | | fvco3 5538 |
. . . . . . . . . . . . 13
⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑢 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑢))) |
74 | 26, 38, 73 | syl2anc 409 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑢))) |
75 | 1 | ad3antrrr 484 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
𝐶 ∈
ℂ) |
76 | 75, 50 | mulcld 7893 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(𝐶 ·
⦋(𝑓‘𝑢) / 𝑘⦌𝐵) ∈ ℂ) |
77 | 71, 76 | eqeltrd 2234 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵) ∈ ℂ) |
78 | | eqid 2157 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) |
79 | 78 | fvmpts 5545 |
. . . . . . . . . . . . 13
⊢ (((𝑓‘𝑢) ∈ 𝐴 ∧ ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵) ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑢)) = ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵)) |
80 | 41, 77, 79 | syl2anc 409 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑢)) = ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵)) |
81 | 72, 74, 80 | 3eqtrd 2194 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵)) |
82 | 35 | iftrued 3512 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢)) |
83 | 82, 40, 53 | 3eqtrd 2194 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0) = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵) |
84 | 83 | oveq2d 5837 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)) = (𝐶 · ⦋(𝑓‘𝑢) / 𝑘⦌𝐵)) |
85 | 71, 81, 84 | 3eqtr4d 2200 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0))) |
86 | 1 | ad3antrrr 484 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ ¬ 𝑢 ≤
(♯‘𝐴)) →
𝐶 ∈
ℂ) |
87 | 86 | mul01d 8263 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ ¬ 𝑢 ≤
(♯‘𝐴)) →
(𝐶 · 0) =
0) |
88 | | simpr 109 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ ¬ 𝑢 ≤
(♯‘𝐴)) →
¬ 𝑢 ≤
(♯‘𝐴)) |
89 | 88 | iffalsed 3515 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ ¬ 𝑢 ≤
(♯‘𝐴)) →
if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0) = 0) |
90 | 89 | oveq2d 5837 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ ¬ 𝑢 ≤
(♯‘𝐴)) →
(𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)) = (𝐶 · 0)) |
91 | 88 | iffalsed 3515 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ ¬ 𝑢 ≤
(♯‘𝐴)) →
if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = 0) |
92 | 87, 90, 91 | 3eqtr4rd 2201 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ ¬ 𝑢 ≤
(♯‘𝐴)) →
if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0))) |
93 | | exmiddc 822 |
. . . . . . . . . . 11
⊢
(DECID 𝑢 ≤ (♯‘𝐴) → (𝑢 ≤ (♯‘𝐴) ∨ ¬ 𝑢 ≤ (♯‘𝐴))) |
94 | 61, 93 | syl 14 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ (𝑢 ≤
(♯‘𝐴) ∨
¬ 𝑢 ≤
(♯‘𝐴))) |
95 | 85, 92, 94 | mpjaodan 788 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0))) |
96 | 80, 77 | eqeltrd 2234 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑢)) ∈ ℂ) |
97 | 74, 96 | eqeltrd 2234 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
∧ 𝑢 ≤
(♯‘𝐴)) →
(((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) ∈ ℂ) |
98 | 97, 56, 61 | ifcldadc 3534 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ if(𝑢 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) ∈ ℂ) |
99 | | fveq2 5467 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑢 → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢)) |
100 | 63, 99 | ifbieq1d 3527 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0)) |
101 | | eqid 2157 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0)) |
102 | 100, 101 | fvmptg 5543 |
. . . . . . . . . 10
⊢ ((𝑢 ∈ ℕ ∧ if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0)) |
103 | 23, 98, 102 | syl2anc 409 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ ((𝑛 ∈ ℕ
↦ if(𝑛 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0)) |
104 | 68 | oveq2d 5837 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ (𝐶 · ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢)) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0))) |
105 | 95, 103, 104 | 3eqtr4d 2200 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ≥‘1))
→ ((𝑛 ∈ ℕ
↦ if(𝑛 ≤
(♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))‘𝑢) = (𝐶 · ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢))) |
106 | | mulcl 7854 |
. . . . . . . . 9
⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ) |
107 | 106 | adantl 275 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 · 𝑣) ∈ ℂ) |
108 | 1 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐶 ∈ ℂ) |
109 | 13, 17, 20, 69, 105, 107, 108 | seq3distr 10407 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)) = (𝐶 · (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)))) |
110 | | fveq2 5467 |
. . . . . . . 8
⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛))) |
111 | | simprr 522 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) |
112 | 1 | adantr 274 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
113 | 112, 42 | mulcld 7893 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ) |
114 | 113 | fmpttd 5621 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ) |
115 | 114 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ) |
116 | 115 | ffvelrnda 5601 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) ∈ ℂ) |
117 | | fvco3 5538 |
. . . . . . . . 9
⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛))) |
118 | 25, 117 | sylan 281 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛))) |
119 | 110, 18, 111, 116, 118 | fsum3 11279 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴))) |
120 | | fveq2 5467 |
. . . . . . . . 9
⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))) |
121 | 42 | fmpttd 5621 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
122 | 121 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
123 | 122 | ffvelrnda 5601 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ) |
124 | | fvco3 5538 |
. . . . . . . . . 10
⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))) |
125 | 25, 124 | sylan 281 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))) |
126 | 120, 18, 111, 123, 125 | fsum3 11279 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴))) |
127 | 126 | oveq2d 5837 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐶 · (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)))) |
128 | 109, 119,
127 | 3eqtr4rd 2201 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚)) |
129 | | sumfct 11266 |
. . . . . . . . 9
⊢
(∀𝑘 ∈
𝐴 𝐵 ∈ ℂ → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵) |
130 | 43, 129 | syl 14 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵) |
131 | 130 | oveq2d 5837 |
. . . . . . 7
⊢ (𝜑 → (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐶 · Σ𝑘 ∈ 𝐴 𝐵)) |
132 | 131 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐶 · Σ𝑘 ∈ 𝐴 𝐵)) |
133 | 113 | ralrimiva 2530 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 · 𝐵) ∈ ℂ) |
134 | | sumfct 11266 |
. . . . . . . 8
⊢
(∀𝑘 ∈
𝐴 (𝐶 · 𝐵) ∈ ℂ → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)) |
135 | 133, 134 | syl 14 |
. . . . . . 7
⊢ (𝜑 → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)) |
136 | 135 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)) |
137 | 128, 132,
136 | 3eqtr3d 2198 |
. . . . 5
⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)) |
138 | 137 | expr 373 |
. . . 4
⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))) |
139 | 138 | exlimdv 1799 |
. . 3
⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) →
(∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))) |
140 | 139 | expimpd 361 |
. 2
⊢ (𝜑 → (((♯‘𝐴) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))) |
141 | | fsummulc2.1 |
. . 3
⊢ (𝜑 → 𝐴 ∈ Fin) |
142 | | fz1f1o 11267 |
. . 3
⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨
((♯‘𝐴) ∈
ℕ ∧ ∃𝑓
𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) |
143 | 141, 142 | syl 14 |
. 2
⊢ (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) |
144 | 11, 140, 143 | mpjaod 708 |
1
⊢ (𝜑 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)) |