Step |
Hyp |
Ref |
Expression |
1 |
|
fsumre.1 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
2 |
|
fsumre.2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
3 |
|
fsumrelem.3 |
⊢ 𝐹 : ℂ ⟶ ℂ |
4 |
|
fsumrelem.4 |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) |
5 |
|
0cn |
⊢ 0 ∈ ℂ |
6 |
3
|
ffvelrni |
⊢ ( 0 ∈ ℂ → ( 𝐹 ‘ 0 ) ∈ ℂ ) |
7 |
5 6
|
ax-mp |
⊢ ( 𝐹 ‘ 0 ) ∈ ℂ |
8 |
7
|
addid1i |
⊢ ( ( 𝐹 ‘ 0 ) + 0 ) = ( 𝐹 ‘ 0 ) |
9 |
|
fvoveq1 |
⊢ ( 𝑥 = 0 → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( 𝐹 ‘ ( 0 + 𝑦 ) ) ) |
10 |
|
fveq2 |
⊢ ( 𝑥 = 0 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 0 ) ) |
11 |
10
|
oveq1d |
⊢ ( 𝑥 = 0 → ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 0 ) + ( 𝐹 ‘ 𝑦 ) ) ) |
12 |
9 11
|
eqeq12d |
⊢ ( 𝑥 = 0 → ( ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( 0 + 𝑦 ) ) = ( ( 𝐹 ‘ 0 ) + ( 𝐹 ‘ 𝑦 ) ) ) ) |
13 |
|
oveq2 |
⊢ ( 𝑦 = 0 → ( 0 + 𝑦 ) = ( 0 + 0 ) ) |
14 |
|
00id |
⊢ ( 0 + 0 ) = 0 |
15 |
13 14
|
eqtrdi |
⊢ ( 𝑦 = 0 → ( 0 + 𝑦 ) = 0 ) |
16 |
15
|
fveq2d |
⊢ ( 𝑦 = 0 → ( 𝐹 ‘ ( 0 + 𝑦 ) ) = ( 𝐹 ‘ 0 ) ) |
17 |
|
fveq2 |
⊢ ( 𝑦 = 0 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 0 ) ) |
18 |
17
|
oveq2d |
⊢ ( 𝑦 = 0 → ( ( 𝐹 ‘ 0 ) + ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 0 ) + ( 𝐹 ‘ 0 ) ) ) |
19 |
16 18
|
eqeq12d |
⊢ ( 𝑦 = 0 → ( ( 𝐹 ‘ ( 0 + 𝑦 ) ) = ( ( 𝐹 ‘ 0 ) + ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ 0 ) = ( ( 𝐹 ‘ 0 ) + ( 𝐹 ‘ 0 ) ) ) ) |
20 |
12 19 4
|
vtocl2ga |
⊢ ( ( 0 ∈ ℂ ∧ 0 ∈ ℂ ) → ( 𝐹 ‘ 0 ) = ( ( 𝐹 ‘ 0 ) + ( 𝐹 ‘ 0 ) ) ) |
21 |
5 5 20
|
mp2an |
⊢ ( 𝐹 ‘ 0 ) = ( ( 𝐹 ‘ 0 ) + ( 𝐹 ‘ 0 ) ) |
22 |
8 21
|
eqtr2i |
⊢ ( ( 𝐹 ‘ 0 ) + ( 𝐹 ‘ 0 ) ) = ( ( 𝐹 ‘ 0 ) + 0 ) |
23 |
7 7 5
|
addcani |
⊢ ( ( ( 𝐹 ‘ 0 ) + ( 𝐹 ‘ 0 ) ) = ( ( 𝐹 ‘ 0 ) + 0 ) ↔ ( 𝐹 ‘ 0 ) = 0 ) |
24 |
22 23
|
mpbi |
⊢ ( 𝐹 ‘ 0 ) = 0 |
25 |
|
sumeq1 |
⊢ ( 𝐴 = ∅ → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ ∅ 𝐵 ) |
26 |
|
sum0 |
⊢ Σ 𝑘 ∈ ∅ 𝐵 = 0 |
27 |
25 26
|
eqtrdi |
⊢ ( 𝐴 = ∅ → Σ 𝑘 ∈ 𝐴 𝐵 = 0 ) |
28 |
27
|
fveq2d |
⊢ ( 𝐴 = ∅ → ( 𝐹 ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = ( 𝐹 ‘ 0 ) ) |
29 |
|
sumeq1 |
⊢ ( 𝐴 = ∅ → Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝐵 ) = Σ 𝑘 ∈ ∅ ( 𝐹 ‘ 𝐵 ) ) |
30 |
|
sum0 |
⊢ Σ 𝑘 ∈ ∅ ( 𝐹 ‘ 𝐵 ) = 0 |
31 |
29 30
|
eqtrdi |
⊢ ( 𝐴 = ∅ → Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝐵 ) = 0 ) |
32 |
24 28 31
|
3eqtr4a |
⊢ ( 𝐴 = ∅ → ( 𝐹 ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝐵 ) ) |
33 |
32
|
a1i |
⊢ ( 𝜑 → ( 𝐴 = ∅ → ( 𝐹 ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝐵 ) ) ) |
34 |
|
addcl |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + 𝑦 ) ∈ ℂ ) |
35 |
34
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 + 𝑦 ) ∈ ℂ ) |
36 |
2
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) |
37 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) |
38 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) |
39 |
|
f1of |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
40 |
38 39
|
syl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
41 |
|
fco |
⊢ ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ ℂ ) |
42 |
37 40 41
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ ℂ ) |
43 |
42
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑥 ) ∈ ℂ ) |
44 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ℕ ) |
45 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
46 |
44 45
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 1 ) ) |
47 |
4
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) + ( 𝐹 ‘ 𝑦 ) ) ) |
48 |
40
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( 𝑓 ‘ 𝑥 ) ∈ 𝐴 ) |
49 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ 𝐴 ) |
50 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) |
51 |
50
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = 𝐵 ) |
52 |
49 2 51
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = 𝐵 ) |
53 |
52
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) ) = ( 𝐹 ‘ 𝐵 ) ) |
54 |
|
fvex |
⊢ ( 𝐹 ‘ 𝐵 ) ∈ V |
55 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) = ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) |
56 |
55
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐵 ) ∈ V ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑘 ) = ( 𝐹 ‘ 𝐵 ) ) |
57 |
49 54 56
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑘 ) = ( 𝐹 ‘ 𝐵 ) ) |
58 |
53 57
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑘 ) ) |
59 |
58
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑘 ) ) |
60 |
59
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ∀ 𝑘 ∈ 𝐴 ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑘 ) ) |
61 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐹 |
62 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) |
63 |
61 62
|
nffv |
⊢ Ⅎ 𝑘 ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) |
64 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ ( 𝑓 ‘ 𝑥 ) ) |
65 |
63 64
|
nfeq |
⊢ Ⅎ 𝑘 ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ ( 𝑓 ‘ 𝑥 ) ) |
66 |
|
2fveq3 |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑥 ) → ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) ) = ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) ) |
67 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑥 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) |
68 |
66 67
|
eqeq12d |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑥 ) → ( ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑘 ) ↔ ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) ) |
69 |
65 68
|
rspc |
⊢ ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐴 → ( ∀ 𝑘 ∈ 𝐴 ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑘 ) → ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) ) |
70 |
48 60 69
|
sylc |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) |
71 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) |
72 |
40 71
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) |
73 |
72
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( 𝐹 ‘ ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑥 ) ) = ( 𝐹 ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) ) |
74 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ∘ 𝑓 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) |
75 |
40 74
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ∘ 𝑓 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) |
76 |
70 73 75
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( 𝐹 ‘ ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑥 ) ) = ( ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ∘ 𝑓 ) ‘ 𝑥 ) ) |
77 |
35 43 46 47 76
|
seqhomo |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝐹 ‘ ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) = ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
78 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑓 ‘ 𝑥 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) |
79 |
37
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) ∈ ℂ ) |
80 |
78 44 38 79 72
|
fsum |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → Σ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
81 |
80
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝐹 ‘ Σ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) ) = ( 𝐹 ‘ ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) ) |
82 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑓 ‘ 𝑥 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑚 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ ( 𝑓 ‘ 𝑥 ) ) ) |
83 |
3
|
ffvelrni |
⊢ ( 𝐵 ∈ ℂ → ( 𝐹 ‘ 𝐵 ) ∈ ℂ ) |
84 |
2 83
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐵 ) ∈ ℂ ) |
85 |
84
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) : 𝐴 ⟶ ℂ ) |
86 |
85
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) : 𝐴 ⟶ ℂ ) |
87 |
86
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑚 ) ∈ ℂ ) |
88 |
82 44 38 87 75
|
fsum |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → Σ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑚 ) = ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
89 |
77 81 88
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝐹 ‘ Σ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) ) = Σ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑚 ) ) |
90 |
|
sumfc |
⊢ Σ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = Σ 𝑘 ∈ 𝐴 𝐵 |
91 |
90
|
fveq2i |
⊢ ( 𝐹 ‘ Σ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) ) = ( 𝐹 ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) |
92 |
|
sumfc |
⊢ Σ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ‘ 𝑚 ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝐵 ) |
93 |
89 91 92
|
3eqtr3g |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝐹 ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝐵 ) ) |
94 |
93
|
expr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ( 𝐹 ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝐵 ) ) ) |
95 |
94
|
exlimdv |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ( 𝐹 ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝐵 ) ) ) |
96 |
95
|
expimpd |
⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( 𝐹 ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝐵 ) ) ) |
97 |
|
fz1f1o |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ) |
98 |
1 97
|
syl |
⊢ ( 𝜑 → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ) |
99 |
33 96 98
|
mpjaod |
⊢ ( 𝜑 → ( 𝐹 ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝐵 ) ) |