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Theorem fsummulc2 10829
Description: A finite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
fsummulc2.1 (𝜑𝐴 ∈ Fin)
fsummulc2.2 (𝜑𝐶 ∈ ℂ)
fsummulc2.3 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Assertion
Ref Expression
fsummulc2 (𝜑 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
Distinct variable groups:   𝐴,𝑘   𝐶,𝑘   𝜑,𝑘
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fsummulc2
Dummy variables 𝑓 𝑚 𝑛 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsummulc2.2 . . . 4 (𝜑𝐶 ∈ ℂ)
21mul01d 7861 . . 3 (𝜑 → (𝐶 · 0) = 0)
3 sumeq1 10731 . . . . . 6 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4 sum0 10767 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
53, 4syl6eq 2136 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = 0)
65oveq2d 5660 . . . 4 (𝐴 = ∅ → (𝐶 · Σ𝑘𝐴 𝐵) = (𝐶 · 0))
7 sumeq1 10731 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵))
8 sum0 10767 . . . . 5 Σ𝑘 ∈ ∅ (𝐶 · 𝐵) = 0
97, 8syl6eq 2136 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐶 · 𝐵) = 0)
106, 9eqeq12d 2102 . . 3 (𝐴 = ∅ → ((𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵) ↔ (𝐶 · 0) = 0))
112, 10syl5ibrcom 155 . 2 (𝜑 → (𝐴 = ∅ → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
12 addcl 7457 . . . . . . . . 9 ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 + 𝑣) ∈ ℂ)
1312adantl 271 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 + 𝑣) ∈ ℂ)
141ad2antrr 472 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝐶 ∈ ℂ)
15 simprl 498 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑢 ∈ ℂ)
16 simprr 499 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑣 ∈ ℂ)
1714, 15, 16adddid 7502 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝐶 · (𝑢 + 𝑣)) = ((𝐶 · 𝑢) + (𝐶 · 𝑣)))
18 simprl 498 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
19 nnuz 9044 . . . . . . . . 9 ℕ = (ℤ‘1)
2018, 19syl6eleq 2180 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
21 elnnuz 9045 . . . . . . . . . . . 12 (𝑢 ∈ ℕ ↔ 𝑢 ∈ (ℤ‘1))
2221biimpri 131 . . . . . . . . . . 11 (𝑢 ∈ (ℤ‘1) → 𝑢 ∈ ℕ)
2322adantl 271 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → 𝑢 ∈ ℕ)
24 f1of 5247 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2524ad2antll 475 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
2625ad2antrr 472 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
27 1zzd 8767 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
2818ad2antrr 472 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
2928nnzd 8857 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
30 eluzelz 9018 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (ℤ‘1) → 𝑢 ∈ ℤ)
3130ad2antlr 473 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ∈ ℤ)
3227, 29, 313jca 1123 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ))
33 eluzle 9021 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (ℤ‘1) → 1 ≤ 𝑢)
3433ad2antlr 473 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 1 ≤ 𝑢)
35 simpr 108 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ≤ (♯‘𝐴))
3634, 35jca 300 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (1 ≤ 𝑢𝑢 ≤ (♯‘𝐴)))
37 elfz2 9421 . . . . . . . . . . . . . 14 (𝑢 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ) ∧ (1 ≤ 𝑢𝑢 ≤ (♯‘𝐴))))
3832, 36, 37sylanbrc 408 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ∈ (1...(♯‘𝐴)))
39 fvco3 5369 . . . . . . . . . . . . 13 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑢 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢) = ((𝑘𝐴𝐵)‘(𝑓𝑢)))
4026, 38, 39syl2anc 403 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢) = ((𝑘𝐴𝐵)‘(𝑓𝑢)))
4126, 38ffvelrnd 5429 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝑓𝑢) ∈ 𝐴)
42 fsummulc2.3 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4342ralrimiva 2446 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
4443ad3antrrr 476 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
45 nfcsb1v 2963 . . . . . . . . . . . . . . . . 17 𝑘(𝑓𝑢) / 𝑘𝐵
4645nfel1 2239 . . . . . . . . . . . . . . . 16 𝑘(𝑓𝑢) / 𝑘𝐵 ∈ ℂ
47 csbeq1a 2941 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑓𝑢) → 𝐵 = (𝑓𝑢) / 𝑘𝐵)
4847eleq1d 2156 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑓𝑢) → (𝐵 ∈ ℂ ↔ (𝑓𝑢) / 𝑘𝐵 ∈ ℂ))
4946, 48rspc 2716 . . . . . . . . . . . . . . 15 ((𝑓𝑢) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑢) / 𝑘𝐵 ∈ ℂ))
5041, 44, 49sylc 61 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝑓𝑢) / 𝑘𝐵 ∈ ℂ)
51 eqid 2088 . . . . . . . . . . . . . . 15 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5251fvmpts 5376 . . . . . . . . . . . . . 14 (((𝑓𝑢) ∈ 𝐴(𝑓𝑢) / 𝑘𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘(𝑓𝑢)) = (𝑓𝑢) / 𝑘𝐵)
5341, 50, 52syl2anc 403 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵)‘(𝑓𝑢)) = (𝑓𝑢) / 𝑘𝐵)
5453, 50eqeltrd 2164 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵)‘(𝑓𝑢)) ∈ ℂ)
5540, 54eqeltrd 2164 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢) ∈ ℂ)
56 0cnd 7471 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
5723nnzd 8857 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → 𝑢 ∈ ℤ)
5818adantr 270 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℕ)
5958nnzd 8857 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℤ)
60 zdcle 8813 . . . . . . . . . . . 12 ((𝑢 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑢 ≤ (♯‘𝐴))
6157, 59, 60syl2anc 403 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → DECID 𝑢 ≤ (♯‘𝐴))
6255, 56, 61ifcldadc 3418 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0) ∈ ℂ)
63 breq1 3846 . . . . . . . . . . . 12 (𝑛 = 𝑢 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑢 ≤ (♯‘𝐴)))
64 fveq2 5299 . . . . . . . . . . . 12 (𝑛 = 𝑢 → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢))
6563, 64ifbieq1d 3411 . . . . . . . . . . 11 (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) = if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0))
66 eqid 2088 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))
6765, 66fvmptg 5374 . . . . . . . . . 10 ((𝑢 ∈ ℕ ∧ if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0))
6823, 62, 67syl2anc 403 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0))
6968, 62eqeltrd 2164 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢) ∈ ℂ)
70 csbov2g 5682 . . . . . . . . . . . 12 ((𝑓𝑢) ∈ 𝐴(𝑓𝑢) / 𝑘(𝐶 · 𝐵) = (𝐶 · (𝑓𝑢) / 𝑘𝐵))
7141, 70syl 14 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝑓𝑢) / 𝑘(𝐶 · 𝐵) = (𝐶 · (𝑓𝑢) / 𝑘𝐵))
7235iftrued 3398 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢))
73 fvco3 5369 . . . . . . . . . . . . 13 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑢 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑢)))
7426, 38, 73syl2anc 403 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑢)))
751ad3antrrr 476 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝐶 ∈ ℂ)
7675, 50mulcld 7498 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · (𝑓𝑢) / 𝑘𝐵) ∈ ℂ)
7771, 76eqeltrd 2164 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝑓𝑢) / 𝑘(𝐶 · 𝐵) ∈ ℂ)
78 eqid 2088 . . . . . . . . . . . . . 14 (𝑘𝐴 ↦ (𝐶 · 𝐵)) = (𝑘𝐴 ↦ (𝐶 · 𝐵))
7978fvmpts 5376 . . . . . . . . . . . . 13 (((𝑓𝑢) ∈ 𝐴(𝑓𝑢) / 𝑘(𝐶 · 𝐵) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑢)) = (𝑓𝑢) / 𝑘(𝐶 · 𝐵))
8041, 77, 79syl2anc 403 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑢)) = (𝑓𝑢) / 𝑘(𝐶 · 𝐵))
8172, 74, 803eqtrd 2124 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (𝑓𝑢) / 𝑘(𝐶 · 𝐵))
8235iftrued 3398 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢))
8382, 40, 533eqtrd 2124 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0) = (𝑓𝑢) / 𝑘𝐵)
8483oveq2d 5660 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0)) = (𝐶 · (𝑓𝑢) / 𝑘𝐵))
8571, 81, 843eqtr4d 2130 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0)))
861ad3antrrr 476 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → 𝐶 ∈ ℂ)
8786mul01d 7861 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · 0) = 0)
88 simpr 108 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → ¬ 𝑢 ≤ (♯‘𝐴))
8988iffalsed 3401 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0) = 0)
9089oveq2d 5660 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0)) = (𝐶 · 0))
9188iffalsed 3401 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = 0)
9287, 90, 913eqtr4rd 2131 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0)))
93 exmiddc 782 . . . . . . . . . . 11 (DECID 𝑢 ≤ (♯‘𝐴) → (𝑢 ≤ (♯‘𝐴) ∨ ¬ 𝑢 ≤ (♯‘𝐴)))
9461, 93syl 14 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → (𝑢 ≤ (♯‘𝐴) ∨ ¬ 𝑢 ≤ (♯‘𝐴)))
9585, 92, 94mpjaodan 747 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0)))
9680, 77eqeltrd 2164 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑢)) ∈ ℂ)
9774, 96eqeltrd 2164 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) ∈ ℂ)
9897, 56, 61ifcldadc 3418 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) ∈ ℂ)
99 fveq2 5299 . . . . . . . . . . . 12 (𝑛 = 𝑢 → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢))
10063, 99ifbieq1d 3411 . . . . . . . . . . 11 (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0) = if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0))
101 eqid 2088 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))
102100, 101fvmptg 5374 . . . . . . . . . 10 ((𝑢 ∈ ℕ ∧ if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0))
10323, 98, 102syl2anc 403 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0))
10468oveq2d 5660 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → (𝐶 · ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢)) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0)))
10595, 103, 1043eqtr4d 2130 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))‘𝑢) = (𝐶 · ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢)))
106 mulcl 7459 . . . . . . . . 9 ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ)
107106adantl 271 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 · 𝑣) ∈ ℂ)
1081adantr 270 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝐶 ∈ ℂ)
10913, 17, 20, 69, 105, 107, 108seq3distr 9934 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)) = (𝐶 · (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴))))
110 fveq2 5299 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
111 simprr 499 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
1121adantr 270 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
113112, 42mulcld 7498 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝐶 · 𝐵) ∈ ℂ)
114113fmpttd 5447 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
115114adantr 270 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
116115ffvelrnda 5428 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) ∈ ℂ)
117 fvco3 5369 . . . . . . . . 9 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
11825, 117sylan 277 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
119110, 18, 111, 116, 118fsum3 10766 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)))
120 fveq2 5299 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
12142fmpttd 5447 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
122121adantr 270 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
123122ffvelrnda 5428 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
124 fvco3 5369 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
12525, 124sylan 277 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
126120, 18, 111, 123, 125fsum3 10766 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)))
127126oveq2d 5660 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴))))
128109, 119, 1273eqtr4rd 2131 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚))
129 sumfct 10750 . . . . . . . . 9 (∀𝑘𝐴 𝐵 ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
13043, 129syl 14 . . . . . . . 8 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
131130oveq2d 5660 . . . . . . 7 (𝜑 → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · Σ𝑘𝐴 𝐵))
132131adantr 270 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · Σ𝑘𝐴 𝐵))
133113ralrimiva 2446 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 (𝐶 · 𝐵) ∈ ℂ)
134 sumfct 10750 . . . . . . . 8 (∀𝑘𝐴 (𝐶 · 𝐵) ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘𝐴 (𝐶 · 𝐵))
135133, 134syl 14 . . . . . . 7 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘𝐴 (𝐶 · 𝐵))
136135adantr 270 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘𝐴 (𝐶 · 𝐵))
137128, 132, 1363eqtr3d 2128 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
138137expr 367 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
139138exlimdv 1747 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
140139expimpd 355 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
141 fsummulc2.1 . . 3 (𝜑𝐴 ∈ Fin)
142 fz1f1o 10751 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
143141, 142syl 14 . 2 (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
14411, 140, 143mpjaod 673 1 (𝜑 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 664  DECID wdc 780  w3a 924   = wceq 1289  wex 1426  wcel 1438  wral 2359  csb 2933  c0 3286  ifcif 3391   class class class wbr 3843  cmpt 3897  ccom 4440  wf 5006  1-1-ontowf1o 5009  cfv 5010  (class class class)co 5644  Fincfn 6447  cc 7338  0cc0 7340  1c1 7341   + caddc 7343   · cmul 7345  cle 7513  cn 8412  cz 8740  cuz 9009  ...cfz 9414  seqcseq 9840  chash 10171  Σcsu 10729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-mulrcl 7434  ax-addcom 7435  ax-mulcom 7436  ax-addass 7437  ax-mulass 7438  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-1rid 7442  ax-0id 7443  ax-rnegex 7444  ax-precex 7445  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-apti 7450  ax-pre-ltadd 7451  ax-pre-mulgt0 7452  ax-pre-mulext 7453  ax-arch 7454  ax-caucvg 7455
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3392  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-ilim 4194  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-isom 5019  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-irdg 6127  df-frec 6148  df-1o 6173  df-oadd 6177  df-er 6282  df-en 6448  df-dom 6449  df-fin 6450  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-reap 8042  df-ap 8049  df-div 8130  df-inn 8413  df-2 8471  df-3 8472  df-4 8473  df-n0 8664  df-z 8741  df-uz 9010  df-q 9095  df-rp 9125  df-fz 9415  df-fzo 9542  df-iseq 9841  df-seq3 9842  df-exp 9943  df-ihash 10172  df-cj 10264  df-re 10265  df-im 10266  df-rsqrt 10419  df-abs 10420  df-clim 10654  df-isum 10730
This theorem is referenced by:  fsummulc1  10830  fsumneg  10832  fsum2mul  10834  cvgratnnlemabsle  10908  mertensabs  10918  eirraplem  11051
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