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Theorem fsummulc2 11340
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 8263 . . 3 (𝜑 → (𝐶 · 0) = 0)
3 sumeq1 11247 . . . . . 6 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4 sum0 11280 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
53, 4eqtrdi 2206 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = 0)
65oveq2d 5837 . . . 4 (𝐴 = ∅ → (𝐶 · Σ𝑘𝐴 𝐵) = (𝐶 · 0))
7 sumeq1 11247 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵))
8 sum0 11280 . . . . 5 Σ𝑘 ∈ ∅ (𝐶 · 𝐵) = 0
97, 8eqtrdi 2206 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐶 · 𝐵) = 0)
106, 9eqeq12d 2172 . . 3 (𝐴 = ∅ → ((𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵) ↔ (𝐶 · 0) = 0))
112, 10syl5ibrcom 156 . 2 (𝜑 → (𝐴 = ∅ → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
12 addcl 7852 . . . . . . . . 9 ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 + 𝑣) ∈ ℂ)
1312adantl 275 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 + 𝑣) ∈ ℂ)
141ad2antrr 480 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝐶 ∈ ℂ)
15 simprl 521 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑢 ∈ ℂ)
16 simprr 522 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑣 ∈ ℂ)
1714, 15, 16adddid 7897 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝐶 · (𝑢 + 𝑣)) = ((𝐶 · 𝑢) + (𝐶 · 𝑣)))
18 simprl 521 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
19 nnuz 9469 . . . . . . . . 9 ℕ = (ℤ‘1)
2018, 19eleqtrdi 2250 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
21 elnnuz 9470 . . . . . . . . . . . 12 (𝑢 ∈ ℕ ↔ 𝑢 ∈ (ℤ‘1))
2221biimpri 132 . . . . . . . . . . 11 (𝑢 ∈ (ℤ‘1) → 𝑢 ∈ ℕ)
2322adantl 275 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → 𝑢 ∈ ℕ)
24 f1of 5413 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2524ad2antll 483 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
2625ad2antrr 480 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
27 1zzd 9189 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
2818ad2antrr 480 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
2928nnzd 9280 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
30 eluzelz 9443 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (ℤ‘1) → 𝑢 ∈ ℤ)
3130ad2antlr 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ∈ ℤ)
3227, 29, 313jca 1162 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ))
33 eluzle 9446 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (ℤ‘1) → 1 ≤ 𝑢)
3433ad2antlr 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 1 ≤ 𝑢)
35 simpr 109 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ≤ (♯‘𝐴))
3634, 35jca 304 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (1 ≤ 𝑢𝑢 ≤ (♯‘𝐴)))
37 elfz2 9914 . . . . . . . . . . . . . 14 (𝑢 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ) ∧ (1 ≤ 𝑢𝑢 ≤ (♯‘𝐴))))
3832, 36, 37sylanbrc 414 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ∈ (1...(♯‘𝐴)))
39 fvco3 5538 . . . . . . . . . . . . 13 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑢 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢) = ((𝑘𝐴𝐵)‘(𝑓𝑢)))
4026, 38, 39syl2anc 409 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢) = ((𝑘𝐴𝐵)‘(𝑓𝑢)))
4126, 38ffvelrnd 5602 . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
4443ad3antrrr 484 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
45 nfcsb1v 3064 . . . . . . . . . . . . . . . . 17 𝑘(𝑓𝑢) / 𝑘𝐵
4645nfel1 2310 . . . . . . . . . . . . . . . 16 𝑘(𝑓𝑢) / 𝑘𝐵 ∈ ℂ
47 csbeq1a 3040 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑓𝑢) → 𝐵 = (𝑓𝑢) / 𝑘𝐵)
4847eleq1d 2226 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑓𝑢) → (𝐵 ∈ ℂ ↔ (𝑓𝑢) / 𝑘𝐵 ∈ ℂ))
4946, 48rspc 2810 . . . . . . . . . . . . . . 15 ((𝑓𝑢) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑢) / 𝑘𝐵 ∈ ℂ))
5041, 44, 49sylc 62 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝑓𝑢) / 𝑘𝐵 ∈ ℂ)
51 eqid 2157 . . . . . . . . . . . . . . 15 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5251fvmpts 5545 . . . . . . . . . . . . . 14 (((𝑓𝑢) ∈ 𝐴(𝑓𝑢) / 𝑘𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘(𝑓𝑢)) = (𝑓𝑢) / 𝑘𝐵)
5341, 50, 52syl2anc 409 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵)‘(𝑓𝑢)) = (𝑓𝑢) / 𝑘𝐵)
5453, 50eqeltrd 2234 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵)‘(𝑓𝑢)) ∈ ℂ)
5540, 54eqeltrd 2234 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢) ∈ ℂ)
56 0cnd 7866 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
5723nnzd 9280 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → 𝑢 ∈ ℤ)
5818adantr 274 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℕ)
5958nnzd 9280 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℤ)
60 zdcle 9235 . . . . . . . . . . . 12 ((𝑢 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑢 ≤ (♯‘𝐴))
6157, 59, 60syl2anc 409 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → DECID 𝑢 ≤ (♯‘𝐴))
6255, 56, 61ifcldadc 3534 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0) ∈ ℂ)
63 breq1 3968 . . . . . . . . . . . 12 (𝑛 = 𝑢 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑢 ≤ (♯‘𝐴)))
64 fveq2 5467 . . . . . . . . . . . 12 (𝑛 = 𝑢 → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢))
6563, 64ifbieq1d 3527 . . . . . . . . . . 11 (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) = if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0))
66 eqid 2157 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))
6765, 66fvmptg 5543 . . . . . . . . . 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 409 . . . . . . . . 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 2234 . . . . . . . 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 5859 . . . . . . . . . . . 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 3512 . . . . . . . . . . . 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 5538 . . . . . . . . . . . . 13 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑢 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑢)))
7426, 38, 73syl2anc 409 . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝐶 ∈ ℂ)
7675, 50mulcld 7893 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · (𝑓𝑢) / 𝑘𝐵) ∈ ℂ)
7771, 76eqeltrd 2234 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝑓𝑢) / 𝑘(𝐶 · 𝐵) ∈ ℂ)
78 eqid 2157 . . . . . . . . . . . . . 14 (𝑘𝐴 ↦ (𝐶 · 𝐵)) = (𝑘𝐴 ↦ (𝐶 · 𝐵))
7978fvmpts 5545 . . . . . . . . . . . . 13 (((𝑓𝑢) ∈ 𝐴(𝑓𝑢) / 𝑘(𝐶 · 𝐵) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑢)) = (𝑓𝑢) / 𝑘(𝐶 · 𝐵))
8041, 77, 79syl2anc 409 . . . . . . . . . . . 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 2194 . . . . . . . . . . 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 3512 . . . . . . . . . . . . 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 2194 . . . . . . . . . . . 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 5837 . . . . . . . . . . 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 2200 . . . . . . . . . 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 484 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → 𝐶 ∈ ℂ)
8786mul01d 8263 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · 0) = 0)
88 simpr 109 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → ¬ 𝑢 ≤ (♯‘𝐴))
8988iffalsed 3515 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑢), 0) = 0)
9089oveq2d 5837 . . . . . . . . . . 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 3515 . . . . . . . . . . 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 2201 . . . . . . . . . 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 822 . . . . . . . . . . 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 788 . . . . . . . . 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 2234 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑢)) ∈ ℂ)
9774, 96eqeltrd 2234 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) ∈ ℂ)
9897, 56, 61ifcldadc 3534 . . . . . . . . . 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 5467 . . . . . . . . . . . 12 (𝑛 = 𝑢 → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢))
10063, 99ifbieq1d 3527 . . . . . . . . . . 11 (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0) = if(𝑢 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0))
101 eqid 2157 . . . . . . . . . . 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 5543 . . . . . . . . . 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 409 . . . . . . . . 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 5837 . . . . . . . . 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 2200 . . . . . . . 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 7854 . . . . . . . . 9 ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ)
107106adantl 275 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 · 𝑣) ∈ ℂ)
1081adantr 274 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝐶 ∈ ℂ)
10913, 17, 20, 69, 105, 107, 108seq3distr 10407 . . . . . . 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 5467 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
111 simprr 522 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
1121adantr 274 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
113112, 42mulcld 7893 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝐶 · 𝐵) ∈ ℂ)
114113fmpttd 5621 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
115114adantr 274 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
116115ffvelrnda 5601 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) ∈ ℂ)
117 fvco3 5538 . . . . . . . . 9 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
11825, 117sylan 281 . . . . . . . 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 11279 . . . . . . 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 5467 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
12142fmpttd 5621 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
122121adantr 274 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
123122ffvelrnda 5601 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
124 fvco3 5538 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
12525, 124sylan 281 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
126120, 18, 111, 123, 125fsum3 11279 . . . . . . . 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 5837 . . . . . . 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 2201 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚))
129 sumfct 11266 . . . . . . . . 9 (∀𝑘𝐴 𝐵 ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
13043, 129syl 14 . . . . . . . 8 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
131130oveq2d 5837 . . . . . . 7 (𝜑 → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · Σ𝑘𝐴 𝐵))
132131adantr 274 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · Σ𝑘𝐴 𝐵))
133113ralrimiva 2530 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 (𝐶 · 𝐵) ∈ ℂ)
134 sumfct 11266 . . . . . . . 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 274 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘𝐴 (𝐶 · 𝐵))
137128, 132, 1363eqtr3d 2198 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
138137expr 373 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
139138exlimdv 1799 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
140139expimpd 361 . 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 11267 . . 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 708 1 (𝜑 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 698  DECID wdc 820  w3a 963   = wceq 1335  wex 1472  wcel 2128  wral 2435  csb 3031  c0 3394  ifcif 3505   class class class wbr 3965  cmpt 4025  ccom 4589  wf 5165  1-1-ontowf1o 5168  cfv 5169  (class class class)co 5821  Fincfn 6682  cc 7725  0cc0 7727  1c1 7728   + caddc 7730   · cmul 7732  cle 7908  cn 8828  cz 9162  cuz 9434  ...cfz 9907  seqcseq 10339  chash 10644  Σcsu 11245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7818  ax-resscn 7819  ax-1cn 7820  ax-1re 7821  ax-icn 7822  ax-addcl 7823  ax-addrcl 7824  ax-mulcl 7825  ax-mulrcl 7826  ax-addcom 7827  ax-mulcom 7828  ax-addass 7829  ax-mulass 7830  ax-distr 7831  ax-i2m1 7832  ax-0lt1 7833  ax-1rid 7834  ax-0id 7835  ax-rnegex 7836  ax-precex 7837  ax-cnre 7838  ax-pre-ltirr 7839  ax-pre-ltwlin 7840  ax-pre-lttrn 7841  ax-pre-apti 7842  ax-pre-ltadd 7843  ax-pre-mulgt0 7844  ax-pre-mulext 7845  ax-arch 7846  ax-caucvg 7847
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-isom 5178  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-frec 6335  df-1o 6360  df-oadd 6364  df-er 6477  df-en 6683  df-dom 6684  df-fin 6685  df-pnf 7909  df-mnf 7910  df-xr 7911  df-ltxr 7912  df-le 7913  df-sub 8043  df-neg 8044  df-reap 8445  df-ap 8452  df-div 8541  df-inn 8829  df-2 8887  df-3 8888  df-4 8889  df-n0 9086  df-z 9163  df-uz 9435  df-q 9524  df-rp 9556  df-fz 9908  df-fzo 10037  df-seqfrec 10340  df-exp 10414  df-ihash 10645  df-cj 10737  df-re 10738  df-im 10739  df-rsqrt 10893  df-abs 10894  df-clim 11171  df-sumdc 11246
This theorem is referenced by:  fsummulc1  11341  fsumneg  11343  fsum2mul  11345  cvgratnnlemabsle  11419  mertensabs  11429  eirraplem  11668
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