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Theorem fsummulc2 14304
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsummulc2.2 . . . 4 (𝜑𝐶 ∈ ℂ)
21mul01d 10086 . . 3 (𝜑 → (𝐶 · 0) = 0)
3 sumeq1 14213 . . . . . 6 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4 sum0 14245 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
53, 4syl6eq 2659 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = 0)
65oveq2d 6543 . . . 4 (𝐴 = ∅ → (𝐶 · Σ𝑘𝐴 𝐵) = (𝐶 · 0))
7 sumeq1 14213 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵))
8 sum0 14245 . . . . 5 Σ𝑘 ∈ ∅ (𝐶 · 𝐵) = 0
97, 8syl6eq 2659 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐶 · 𝐵) = 0)
106, 9eqeq12d 2624 . . 3 (𝐴 = ∅ → ((𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵) ↔ (𝐶 · 0) = 0))
112, 10syl5ibrcom 235 . 2 (𝜑 → (𝐴 = ∅ → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
12 addcl 9874 . . . . . . . . 9 ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝑛 + 𝑚) ∈ ℂ)
1312adantl 480 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝑛 + 𝑚) ∈ ℂ)
141adantr 479 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝐶 ∈ ℂ)
15 adddi 9881 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
16153expb 1257 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
1714, 16sylan 486 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
18 simprl 789 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ ℕ)
19 nnuz 11555 . . . . . . . . 9 ℕ = (ℤ‘1)
2018, 19syl6eleq 2697 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ (ℤ‘1))
21 fsummulc2.3 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
22 eqid 2609 . . . . . . . . . . . 12 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
2321, 22fmptd 6277 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
2423ad2antrr 757 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (𝑘𝐴𝐵):𝐴⟶ℂ)
25 simprr 791 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)
2625adantr 479 . . . . . . . . . . 11 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)
27 f1of 6035 . . . . . . . . . . 11 (𝑓:(1...(#‘𝐴))–1-1-onto𝐴𝑓:(1...(#‘𝐴))⟶𝐴)
2826, 27syl 17 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 𝑓:(1...(#‘𝐴))⟶𝐴)
29 fco 5957 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(#‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(#‘𝐴))⟶ℂ)
3024, 28, 29syl2anc 690 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(#‘𝐴))⟶ℂ)
31 simpr 475 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 𝑛 ∈ (1...(#‘𝐴)))
3230, 31ffvelrnd 6253 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
3328, 31ffvelrnd 6253 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
34 simpr 475 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝑘𝐴)
351adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
3635, 21mulcld 9916 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → (𝐶 · 𝐵) ∈ ℂ)
37 eqid 2609 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐶 · 𝐵)) = (𝑘𝐴 ↦ (𝐶 · 𝐵))
3837fvmpt2 6185 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐶 · 𝐵) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · 𝐵))
3934, 36, 38syl2anc 690 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · 𝐵))
4022fvmpt2 6185 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
4134, 21, 40syl2anc 690 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
4241oveq2d 6543 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) = (𝐶 · 𝐵))
4339, 42eqtr4d 2646 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)))
4443ralrimiva 2948 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)))
4544ad2antrr 757 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)))
46 nffvmpt1 6096 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛))
47 nfcv 2750 . . . . . . . . . . . . 13 𝑘𝐶
48 nfcv 2750 . . . . . . . . . . . . 13 𝑘 ·
49 nffvmpt1 6096 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
5047, 48, 49nfov 6553 . . . . . . . . . . . 12 𝑘(𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))
5146, 50nfeq 2761 . . . . . . . . . . 11 𝑘((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))
52 fveq2 6088 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
53 fveq2 6088 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
5453oveq2d 6543 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛))))
5552, 54eqeq12d 2624 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))))
5651, 55rspc 3275 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))))
5733, 45, 56sylc 62 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛))))
5827ad2antll 760 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(#‘𝐴))⟶𝐴)
59 fvco3 6170 . . . . . . . . . 10 ((𝑓:(1...(#‘𝐴))⟶𝐴𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
6058, 59sylan 486 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
61 fvco3 6170 . . . . . . . . . . 11 ((𝑓:(1...(#‘𝐴))⟶𝐴𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6258, 61sylan 486 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6362oveq2d 6543 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (𝐶 · (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛))))
6457, 60, 633eqtr4d 2653 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = (𝐶 · (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛)))
6513, 17, 20, 32, 64seqdistr 12669 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (seq1( + , ((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓))‘(#‘𝐴)) = (𝐶 · (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴))))
66 fveq2 6088 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
6736, 37fmptd 6277 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
6867adantr 479 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
6968ffvelrnda 6252 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) ∈ ℂ)
7066, 18, 25, 69, 60fsum 14244 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = (seq1( + , ((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓))‘(#‘𝐴)))
71 fveq2 6088 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7223adantr 479 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
7372ffvelrnda 6252 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7471, 18, 25, 73, 62fsum 14244 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴)))
7574oveq2d 6543 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴))))
7665, 70, 753eqtr4rd 2654 . . . . . 6 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚))
77 sumfc 14233 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵
7877oveq2i 6538 . . . . . 6 (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · Σ𝑘𝐴 𝐵)
79 sumfc 14233 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘𝐴 (𝐶 · 𝐵)
8076, 78, 793eqtr3g 2666 . . . . 5 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
8180expr 640 . . . 4 ((𝜑 ∧ (#‘𝐴) ∈ ℕ) → (𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
8281exlimdv 1847 . . 3 ((𝜑 ∧ (#‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
8382expimpd 626 . 2 (𝜑 → (((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴) → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
84 fsummulc2.1 . . 3 (𝜑𝐴 ∈ Fin)
85 fz1f1o 14234 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)))
8684, 85syl 17 . 2 (𝜑 → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)))
8711, 83, 86mpjaod 394 1 (𝜑 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wa 382   = wceq 1474  wex 1694  wcel 1976  wral 2895  c0 3873  cmpt 4637  ccom 5032  wf 5786  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  Fincfn 7818  cc 9790  0cc0 9792  1c1 9793   + caddc 9795   · cmul 9797  cn 10867  cuz 11519  ...cfz 12152  seqcseq 12618  #chash 12934  Σcsu 14210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-rp 11665  df-fz 12153  df-fzo 12290  df-seq 12619  df-exp 12678  df-hash 12935  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-clim 14013  df-sum 14211
This theorem is referenced by:  fsummulc1  14305  fsumneg  14307  fsum2mul  14309  incexc2  14355  mertens  14403  binomrisefac  14558  fsumkthpow  14572  eirrlem  14717  pwp1fsum  14898  csbren  22907  trirn  22908  itg1addlem4  23189  itg1addlem5  23190  itg1mulc  23194  elqaalem3  23797  advlogexp  24118  fsumharmonic  24455  basellem8  24531  muinv  24636  fsumdvdsmul  24638  logfaclbnd  24664  dchrsum2  24710  sumdchr2  24712  rplogsumlem2  24891  rpvmasumlem  24893  dchrmusum2  24900  dchrvmasumlem1  24901  dchrvmasum2lem  24902  dchrvmasumlem2  24904  dchrvmasumiflem1  24907  rpvmasum2  24918  dchrisum0lem2  24924  mudivsum  24936  mulogsum  24938  mulog2sumlem1  24940  mulog2sumlem2  24941  mulog2sumlem3  24942  vmalogdivsum2  24944  logsqvma  24948  selberglem1  24951  selberglem2  24952  selberg  24954  selberg3lem1  24963  selberg4lem1  24966  selberg4  24967  selbergr  24974  selberg3r  24975  selberg34r  24977  pntsval2  24982  pntrlog2bndlem2  24984  pntrlog2bndlem3  24985  pntrlog2bndlem4  24986  pntrlog2bndlem6  24989  pntpbnd2  24993  pntlemk  25012  axsegconlem9  25523  ax5seglem1  25526  ax5seglem2  25527  ax5seglem9  25535  knoppndvlem11  31489  jm2.22  36376  dvnprodlem2  38634  stoweidlem26  38716  stirlinglem12  38775  fourierdlem83  38879  etransclem46  38970  pwdif  39837  altgsumbcALT  41919  aacllem  42312
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