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Theorem fsummulc2 14802
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 10489 . . 3 (𝜑 → (𝐶 · 0) = 0)
3 sumeq1 14706 . . . . . 6 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4 sum0 14739 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
53, 4syl6eq 2815 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = 0)
65oveq2d 6858 . . . 4 (𝐴 = ∅ → (𝐶 · Σ𝑘𝐴 𝐵) = (𝐶 · 0))
7 sumeq1 14706 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵))
8 sum0 14739 . . . . 5 Σ𝑘 ∈ ∅ (𝐶 · 𝐵) = 0
97, 8syl6eq 2815 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐶 · 𝐵) = 0)
106, 9eqeq12d 2780 . . 3 (𝐴 = ∅ → ((𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵) ↔ (𝐶 · 0) = 0))
112, 10syl5ibrcom 238 . 2 (𝜑 → (𝐴 = ∅ → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
12 addcl 10271 . . . . . . . . 9 ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝑛 + 𝑚) ∈ ℂ)
1312adantl 473 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝑛 + 𝑚) ∈ ℂ)
141adantr 472 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝐶 ∈ ℂ)
15 adddi 10278 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
16153expb 1149 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
1714, 16sylan 575 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
18 simprl 787 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
19 nnuz 11923 . . . . . . . . 9 ℕ = (ℤ‘1)
2018, 19syl6eleq 2854 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
21 fsummulc2.3 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2221fmpttd 6575 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
2322ad2antrr 717 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑘𝐴𝐵):𝐴⟶ℂ)
24 simprr 789 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
2524adantr 472 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
26 f1of 6320 . . . . . . . . . . 11 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2725, 26syl 17 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
28 fco 6240 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2923, 27, 28syl2anc 579 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
30 simpr 477 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑛 ∈ (1...(♯‘𝐴)))
3129, 30ffvelrnd 6550 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
3227, 30ffvelrnd 6550 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
33 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝑘𝐴)
341adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
3534, 21mulcld 10314 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → (𝐶 · 𝐵) ∈ ℂ)
36 eqid 2765 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐶 · 𝐵)) = (𝑘𝐴 ↦ (𝐶 · 𝐵))
3736fvmpt2 6480 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐶 · 𝐵) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · 𝐵))
3833, 35, 37syl2anc 579 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · 𝐵))
39 eqid 2765 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
4039fvmpt2 6480 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
4133, 21, 40syl2anc 579 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
4241oveq2d 6858 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) = (𝐶 · 𝐵))
4338, 42eqtr4d 2802 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)))
4443ralrimiva 3113 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)))
4544ad2antrr 717 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)))
46 nffvmpt1 6386 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛))
47 nfcv 2907 . . . . . . . . . . . . 13 𝑘𝐶
48 nfcv 2907 . . . . . . . . . . . . 13 𝑘 ·
49 nffvmpt1 6386 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
5047, 48, 49nfov 6872 . . . . . . . . . . . 12 𝑘(𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))
5146, 50nfeq 2919 . . . . . . . . . . 11 𝑘((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))
52 fveq2 6375 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
53 fveq2 6375 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
5453oveq2d 6858 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛))))
5552, 54eqeq12d 2780 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))))
5651, 55rspc 3455 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))))
5732, 45, 56sylc 65 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛))))
5826ad2antll 720 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
59 fvco3 6464 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
6058, 59sylan 575 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
61 fvco3 6464 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6258, 61sylan 575 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6362oveq2d 6858 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝐶 · (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛))))
6457, 60, 633eqtr4d 2809 . . . . . . . 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, 31, 64seqdistr 13059 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( + , ((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓))‘(♯‘𝐴)) = (𝐶 · (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴))))
66 fveq2 6375 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
6735fmpttd 6575 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
6867adantr 472 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
6968ffvelrnda 6549 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) ∈ ℂ)
7066, 18, 24, 69, 60fsum 14738 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = (seq1( + , ((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
71 fveq2 6375 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7222adantr 472 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
7372ffvelrnda 6549 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7471, 18, 24, 73, 62fsum 14738 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)))
7574oveq2d 6858 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴))))
7665, 70, 753eqtr4rd 2810 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚))
77 sumfc 14727 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵
7877oveq2i 6853 . . . . . 6 (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · Σ𝑘𝐴 𝐵)
79 sumfc 14727 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘𝐴 (𝐶 · 𝐵)
8076, 78, 793eqtr3g 2822 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
8180expr 448 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
8281exlimdv 2028 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
8382expimpd 445 . 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 14728 . . 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 886 1 (𝜑 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wo 873   = wceq 1652  wex 1874  wcel 2155  wral 3055  c0 4079  cmpt 4888  ccom 5281  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  Fincfn 8160  cc 10187  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194  cn 11274  cuz 11886  ...cfz 12533  seqcseq 13008  chash 13321  Σcsu 14703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-clim 14506  df-sum 14704
This theorem is referenced by:  fsummulc1  14803  fsumneg  14805  fsum2mul  14807  incexc2  14856  mertens  14903  binomrisefac  15057  fsumkthpow  15071  eirrlem  15216  pwp1fsum  15398  csbren  23471  trirn  23472  itg1addlem4  23757  itg1addlem5  23758  itg1mulc  23762  elqaalem3  24367  advlogexp  24692  fsumharmonic  25029  basellem8  25105  muinv  25210  fsumdvdsmul  25212  logfaclbnd  25238  dchrsum2  25284  sumdchr2  25286  rplogsumlem2  25465  rpvmasumlem  25467  dchrmusum2  25474  dchrvmasumlem1  25475  dchrvmasum2lem  25476  dchrvmasumlem2  25478  dchrvmasumiflem1  25481  rpvmasum2  25492  dchrisum0lem2  25498  mudivsum  25510  mulogsum  25512  mulog2sumlem1  25514  mulog2sumlem2  25515  mulog2sumlem3  25516  vmalogdivsum2  25518  logsqvma  25522  selberglem1  25525  selberglem2  25526  selberg  25528  selberg3lem1  25537  selberg4lem1  25540  selberg4  25541  selbergr  25548  selberg3r  25549  selberg34r  25551  pntsval2  25556  pntrlog2bndlem2  25558  pntrlog2bndlem3  25559  pntrlog2bndlem4  25560  pntrlog2bndlem6  25563  pntpbnd2  25567  pntlemk  25586  axsegconlem9  26096  ax5seglem1  26099  ax5seglem2  26100  ax5seglem9  26108  hgt750lemf  31114  hgt750lemb  31117  knoppndvlem11  32884  jm2.22  38171  dvnprodlem2  40732  stoweidlem26  40812  stirlinglem12  40871  fourierdlem83  40975  etransclem46  41066  pwdif  42109  altgsumbcALT  42732  aacllem  43151
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