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Theorem fsummulc2 15820
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 11460 . . 3 (𝜑 → (𝐶 · 0) = 0)
3 sumeq1 15725 . . . . . 6 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4 sum0 15757 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
53, 4eqtrdi 2793 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = 0)
65oveq2d 7447 . . . 4 (𝐴 = ∅ → (𝐶 · Σ𝑘𝐴 𝐵) = (𝐶 · 0))
7 sumeq1 15725 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵))
8 sum0 15757 . . . . 5 Σ𝑘 ∈ ∅ (𝐶 · 𝐵) = 0
97, 8eqtrdi 2793 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐶 · 𝐵) = 0)
106, 9eqeq12d 2753 . . 3 (𝐴 = ∅ → ((𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵) ↔ (𝐶 · 0) = 0))
112, 10syl5ibrcom 247 . 2 (𝜑 → (𝐴 = ∅ → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
12 addcl 11237 . . . . . . . . 9 ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝑛 + 𝑚) ∈ ℂ)
1312adantl 481 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝑛 + 𝑚) ∈ ℂ)
141adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝐶 ∈ ℂ)
15 adddi 11244 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
16153expb 1121 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
1714, 16sylan 580 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
18 simprl 771 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
19 nnuz 12921 . . . . . . . . 9 ℕ = (ℤ‘1)
2018, 19eleqtrdi 2851 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
21 fsummulc2.3 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2221fmpttd 7135 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
2322ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑘𝐴𝐵):𝐴⟶ℂ)
24 simprr 773 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
2524adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
26 f1of 6848 . . . . . . . . . . 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 6760 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2923, 27, 28syl2anc 584 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
30 simpr 484 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑛 ∈ (1...(♯‘𝐴)))
3129, 30ffvelcdmd 7105 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
3227, 30ffvelcdmd 7105 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
33 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝑘𝐴)
341adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
3534, 21mulcld 11281 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → (𝐶 · 𝐵) ∈ ℂ)
36 eqid 2737 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐶 · 𝐵)) = (𝑘𝐴 ↦ (𝐶 · 𝐵))
3736fvmpt2 7027 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐶 · 𝐵) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · 𝐵))
3833, 35, 37syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · 𝐵))
39 eqid 2737 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
4039fvmpt2 7027 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
4133, 21, 40syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
4241oveq2d 7447 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) = (𝐶 · 𝐵))
4338, 42eqtr4d 2780 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)))
4443ralrimiva 3146 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)))
4544ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)))
46 nffvmpt1 6917 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛))
47 nfcv 2905 . . . . . . . . . . . . 13 𝑘𝐶
48 nfcv 2905 . . . . . . . . . . . . 13 𝑘 ·
49 nffvmpt1 6917 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
5047, 48, 49nfov 7461 . . . . . . . . . . . 12 𝑘(𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))
5146, 50nfeq 2919 . . . . . . . . . . 11 𝑘((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))
52 fveq2 6906 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
53 fveq2 6906 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
5453oveq2d 7447 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛))))
5552, 54eqeq12d 2753 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘𝐴𝐵)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛)))))
5651, 55rspc 3610 . . . . . . . . . 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 729 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
59 fvco3 7008 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
6058, 59sylan 580 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
61 fvco3 7008 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6258, 61sylan 580 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6362oveq2d 7447 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝐶 · (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛)) = (𝐶 · ((𝑘𝐴𝐵)‘(𝑓𝑛))))
6457, 60, 633eqtr4d 2787 . . . . . . . 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 14094 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( + , ((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓))‘(♯‘𝐴)) = (𝐶 · (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴))))
66 fveq2 6906 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘(𝑓𝑛)))
6735fmpttd 7135 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
6867adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
6968ffvelcdmda 7104 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) ∈ ℂ)
7066, 18, 24, 69, 60fsum 15756 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = (seq1( + , ((𝑘𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
71 fveq2 6906 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7222adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
7372ffvelcdmda 7104 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7471, 18, 24, 73, 62fsum 15756 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)))
7574oveq2d 7447 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴))))
7665, 70, 753eqtr4rd 2788 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚))
77 sumfc 15745 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵
7877oveq2i 7442 . . . . . 6 (𝐶 · Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐶 · Σ𝑘𝐴 𝐵)
79 sumfc 15745 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘𝐴 (𝐶 · 𝐵)
8076, 78, 793eqtr3g 2800 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
8180expr 456 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
8281exlimdv 1933 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵)))
8382expimpd 453 . 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 15746 . . 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 861 1 (𝜑 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  wral 3061  c0 4333  cmpt 5225  ccom 5689  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Fincfn 8985  cc 11153  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  cn 12266  cuz 12878  ...cfz 13547  seqcseq 14042  chash 14369  Σcsu 15722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723
This theorem is referenced by:  fsummulc1  15821  fsumneg  15823  fsum2mul  15825  incexc2  15874  pwdif  15904  mertens  15922  binomrisefac  16078  fsumkthpow  16092  eirrlem  16240  pwp1fsum  16428  csbren  25433  trirn  25434  itg1addlem4  25734  itg1addlem5  25735  itg1mulc  25739  elqaalem3  26363  advlogexp  26697  fsumharmonic  27055  basellem8  27131  muinv  27236  fsumdvdsmul  27238  fsumdvdsmulOLD  27240  logfaclbnd  27266  dchrsum2  27312  sumdchr2  27314  rplogsumlem2  27529  rpvmasumlem  27531  dchrmusum2  27538  dchrvmasumlem1  27539  dchrvmasum2lem  27540  dchrvmasumlem2  27542  dchrvmasumiflem1  27545  rpvmasum2  27556  dchrisum0lem2  27562  mudivsum  27574  mulogsum  27576  mulog2sumlem1  27578  mulog2sumlem2  27579  mulog2sumlem3  27580  vmalogdivsum2  27582  logsqvma  27586  selberglem1  27589  selberglem2  27590  selberg  27592  selberg3lem1  27601  selberg4lem1  27604  selberg4  27605  selbergr  27612  selberg3r  27613  selberg34r  27615  pntsval2  27620  pntrlog2bndlem2  27622  pntrlog2bndlem3  27623  pntrlog2bndlem4  27624  pntrlog2bndlem6  27627  pntpbnd2  27631  pntlemk  27650  axsegconlem9  28940  ax5seglem1  28943  ax5seglem2  28944  ax5seglem9  28952  hgt750lemf  34668  hgt750lemb  34671  knoppndvlem11  36523  3factsumint4  42025  lcmineqlem6  42035  oddnumth  42345  jm2.22  43007  dvnprodlem2  45962  stoweidlem26  46041  stirlinglem12  46100  fourierdlem83  46204  etransclem46  46295  altgsumbcALT  48269  aacllem  49320
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