Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fsum2dsub Structured version   Visualization version   GIF version

Theorem fsum2dsub 31136
Description: Lemma for breprexp 31162- Re-index a double sum, using difference of the initial indices. (Contributed by Thierry Arnoux, 7-Dec-2021.)
Hypotheses
Ref Expression
fzsum2sub.m (𝜑𝑀 ∈ ℕ0)
fzsum2sub.n (𝜑𝑁 ∈ ℕ0)
fzsum2sub.1 (𝑖 = (𝑘𝑗) → 𝐴 = 𝐵)
fzsum2sub.2 ((𝜑𝑖 ∈ (ℤ‘-𝑗) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ)
fzsum2sub.3 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) → 𝐵 = 0)
fzsum2sub.4 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0..^𝑗)) → 𝐵 = 0)
Assertion
Ref Expression
fsum2dsub (𝜑 → Σ𝑖 ∈ (0...𝑀𝑗 ∈ (1...𝑁)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑖   𝑖,𝑀,𝑗,𝑘   𝑖,𝑁,𝑗,𝑘   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐵(𝑗,𝑘)

Proof of Theorem fsum2dsub
StepHypRef Expression
1 fzssz 12550 . . . . . 6 (1...𝑁) ⊆ ℤ
2 simpr 477 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
31, 2sseldi 3759 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℤ)
4 0zd 11636 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → 0 ∈ ℤ)
5 fzsum2sub.m . . . . . . 7 (𝜑𝑀 ∈ ℕ0)
65nn0zd 11727 . . . . . 6 (𝜑𝑀 ∈ ℤ)
76adantr 472 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℤ)
8 simpll 783 . . . . . 6 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝜑)
9 fz1ssnn 12579 . . . . . . . . . . . 12 (1...𝑁) ⊆ ℕ
10 nnssnn0 11541 . . . . . . . . . . . 12 ℕ ⊆ ℕ0
119, 10sstri 3770 . . . . . . . . . . 11 (1...𝑁) ⊆ ℕ0
1211, 2sseldi 3759 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ0)
13 nn0uz 11922 . . . . . . . . . 10 0 = (ℤ‘0)
1412, 13syl6eleq 2854 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (ℤ‘0))
15 neg0 10581 . . . . . . . . . 10 -0 = 0
16 uzneg 11905 . . . . . . . . . 10 (𝑗 ∈ (ℤ‘0) → -0 ∈ (ℤ‘-𝑗))
1715, 16syl5eqelr 2849 . . . . . . . . 9 (𝑗 ∈ (ℤ‘0) → 0 ∈ (ℤ‘-𝑗))
18 fzss1 12587 . . . . . . . . 9 (0 ∈ (ℤ‘-𝑗) → (0...𝑀) ⊆ (-𝑗...𝑀))
1914, 17, 183syl 18 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (0...𝑀) ⊆ (-𝑗...𝑀))
20 fzssuz 12589 . . . . . . . 8 (-𝑗...𝑀) ⊆ (ℤ‘-𝑗)
2119, 20syl6ss 3773 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (0...𝑀) ⊆ (ℤ‘-𝑗))
2221sselda 3761 . . . . . 6 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ‘-𝑗))
232adantr 472 . . . . . 6 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝑗 ∈ (1...𝑁))
24 fzsum2sub.2 . . . . . 6 ((𝜑𝑖 ∈ (ℤ‘-𝑗) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ)
258, 22, 23, 24syl3anc 1490 . . . . 5 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℂ)
26 fzsum2sub.1 . . . . 5 (𝑖 = (𝑘𝑗) → 𝐴 = 𝐵)
273, 4, 7, 25, 26fsumshft 14796 . . . 4 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
285adantr 472 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℕ0)
299, 2sseldi 3759 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ)
3029nnnn0d 11598 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ0)
3128, 30nn0addcld 11602 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℕ0)
3231nn0red 11599 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℝ)
3332ltp1d 11208 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) < ((𝑀 + 𝑗) + 1))
34 fzdisj 12575 . . . . . . . 8 ((𝑀 + 𝑗) < ((𝑀 + 𝑗) + 1) → ((𝑗...(𝑀 + 𝑗)) ∩ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) = ∅)
3533, 34syl 17 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → ((𝑗...(𝑀 + 𝑗)) ∩ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) = ∅)
36 fzsum2sub.n . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
3736nn0zd 11727 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℤ)
386, 37zaddcld 11733 . . . . . . . . . 10 (𝜑 → (𝑀 + 𝑁) ∈ ℤ)
3938adantr 472 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ ℤ)
4031nn0zd 11727 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℤ)
4129nnred 11291 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℝ)
42 nn0addge2 11587 . . . . . . . . . 10 ((𝑗 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → 𝑗 ≤ (𝑀 + 𝑗))
4341, 28, 42syl2anc 579 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ≤ (𝑀 + 𝑗))
4436nn0red 11599 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℝ)
4544adantr 472 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
4628nn0red 11599 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
47 elfzle2 12552 . . . . . . . . . . 11 (𝑗 ∈ (1...𝑁) → 𝑗𝑁)
4847adantl 473 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗𝑁)
4941, 45, 46, 48leadd2dd 10896 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ≤ (𝑀 + 𝑁))
50 elfz4 12542 . . . . . . . . 9 (((𝑗 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ (𝑀 + 𝑗) ∈ ℤ) ∧ (𝑗 ≤ (𝑀 + 𝑗) ∧ (𝑀 + 𝑗) ≤ (𝑀 + 𝑁))) → (𝑀 + 𝑗) ∈ (𝑗...(𝑀 + 𝑁)))
513, 39, 40, 43, 49, 50syl32anc 1497 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ (𝑗...(𝑀 + 𝑁)))
52 fzsplit 12574 . . . . . . . 8 ((𝑀 + 𝑗) ∈ (𝑗...(𝑀 + 𝑁)) → (𝑗...(𝑀 + 𝑁)) = ((𝑗...(𝑀 + 𝑗)) ∪ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))))
5351, 52syl 17 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) = ((𝑗...(𝑀 + 𝑗)) ∪ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))))
54 fzfid 12980 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) ∈ Fin)
55 simpll 783 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝜑)
562adantr 472 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑗 ∈ (1...𝑁))
5711, 56sseldi 3759 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑗 ∈ ℕ0)
58 fz2ssnn0 29996 . . . . . . . . . 10 (𝑗 ∈ ℕ0 → (𝑗...(𝑀 + 𝑁)) ⊆ ℕ0)
5957, 58syl 17 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → (𝑗...(𝑀 + 𝑁)) ⊆ ℕ0)
60 simpr 477 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑁)))
6159, 60sseldd 3762 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑘 ∈ ℕ0)
6226eleq1d 2829 . . . . . . . . 9 (𝑖 = (𝑘𝑗) → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
63 simpll 783 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝜑)
64 simplr 785 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (ℤ‘-𝑗))
65 simpr 477 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
6663, 64, 65, 24syl3anc 1490 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ)
6766an32s 642 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (ℤ‘-𝑗)) → 𝐴 ∈ ℂ)
6867ralrimiva 3113 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → ∀𝑖 ∈ (ℤ‘-𝑗)𝐴 ∈ ℂ)
6968adantr 472 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → ∀𝑖 ∈ (ℤ‘-𝑗)𝐴 ∈ ℂ)
70 nnsscn 11279 . . . . . . . . . . . . 13 ℕ ⊆ ℂ
719, 70sstri 3770 . . . . . . . . . . . 12 (1...𝑁) ⊆ ℂ
72 simplr 785 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (1...𝑁))
7371, 72sseldi 3759 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ ℂ)
74 simpr 477 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
7574nn0cnd 11600 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
7673, 75negsubdi2d 10662 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → -(𝑗𝑘) = (𝑘𝑗))
771, 72sseldi 3759 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ ℤ)
78 eluzmn 11893 . . . . . . . . . . . 12 ((𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (ℤ‘(𝑗𝑘)))
7977, 74, 78syl2anc 579 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (ℤ‘(𝑗𝑘)))
80 uzneg 11905 . . . . . . . . . . 11 (𝑗 ∈ (ℤ‘(𝑗𝑘)) → -(𝑗𝑘) ∈ (ℤ‘-𝑗))
8179, 80syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → -(𝑗𝑘) ∈ (ℤ‘-𝑗))
8276, 81eqeltrrd 2845 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → (𝑘𝑗) ∈ (ℤ‘-𝑗))
8362, 69, 82rspcdva 3467 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
8455, 56, 61, 83syl21anc 866 . . . . . . 7 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝐵 ∈ ℂ)
8535, 53, 54, 84fsumsplit 14756 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 = (Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 + Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵))
863zcnd 11730 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℂ)
8786addid2d 10491 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → (0 + 𝑗) = 𝑗)
8887oveq1d 6857 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → ((0 + 𝑗)...(𝑀 + 𝑗)) = (𝑗...(𝑀 + 𝑗)))
8988eqcomd 2771 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑗)) = ((0 + 𝑗)...(𝑀 + 𝑗)))
9089sumeq1d 14716 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
91 fzsum2sub.3 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) → 𝐵 = 0)
9291sumeq2dv 14718 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵 = Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0)
93 fzfi 12979 . . . . . . . . 9 (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin
94 sumz 14738 . . . . . . . . . 10 (((((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ⊆ (ℤ‘0) ∨ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0)
9594olcs 902 . . . . . . . . 9 ((((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0)
9693, 95ax-mp 5 . . . . . . . 8 Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0
9792, 96syl6eq 2815 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵 = 0)
9890, 97oveq12d 6860 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 + Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵) = (Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 + 0))
99 fzfid 12980 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → ((0 + 𝑗)...(𝑀 + 𝑗)) ∈ Fin)
100 simpll 783 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝜑)
1012adantr 472 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑗 ∈ (1...𝑁))
102 elfzuz3 12546 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (1...𝑁) → 𝑁 ∈ (ℤ𝑗))
103102adantl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ𝑗))
104 eluzadd 11915 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ𝑗) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈ (ℤ‘(𝑗 + 𝑀)))
105103, 7, 104syl2anc 579 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑁 + 𝑀) ∈ (ℤ‘(𝑗 + 𝑀)))
10636nn0cnd 11600 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℂ)
107106adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
108 zsscn 11632 . . . . . . . . . . . . . . . 16 ℤ ⊆ ℂ
109108, 7sseldi 3759 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℂ)
110107, 109addcomd 10492 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑁 + 𝑀) = (𝑀 + 𝑁))
11186, 109addcomd 10492 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗 + 𝑀) = (𝑀 + 𝑗))
112111fveq2d 6379 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑁)) → (ℤ‘(𝑗 + 𝑀)) = (ℤ‘(𝑀 + 𝑗)))
113105, 110, 1123eltr3d 2858 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝑀 + 𝑗)))
114113adantr 472 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → (𝑀 + 𝑁) ∈ (ℤ‘(𝑀 + 𝑗)))
115 fzss2 12588 . . . . . . . . . . . 12 ((𝑀 + 𝑁) ∈ (ℤ‘(𝑀 + 𝑗)) → (𝑗...(𝑀 + 𝑗)) ⊆ (𝑗...(𝑀 + 𝑁)))
116114, 115syl 17 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → (𝑗...(𝑀 + 𝑗)) ⊆ (𝑗...(𝑀 + 𝑁)))
117 simpr 477 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗)))
11888adantr 472 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → ((0 + 𝑗)...(𝑀 + 𝑗)) = (𝑗...(𝑀 + 𝑗)))
119117, 118eleqtrd 2846 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑗)))
120116, 119sseldd 3762 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑁)))
121100, 101, 120, 61syl21anc 866 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ ℕ0)
122100, 101, 121, 83syl21anc 866 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝐵 ∈ ℂ)
12399, 122fsumcl 14749 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 ∈ ℂ)
124123addid1d 10490 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 + 0) = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
12585, 98, 1243eqtrrd 2804 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵)
126 fzval3 12745 . . . . . . . . . 10 ((𝑀 + 𝑁) ∈ ℤ → (𝑗...(𝑀 + 𝑁)) = (𝑗..^((𝑀 + 𝑁) + 1)))
12739, 126syl 17 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) = (𝑗..^((𝑀 + 𝑁) + 1)))
128127ineq2d 3976 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∩ (𝑗...(𝑀 + 𝑁))) = ((0..^𝑗) ∩ (𝑗..^((𝑀 + 𝑁) + 1))))
129 fzodisj 12710 . . . . . . . 8 ((0..^𝑗) ∩ (𝑗..^((𝑀 + 𝑁) + 1))) = ∅
130128, 129syl6eq 2815 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∩ (𝑗...(𝑀 + 𝑁))) = ∅)
13139peano2zd 11732 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → ((𝑀 + 𝑁) + 1) ∈ ℤ)
13230nn0ge0d 11601 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 0 ≤ 𝑗)
133131zred 11729 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → ((𝑀 + 𝑁) + 1) ∈ ℝ)
13439zred 11729 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ ℝ)
135 nn0addge2 11587 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → 𝑁 ≤ (𝑀 + 𝑁))
13644, 5, 135syl2anc 579 . . . . . . . . . . . . 13 (𝜑𝑁 ≤ (𝑀 + 𝑁))
137136adantr 472 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ≤ (𝑀 + 𝑁))
138134lep1d 11209 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ≤ ((𝑀 + 𝑁) + 1))
13945, 134, 133, 137, 138letrd 10448 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ≤ ((𝑀 + 𝑁) + 1))
14041, 45, 133, 48, 139letrd 10448 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ≤ ((𝑀 + 𝑁) + 1))
141 elfz4 12542 . . . . . . . . . 10 (((0 ∈ ℤ ∧ ((𝑀 + 𝑁) + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗 ≤ ((𝑀 + 𝑁) + 1))) → 𝑗 ∈ (0...((𝑀 + 𝑁) + 1)))
1424, 131, 3, 132, 140, 141syl32anc 1497 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (0...((𝑀 + 𝑁) + 1)))
143 fzosplit 12709 . . . . . . . . 9 (𝑗 ∈ (0...((𝑀 + 𝑁) + 1)) → (0..^((𝑀 + 𝑁) + 1)) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1))))
144142, 143syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (0..^((𝑀 + 𝑁) + 1)) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1))))
145 fzval3 12745 . . . . . . . . 9 ((𝑀 + 𝑁) ∈ ℤ → (0...(𝑀 + 𝑁)) = (0..^((𝑀 + 𝑁) + 1)))
14639, 145syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) = (0..^((𝑀 + 𝑁) + 1)))
147127uneq2d 3929 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∪ (𝑗...(𝑀 + 𝑁))) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1))))
148144, 146, 1473eqtr4d 2809 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) = ((0..^𝑗) ∪ (𝑗...(𝑀 + 𝑁))))
149 fzfid 12980 . . . . . . . 8 (𝜑 → (0...(𝑀 + 𝑁)) ∈ Fin)
150149adantr 472 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) ∈ Fin)
151 simpl 474 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝜑)
1522adantrl 707 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑗 ∈ (1...𝑁))
153 fz0ssnn0 12642 . . . . . . . . . 10 (0...(𝑀 + 𝑁)) ⊆ ℕ0
154 simprl 787 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
155153, 154sseldi 3759 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑘 ∈ ℕ0)
156151, 152, 155, 83syl21anc 866 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝐵 ∈ ℂ)
157156anass1rs 645 . . . . . . 7 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → 𝐵 ∈ ℂ)
158130, 148, 150, 157fsumsplit 14756 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵 = (Σ𝑘 ∈ (0..^𝑗)𝐵 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵))
159 fzsum2sub.4 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0..^𝑗)) → 𝐵 = 0)
160159sumeq2dv 14718 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0..^𝑗)𝐵 = Σ𝑘 ∈ (0..^𝑗)0)
161 fzofi 12981 . . . . . . . . 9 (0..^𝑗) ∈ Fin
162 sumz 14738 . . . . . . . . . 10 (((0..^𝑗) ⊆ (ℤ‘0) ∨ (0..^𝑗) ∈ Fin) → Σ𝑘 ∈ (0..^𝑗)0 = 0)
163162olcs 902 . . . . . . . . 9 ((0..^𝑗) ∈ Fin → Σ𝑘 ∈ (0..^𝑗)0 = 0)
164161, 163ax-mp 5 . . . . . . . 8 Σ𝑘 ∈ (0..^𝑗)0 = 0
165160, 164syl6eq 2815 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0..^𝑗)𝐵 = 0)
166165oveq1d 6857 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0..^𝑗)𝐵 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) = (0 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵))
16754, 84fsumcl 14749 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 ∈ ℂ)
168167addid2d 10491 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (0 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) = Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵)
169158, 166, 1683eqtrrd 2804 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
170125, 169eqtrd 2799 . . . 4 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
17127, 170eqtrd 2799 . . 3 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
172171sumeq2dv 14718 . 2 (𝜑 → Σ𝑗 ∈ (1...𝑁𝑖 ∈ (0...𝑀)𝐴 = Σ𝑗 ∈ (1...𝑁𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
173 fzfid 12980 . . 3 (𝜑 → (0...𝑀) ∈ Fin)
174 fzfid 12980 . . 3 (𝜑 → (1...𝑁) ∈ Fin)
17525anasss 458 . . . 4 ((𝜑 ∧ (𝑗 ∈ (1...𝑁) ∧ 𝑖 ∈ (0...𝑀))) → 𝐴 ∈ ℂ)
176175ancom2s 640 . . 3 ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (1...𝑁))) → 𝐴 ∈ ℂ)
177173, 174, 176fsumcom 14791 . 2 (𝜑 → Σ𝑖 ∈ (0...𝑀𝑗 ∈ (1...𝑁)𝐴 = Σ𝑗 ∈ (1...𝑁𝑖 ∈ (0...𝑀)𝐴)
178149, 174, 156fsumcom 14791 . 2 (𝜑 → Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵 = Σ𝑗 ∈ (1...𝑁𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
179172, 177, 1783eqtr4d 2809 1 (𝜑 → Σ𝑖 ∈ (0...𝑀𝑗 ∈ (1...𝑁)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  cun 3730  cin 3731  wss 3732  c0 4079   class class class wbr 4809  cfv 6068  (class class class)co 6842  Fincfn 8160  cc 10187  cr 10188  0cc0 10189  1c1 10190   + caddc 10192   < clt 10328  cle 10329  cmin 10520  -cneg 10521  cn 11274  0cn0 11538  cz 11624  cuz 11886  ...cfz 12533  ..^cfzo 12673  Σcsu 14701
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 2070  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 2063  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 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-sum 14702
This theorem is referenced by:  breprexplemc  31161
  Copyright terms: Public domain W3C validator