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Theorem fsum2dsub 34622
Description: Lemma for breprexp 34648- 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 simpr 484 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
21elfzelzd 13565 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℤ)
3 0zd 12625 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → 0 ∈ ℤ)
4 fzsum2sub.m . . . . . . 7 (𝜑𝑀 ∈ ℕ0)
54nn0zd 12639 . . . . . 6 (𝜑𝑀 ∈ ℤ)
65adantr 480 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℤ)
7 simpll 767 . . . . . 6 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝜑)
8 fz1ssnn 13595 . . . . . . . . . . . 12 (1...𝑁) ⊆ ℕ
9 nnssnn0 12529 . . . . . . . . . . . 12 ℕ ⊆ ℕ0
108, 9sstri 3993 . . . . . . . . . . 11 (1...𝑁) ⊆ ℕ0
1110, 1sselid 3981 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ0)
12 nn0uz 12920 . . . . . . . . . 10 0 = (ℤ‘0)
1311, 12eleqtrdi 2851 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (ℤ‘0))
14 neg0 11555 . . . . . . . . . 10 -0 = 0
15 uzneg 12898 . . . . . . . . . 10 (𝑗 ∈ (ℤ‘0) → -0 ∈ (ℤ‘-𝑗))
1614, 15eqeltrrid 2846 . . . . . . . . 9 (𝑗 ∈ (ℤ‘0) → 0 ∈ (ℤ‘-𝑗))
17 fzss1 13603 . . . . . . . . 9 (0 ∈ (ℤ‘-𝑗) → (0...𝑀) ⊆ (-𝑗...𝑀))
1813, 16, 173syl 18 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (0...𝑀) ⊆ (-𝑗...𝑀))
19 fzssuz 13605 . . . . . . . 8 (-𝑗...𝑀) ⊆ (ℤ‘-𝑗)
2018, 19sstrdi 3996 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (0...𝑀) ⊆ (ℤ‘-𝑗))
2120sselda 3983 . . . . . 6 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ‘-𝑗))
221adantr 480 . . . . . 6 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝑗 ∈ (1...𝑁))
23 fzsum2sub.2 . . . . . 6 ((𝜑𝑖 ∈ (ℤ‘-𝑗) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ)
247, 21, 22, 23syl3anc 1373 . . . . 5 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℂ)
25 fzsum2sub.1 . . . . 5 (𝑖 = (𝑘𝑗) → 𝐴 = 𝐵)
262, 3, 6, 24, 25fsumshft 15816 . . . 4 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
274adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℕ0)
288, 1sselid 3981 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ)
2928nnnn0d 12587 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ0)
3027, 29nn0addcld 12591 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℕ0)
3130nn0red 12588 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℝ)
3231ltp1d 12198 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) < ((𝑀 + 𝑗) + 1))
33 fzdisj 13591 . . . . . . . 8 ((𝑀 + 𝑗) < ((𝑀 + 𝑗) + 1) → ((𝑗...(𝑀 + 𝑗)) ∩ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) = ∅)
3432, 33syl 17 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → ((𝑗...(𝑀 + 𝑗)) ∩ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) = ∅)
35 fzsum2sub.n . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
3635nn0zd 12639 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℤ)
375, 36zaddcld 12726 . . . . . . . . . 10 (𝜑 → (𝑀 + 𝑁) ∈ ℤ)
3837adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ ℤ)
3930nn0zd 12639 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℤ)
4028nnred 12281 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℝ)
41 nn0addge2 12573 . . . . . . . . . 10 ((𝑗 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → 𝑗 ≤ (𝑀 + 𝑗))
4240, 27, 41syl2anc 584 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ≤ (𝑀 + 𝑗))
4335nn0red 12588 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℝ)
4443adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
4527nn0red 12588 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
46 elfzle2 13568 . . . . . . . . . . 11 (𝑗 ∈ (1...𝑁) → 𝑗𝑁)
4746adantl 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗𝑁)
4840, 44, 45, 47leadd2dd 11878 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ≤ (𝑀 + 𝑁))
492, 38, 39, 42, 48elfzd 13555 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ (𝑗...(𝑀 + 𝑁)))
50 fzsplit 13590 . . . . . . . 8 ((𝑀 + 𝑗) ∈ (𝑗...(𝑀 + 𝑁)) → (𝑗...(𝑀 + 𝑁)) = ((𝑗...(𝑀 + 𝑗)) ∪ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))))
5149, 50syl 17 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) = ((𝑗...(𝑀 + 𝑗)) ∪ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))))
52 fzfid 14014 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) ∈ Fin)
53 simpll 767 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝜑)
541adantr 480 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑗 ∈ (1...𝑁))
5510, 54sselid 3981 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑗 ∈ ℕ0)
56 fz2ssnn0 32787 . . . . . . . . . 10 (𝑗 ∈ ℕ0 → (𝑗...(𝑀 + 𝑁)) ⊆ ℕ0)
5755, 56syl 17 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → (𝑗...(𝑀 + 𝑁)) ⊆ ℕ0)
58 simpr 484 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑁)))
5957, 58sseldd 3984 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑘 ∈ ℕ0)
6025eleq1d 2826 . . . . . . . . 9 (𝑖 = (𝑘𝑗) → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
61 simpll 767 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝜑)
62 simplr 769 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (ℤ‘-𝑗))
63 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
6461, 62, 63, 23syl3anc 1373 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ)
6564an32s 652 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (ℤ‘-𝑗)) → 𝐴 ∈ ℂ)
6665ralrimiva 3146 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → ∀𝑖 ∈ (ℤ‘-𝑗)𝐴 ∈ ℂ)
6766adantr 480 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → ∀𝑖 ∈ (ℤ‘-𝑗)𝐴 ∈ ℂ)
68 nnsscn 12271 . . . . . . . . . . . . 13 ℕ ⊆ ℂ
698, 68sstri 3993 . . . . . . . . . . . 12 (1...𝑁) ⊆ ℂ
70 simplr 769 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (1...𝑁))
7169, 70sselid 3981 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ ℂ)
72 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
7372nn0cnd 12589 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
7471, 73negsubdi2d 11636 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → -(𝑗𝑘) = (𝑘𝑗))
7570elfzelzd 13565 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ ℤ)
76 eluzmn 12885 . . . . . . . . . . . 12 ((𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (ℤ‘(𝑗𝑘)))
7775, 72, 76syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (ℤ‘(𝑗𝑘)))
78 uzneg 12898 . . . . . . . . . . 11 (𝑗 ∈ (ℤ‘(𝑗𝑘)) → -(𝑗𝑘) ∈ (ℤ‘-𝑗))
7977, 78syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → -(𝑗𝑘) ∈ (ℤ‘-𝑗))
8074, 79eqeltrrd 2842 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → (𝑘𝑗) ∈ (ℤ‘-𝑗))
8160, 67, 80rspcdva 3623 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
8253, 54, 59, 81syl21anc 838 . . . . . . 7 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝐵 ∈ ℂ)
8334, 51, 52, 82fsumsplit 15777 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 = (Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 + Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵))
842zcnd 12723 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℂ)
8584addlidd 11462 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → (0 + 𝑗) = 𝑗)
8685oveq1d 7446 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → ((0 + 𝑗)...(𝑀 + 𝑗)) = (𝑗...(𝑀 + 𝑗)))
8786eqcomd 2743 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑗)) = ((0 + 𝑗)...(𝑀 + 𝑗)))
8887sumeq1d 15736 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
89 fzsum2sub.3 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) → 𝐵 = 0)
9089sumeq2dv 15738 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵 = Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0)
91 fzfi 14013 . . . . . . . . 9 (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin
92 sumz 15758 . . . . . . . . . 10 (((((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ⊆ (ℤ‘0) ∨ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0)
9392olcs 877 . . . . . . . . 9 ((((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0)
9491, 93ax-mp 5 . . . . . . . 8 Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0
9590, 94eqtrdi 2793 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵 = 0)
9688, 95oveq12d 7449 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 + Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵) = (Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 + 0))
97 fzfid 14014 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → ((0 + 𝑗)...(𝑀 + 𝑗)) ∈ Fin)
98 simpll 767 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝜑)
991adantr 480 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑗 ∈ (1...𝑁))
100 elfzuz3 13561 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (1...𝑁) → 𝑁 ∈ (ℤ𝑗))
101100adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ𝑗))
102 eluzadd 12907 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ𝑗) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈ (ℤ‘(𝑗 + 𝑀)))
103101, 6, 102syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑁 + 𝑀) ∈ (ℤ‘(𝑗 + 𝑀)))
10435nn0cnd 12589 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℂ)
105104adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
106 zsscn 12621 . . . . . . . . . . . . . . . 16 ℤ ⊆ ℂ
107106, 6sselid 3981 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℂ)
108105, 107addcomd 11463 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑁 + 𝑀) = (𝑀 + 𝑁))
10984, 107addcomd 11463 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗 + 𝑀) = (𝑀 + 𝑗))
110109fveq2d 6910 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑁)) → (ℤ‘(𝑗 + 𝑀)) = (ℤ‘(𝑀 + 𝑗)))
111103, 108, 1103eltr3d 2855 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝑀 + 𝑗)))
112111adantr 480 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → (𝑀 + 𝑁) ∈ (ℤ‘(𝑀 + 𝑗)))
113 fzss2 13604 . . . . . . . . . . . 12 ((𝑀 + 𝑁) ∈ (ℤ‘(𝑀 + 𝑗)) → (𝑗...(𝑀 + 𝑗)) ⊆ (𝑗...(𝑀 + 𝑁)))
114112, 113syl 17 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → (𝑗...(𝑀 + 𝑗)) ⊆ (𝑗...(𝑀 + 𝑁)))
115 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗)))
11686adantr 480 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → ((0 + 𝑗)...(𝑀 + 𝑗)) = (𝑗...(𝑀 + 𝑗)))
117115, 116eleqtrd 2843 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑗)))
118114, 117sseldd 3984 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑁)))
11998, 99, 118, 59syl21anc 838 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ ℕ0)
12098, 99, 119, 81syl21anc 838 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝐵 ∈ ℂ)
12197, 120fsumcl 15769 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 ∈ ℂ)
122121addridd 11461 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 + 0) = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
12383, 96, 1223eqtrrd 2782 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵)
124 fzval3 13773 . . . . . . . . . 10 ((𝑀 + 𝑁) ∈ ℤ → (𝑗...(𝑀 + 𝑁)) = (𝑗..^((𝑀 + 𝑁) + 1)))
12538, 124syl 17 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) = (𝑗..^((𝑀 + 𝑁) + 1)))
126125ineq2d 4220 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∩ (𝑗...(𝑀 + 𝑁))) = ((0..^𝑗) ∩ (𝑗..^((𝑀 + 𝑁) + 1))))
127 fzodisj 13733 . . . . . . . 8 ((0..^𝑗) ∩ (𝑗..^((𝑀 + 𝑁) + 1))) = ∅
128126, 127eqtrdi 2793 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∩ (𝑗...(𝑀 + 𝑁))) = ∅)
12938peano2zd 12725 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → ((𝑀 + 𝑁) + 1) ∈ ℤ)
13029nn0ge0d 12590 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 0 ≤ 𝑗)
131129zred 12722 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → ((𝑀 + 𝑁) + 1) ∈ ℝ)
13238zred 12722 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ ℝ)
133 nn0addge2 12573 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → 𝑁 ≤ (𝑀 + 𝑁))
13443, 4, 133syl2anc 584 . . . . . . . . . . . . 13 (𝜑𝑁 ≤ (𝑀 + 𝑁))
135134adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ≤ (𝑀 + 𝑁))
136132lep1d 12199 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ≤ ((𝑀 + 𝑁) + 1))
13744, 132, 131, 135, 136letrd 11418 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ≤ ((𝑀 + 𝑁) + 1))
13840, 44, 131, 47, 137letrd 11418 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ≤ ((𝑀 + 𝑁) + 1))
1393, 129, 2, 130, 138elfzd 13555 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (0...((𝑀 + 𝑁) + 1)))
140 fzosplit 13732 . . . . . . . . 9 (𝑗 ∈ (0...((𝑀 + 𝑁) + 1)) → (0..^((𝑀 + 𝑁) + 1)) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1))))
141139, 140syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (0..^((𝑀 + 𝑁) + 1)) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1))))
142 fzval3 13773 . . . . . . . . 9 ((𝑀 + 𝑁) ∈ ℤ → (0...(𝑀 + 𝑁)) = (0..^((𝑀 + 𝑁) + 1)))
14338, 142syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) = (0..^((𝑀 + 𝑁) + 1)))
144125uneq2d 4168 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∪ (𝑗...(𝑀 + 𝑁))) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1))))
145141, 143, 1443eqtr4d 2787 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) = ((0..^𝑗) ∪ (𝑗...(𝑀 + 𝑁))))
146 fzfid 14014 . . . . . . . 8 (𝜑 → (0...(𝑀 + 𝑁)) ∈ Fin)
147146adantr 480 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) ∈ Fin)
148 simpl 482 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝜑)
1491adantrl 716 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑗 ∈ (1...𝑁))
150 fz0ssnn0 13662 . . . . . . . . . 10 (0...(𝑀 + 𝑁)) ⊆ ℕ0
151 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
152150, 151sselid 3981 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑘 ∈ ℕ0)
153148, 149, 152, 81syl21anc 838 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝐵 ∈ ℂ)
154153anass1rs 655 . . . . . . 7 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → 𝐵 ∈ ℂ)
155128, 145, 147, 154fsumsplit 15777 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵 = (Σ𝑘 ∈ (0..^𝑗)𝐵 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵))
156 fzsum2sub.4 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0..^𝑗)) → 𝐵 = 0)
157156sumeq2dv 15738 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0..^𝑗)𝐵 = Σ𝑘 ∈ (0..^𝑗)0)
158 fzofi 14015 . . . . . . . . 9 (0..^𝑗) ∈ Fin
159 sumz 15758 . . . . . . . . . 10 (((0..^𝑗) ⊆ (ℤ‘0) ∨ (0..^𝑗) ∈ Fin) → Σ𝑘 ∈ (0..^𝑗)0 = 0)
160159olcs 877 . . . . . . . . 9 ((0..^𝑗) ∈ Fin → Σ𝑘 ∈ (0..^𝑗)0 = 0)
161158, 160ax-mp 5 . . . . . . . 8 Σ𝑘 ∈ (0..^𝑗)0 = 0
162157, 161eqtrdi 2793 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0..^𝑗)𝐵 = 0)
163162oveq1d 7446 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0..^𝑗)𝐵 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) = (0 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵))
16452, 82fsumcl 15769 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 ∈ ℂ)
165164addlidd 11462 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (0 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) = Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵)
166155, 163, 1653eqtrrd 2782 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
167123, 166eqtrd 2777 . . . 4 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
16826, 167eqtrd 2777 . . 3 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
169168sumeq2dv 15738 . 2 (𝜑 → Σ𝑗 ∈ (1...𝑁𝑖 ∈ (0...𝑀)𝐴 = Σ𝑗 ∈ (1...𝑁𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
170 fzfid 14014 . . 3 (𝜑 → (0...𝑀) ∈ Fin)
171 fzfid 14014 . . 3 (𝜑 → (1...𝑁) ∈ Fin)
17224anasss 466 . . . 4 ((𝜑 ∧ (𝑗 ∈ (1...𝑁) ∧ 𝑖 ∈ (0...𝑀))) → 𝐴 ∈ ℂ)
173172ancom2s 650 . . 3 ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (1...𝑁))) → 𝐴 ∈ ℂ)
174170, 171, 173fsumcom 15811 . 2 (𝜑 → Σ𝑖 ∈ (0...𝑀𝑗 ∈ (1...𝑁)𝐴 = Σ𝑗 ∈ (1...𝑁𝑖 ∈ (0...𝑀)𝐴)
175146, 171, 153fsumcom 15811 . 2 (𝜑 → Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵 = Σ𝑗 ∈ (1...𝑁𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
176169, 174, 1753eqtr4d 2787 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 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cun 3949  cin 3950  wss 3951  c0 4333   class class class wbr 5143  cfv 6561  (class class class)co 7431  Fincfn 8985  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cmin 11492  -cneg 11493  cn 12266  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  ..^cfzo 13694  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:  breprexplemc  34647
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