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Theorem fsum2dsub 30813
Description: Lemma for breprexp 30839- 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 12381 . . . . . 6 (1...𝑁) ⊆ ℤ
2 simpr 476 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
31, 2sseldi 3634 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℤ)
4 0zd 11427 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → 0 ∈ ℤ)
5 fzsum2sub.m . . . . . . 7 (𝜑𝑀 ∈ ℕ0)
65nn0zd 11518 . . . . . 6 (𝜑𝑀 ∈ ℤ)
76adantr 480 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℤ)
8 simpll 805 . . . . . 6 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝜑)
9 fz1ssnn 12410 . . . . . . . . . . . 12 (1...𝑁) ⊆ ℕ
10 nnssnn0 11333 . . . . . . . . . . . 12 ℕ ⊆ ℕ0
119, 10sstri 3645 . . . . . . . . . . 11 (1...𝑁) ⊆ ℕ0
1211, 2sseldi 3634 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ0)
13 nn0uz 11760 . . . . . . . . . 10 0 = (ℤ‘0)
1412, 13syl6eleq 2740 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (ℤ‘0))
15 neg0 10365 . . . . . . . . . 10 -0 = 0
16 uzneg 11744 . . . . . . . . . 10 (𝑗 ∈ (ℤ‘0) → -0 ∈ (ℤ‘-𝑗))
1715, 16syl5eqelr 2735 . . . . . . . . 9 (𝑗 ∈ (ℤ‘0) → 0 ∈ (ℤ‘-𝑗))
18 fzss1 12418 . . . . . . . . 9 (0 ∈ (ℤ‘-𝑗) → (0...𝑀) ⊆ (-𝑗...𝑀))
1914, 17, 183syl 18 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (0...𝑀) ⊆ (-𝑗...𝑀))
20 fzssuz 12420 . . . . . . . 8 (-𝑗...𝑀) ⊆ (ℤ‘-𝑗)
2119, 20syl6ss 3648 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (0...𝑀) ⊆ (ℤ‘-𝑗))
2221sselda 3636 . . . . . 6 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ‘-𝑗))
232adantr 480 . . . . . 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 1366 . . . . 5 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℂ)
26 fzsum2sub.1 . . . . 5 (𝑖 = (𝑘𝑗) → 𝐴 = 𝐵)
273, 4, 7, 25, 26fsumshft 14556 . . . 4 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
285adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℕ0)
299, 2sseldi 3634 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ)
3029nnnn0d 11389 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ0)
3128, 30nn0addcld 11393 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℕ0)
3231nn0red 11390 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℝ)
3332ltp1d 10992 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) < ((𝑀 + 𝑗) + 1))
34 fzdisj 12406 . . . . . . . 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 11518 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℤ)
386, 37zaddcld 11524 . . . . . . . . . 10 (𝜑 → (𝑀 + 𝑁) ∈ ℤ)
3938adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ ℤ)
4031nn0zd 11518 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℤ)
4129nnred 11073 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℝ)
42 nn0addge2 11378 . . . . . . . . . 10 ((𝑗 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → 𝑗 ≤ (𝑀 + 𝑗))
4341, 28, 42syl2anc 694 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ≤ (𝑀 + 𝑗))
4436nn0red 11390 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℝ)
4544adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
4628nn0red 11390 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
47 elfzle2 12383 . . . . . . . . . . 11 (𝑗 ∈ (1...𝑁) → 𝑗𝑁)
4847adantl 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗𝑁)
4941, 45, 46, 48leadd2dd 10680 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ≤ (𝑀 + 𝑁))
50 elfz4 12373 . . . . . . . . 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 1374 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ (𝑗...(𝑀 + 𝑁)))
52 fzsplit 12405 . . . . . . . 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 12812 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) ∈ Fin)
55 simpll 805 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝜑)
562adantr 480 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑗 ∈ (1...𝑁))
5711, 56sseldi 3634 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑗 ∈ ℕ0)
58 fz2ssnn0 29675 . . . . . . . . . 10 (𝑗 ∈ ℕ0 → (𝑗...(𝑀 + 𝑁)) ⊆ ℕ0)
5957, 58syl 17 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → (𝑗...(𝑀 + 𝑁)) ⊆ ℕ0)
60 simpr 476 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑁)))
6159, 60sseldd 3637 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑘 ∈ ℕ0)
6226eleq1d 2715 . . . . . . . . 9 (𝑖 = (𝑘𝑗) → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
63 simpll 805 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝜑)
64 simplr 807 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (ℤ‘-𝑗))
65 simpr 476 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
6663, 64, 65, 24syl3anc 1366 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (ℤ‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ)
6766an32s 863 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (ℤ‘-𝑗)) → 𝐴 ∈ ℂ)
6867ralrimiva 2995 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → ∀𝑖 ∈ (ℤ‘-𝑗)𝐴 ∈ ℂ)
6968adantr 480 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → ∀𝑖 ∈ (ℤ‘-𝑗)𝐴 ∈ ℂ)
70 nnsscn 11063 . . . . . . . . . . . . 13 ℕ ⊆ ℂ
719, 70sstri 3645 . . . . . . . . . . . 12 (1...𝑁) ⊆ ℂ
72 simplr 807 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (1...𝑁))
7371, 72sseldi 3634 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ ℂ)
74 simpr 476 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
7574nn0cnd 11391 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
7673, 75negsubdi2d 10446 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → -(𝑗𝑘) = (𝑘𝑗))
771, 72sseldi 3634 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ ℤ)
78 eluzmn 11732 . . . . . . . . . . . 12 ((𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (ℤ‘(𝑗𝑘)))
7977, 74, 78syl2anc 694 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (ℤ‘(𝑗𝑘)))
80 uzneg 11744 . . . . . . . . . . 11 (𝑗 ∈ (ℤ‘(𝑗𝑘)) → -(𝑗𝑘) ∈ (ℤ‘-𝑗))
8179, 80syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → -(𝑗𝑘) ∈ (ℤ‘-𝑗))
8276, 81eqeltrrd 2731 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → (𝑘𝑗) ∈ (ℤ‘-𝑗))
8362, 69, 82rspcdva 3347 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
8455, 56, 61, 83syl21anc 1365 . . . . . . 7 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝐵 ∈ ℂ)
8535, 53, 54, 84fsumsplit 14515 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 = (Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 + Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵))
863zcnd 11521 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℂ)
8786addid2d 10275 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → (0 + 𝑗) = 𝑗)
8887oveq1d 6705 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → ((0 + 𝑗)...(𝑀 + 𝑗)) = (𝑗...(𝑀 + 𝑗)))
8988eqcomd 2657 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑗)) = ((0 + 𝑗)...(𝑀 + 𝑗)))
9089sumeq1d 14475 . . . . . . 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 14477 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵 = Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0)
93 fzfi 12811 . . . . . . . . 9 (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin
94 sumz 14497 . . . . . . . . . 10 (((((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ⊆ (ℤ‘0) ∨ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0)
9594olcs 409 . . . . . . . . 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 2701 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵 = 0)
9890, 97oveq12d 6708 . . . . . 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 12812 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → ((0 + 𝑗)...(𝑀 + 𝑗)) ∈ Fin)
100 simpll 805 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝜑)
1012adantr 480 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑗 ∈ (1...𝑁))
102 elfzuz3 12377 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (1...𝑁) → 𝑁 ∈ (ℤ𝑗))
103102adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ𝑗))
104 eluzadd 11754 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ𝑗) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈ (ℤ‘(𝑗 + 𝑀)))
105103, 7, 104syl2anc 694 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑁 + 𝑀) ∈ (ℤ‘(𝑗 + 𝑀)))
10636nn0cnd 11391 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℂ)
107106adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
108 zsscn 11423 . . . . . . . . . . . . . . . 16 ℤ ⊆ ℂ
109108, 7sseldi 3634 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℂ)
110107, 109addcomd 10276 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑁 + 𝑀) = (𝑀 + 𝑁))
11186, 109addcomd 10276 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑗 + 𝑀) = (𝑀 + 𝑗))
112111fveq2d 6233 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑁)) → (ℤ‘(𝑗 + 𝑀)) = (ℤ‘(𝑀 + 𝑗)))
113105, 110, 1123eltr3d 2744 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝑀 + 𝑗)))
114113adantr 480 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → (𝑀 + 𝑁) ∈ (ℤ‘(𝑀 + 𝑗)))
115 fzss2 12419 . . . . . . . . . . . 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 476 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗)))
11888adantr 480 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → ((0 + 𝑗)...(𝑀 + 𝑗)) = (𝑗...(𝑀 + 𝑗)))
119117, 118eleqtrd 2732 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑗)))
120116, 119sseldd 3637 . . . . . . . . . 10 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑁)))
121100, 101, 120, 61syl21anc 1365 . . . . . . . . 9 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ ℕ0)
122100, 101, 121, 83syl21anc 1365 . . . . . . . 8 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝐵 ∈ ℂ)
12399, 122fsumcl 14508 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 ∈ ℂ)
124123addid1d 10274 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 + 0) = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
12585, 98, 1243eqtrrd 2690 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵)
126 fzval3 12576 . . . . . . . . . 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 3847 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∩ (𝑗...(𝑀 + 𝑁))) = ((0..^𝑗) ∩ (𝑗..^((𝑀 + 𝑁) + 1))))
129 fzodisj 12541 . . . . . . . 8 ((0..^𝑗) ∩ (𝑗..^((𝑀 + 𝑁) + 1))) = ∅
130128, 129syl6eq 2701 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∩ (𝑗...(𝑀 + 𝑁))) = ∅)
13139peano2zd 11523 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → ((𝑀 + 𝑁) + 1) ∈ ℤ)
13230nn0ge0d 11392 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 0 ≤ 𝑗)
133131zred 11520 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → ((𝑀 + 𝑁) + 1) ∈ ℝ)
13439zred 11520 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ ℝ)
135 nn0addge2 11378 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → 𝑁 ≤ (𝑀 + 𝑁))
13644, 5, 135syl2anc 694 . . . . . . . . . . . . 13 (𝜑𝑁 ≤ (𝑀 + 𝑁))
137136adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ≤ (𝑀 + 𝑁))
138134lep1d 10993 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ≤ ((𝑀 + 𝑁) + 1))
13945, 134, 133, 137, 138letrd 10232 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑁 ≤ ((𝑀 + 𝑁) + 1))
14041, 45, 133, 48, 139letrd 10232 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ≤ ((𝑀 + 𝑁) + 1))
141 elfz4 12373 . . . . . . . . . 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 1374 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (0...((𝑀 + 𝑁) + 1)))
143 fzosplit 12540 . . . . . . . . 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 12576 . . . . . . . . 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 3800 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∪ (𝑗...(𝑀 + 𝑁))) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1))))
148144, 146, 1473eqtr4d 2695 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) = ((0..^𝑗) ∪ (𝑗...(𝑀 + 𝑁))))
149 fzfid 12812 . . . . . . . 8 (𝜑 → (0...(𝑀 + 𝑁)) ∈ Fin)
150149adantr 480 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) ∈ Fin)
151 simpl 472 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝜑)
1522adantrl 752 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑗 ∈ (1...𝑁))
153 fz0ssnn0 12473 . . . . . . . . . 10 (0...(𝑀 + 𝑁)) ⊆ ℕ0
154 simprl 809 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
155153, 154sseldi 3634 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑘 ∈ ℕ0)
156151, 152, 155, 83syl21anc 1365 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝐵 ∈ ℂ)
157156anass1rs 866 . . . . . . 7 (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → 𝐵 ∈ ℂ)
158130, 148, 150, 157fsumsplit 14515 . . . . . 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 14477 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0..^𝑗)𝐵 = Σ𝑘 ∈ (0..^𝑗)0)
161 fzofi 12813 . . . . . . . . 9 (0..^𝑗) ∈ Fin
162 sumz 14497 . . . . . . . . . 10 (((0..^𝑗) ⊆ (ℤ‘0) ∨ (0..^𝑗) ∈ Fin) → Σ𝑘 ∈ (0..^𝑗)0 = 0)
163162olcs 409 . . . . . . . . 9 ((0..^𝑗) ∈ Fin → Σ𝑘 ∈ (0..^𝑗)0 = 0)
164161, 163ax-mp 5 . . . . . . . 8 Σ𝑘 ∈ (0..^𝑗)0 = 0
165160, 164syl6eq 2701 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0..^𝑗)𝐵 = 0)
166165oveq1d 6705 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0..^𝑗)𝐵 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) = (0 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵))
16754, 84fsumcl 14508 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 ∈ ℂ)
168167addid2d 10275 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑁)) → (0 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) = Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵)
169158, 166, 1683eqtrrd 2690 . . . . 5 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
170125, 169eqtrd 2685 . . . 4 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
17127, 170eqtrd 2685 . . 3 ((𝜑𝑗 ∈ (1...𝑁)) → Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
172171sumeq2dv 14477 . 2 (𝜑 → Σ𝑗 ∈ (1...𝑁𝑖 ∈ (0...𝑀)𝐴 = Σ𝑗 ∈ (1...𝑁𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
173 fzfid 12812 . . 3 (𝜑 → (0...𝑀) ∈ Fin)
174 fzfid 12812 . . 3 (𝜑 → (1...𝑁) ∈ Fin)
17525anasss 680 . . . 4 ((𝜑 ∧ (𝑗 ∈ (1...𝑁) ∧ 𝑖 ∈ (0...𝑀))) → 𝐴 ∈ ℂ)
176175ancom2s 861 . . 3 ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (1...𝑁))) → 𝐴 ∈ ℂ)
177173, 174, 176fsumcom 14551 . 2 (𝜑 → Σ𝑖 ∈ (0...𝑀𝑗 ∈ (1...𝑁)𝐴 = Σ𝑗 ∈ (1...𝑁𝑖 ∈ (0...𝑀)𝐴)
178149, 174, 156fsumcom 14551 . 2 (𝜑 → Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵 = Σ𝑗 ∈ (1...𝑁𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
179172, 177, 1783eqtr4d 2695 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 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cun 3605  cin 3606  wss 3607  c0 3948   class class class wbr 4685  cfv 5926  (class class class)co 6690  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304  -cneg 10305  cn 11058  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  ..^cfzo 12504  Σcsu 14460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461
This theorem is referenced by:  breprexplemc  30838
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