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Theorem fprodefsum 15032
Description: Move the exponential function from inside a finite product to outside a finite sum. (Contributed by Scott Fenton, 26-Dec-2017.)
Hypotheses
Ref Expression
fprodefsum.1 𝑍 = (ℤ𝑀)
fprodefsum.2 (𝜑𝑁𝑍)
fprodefsum.3 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
Assertion
Ref Expression
fprodefsum (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))
Distinct variable groups:   𝜑,𝑘   𝑘,𝑀   𝑘,𝑁   𝑘,𝑍
Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem fprodefsum
Dummy variables 𝑎 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fprodefsum.2 . . . 4 (𝜑𝑁𝑍)
2 fprodefsum.1 . . . 4 𝑍 = (ℤ𝑀)
31, 2syl6eleq 2860 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
4 oveq2 6802 . . . . . . 7 (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
54prodeq1d 14859 . . . . . 6 (𝑎 = 𝑀 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
64sumeq1d 14640 . . . . . . 7 (𝑎 = 𝑀 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚))
76fveq2d 6337 . . . . . 6 (𝑎 = 𝑀 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)))
85, 7eqeq12d 2786 . . . . 5 (𝑎 = 𝑀 → (∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) ↔ ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚))))
98imbi2d 329 . . . 4 (𝑎 = 𝑀 → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚))) ↔ (𝜑 → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)))))
10 oveq2 6802 . . . . . . 7 (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
1110prodeq1d 14859 . . . . . 6 (𝑎 = 𝑛 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
1210sumeq1d 14640 . . . . . . 7 (𝑎 = 𝑛 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))
1312fveq2d 6337 . . . . . 6 (𝑎 = 𝑛 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)))
1411, 13eqeq12d 2786 . . . . 5 (𝑎 = 𝑛 → (∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) ↔ ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))))
1514imbi2d 329 . . . 4 (𝑎 = 𝑛 → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚))) ↔ (𝜑 → ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)))))
16 oveq2 6802 . . . . . . 7 (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
1716prodeq1d 14859 . . . . . 6 (𝑎 = (𝑛 + 1) → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
1816sumeq1d 14640 . . . . . . 7 (𝑎 = (𝑛 + 1) → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚))
1918fveq2d 6337 . . . . . 6 (𝑎 = (𝑛 + 1) → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)))
2017, 19eqeq12d 2786 . . . . 5 (𝑎 = (𝑛 + 1) → (∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) ↔ ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚))))
2120imbi2d 329 . . . 4 (𝑎 = (𝑛 + 1) → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚))) ↔ (𝜑 → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)))))
22 oveq2 6802 . . . . . . 7 (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
2322prodeq1d 14859 . . . . . 6 (𝑎 = 𝑁 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
2422sumeq1d 14640 . . . . . . 7 (𝑎 = 𝑁 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚))
2524fveq2d 6337 . . . . . 6 (𝑎 = 𝑁 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚)))
2623, 25eqeq12d 2786 . . . . 5 (𝑎 = 𝑁 → (∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) ↔ ∏𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚))))
2726imbi2d 329 . . . 4 (𝑎 = 𝑁 → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚))) ↔ (𝜑 → ∏𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚)))))
28 fzsn 12591 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
2928adantl 467 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
3029prodeq1d 14859 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
31 simpr 471 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
32 uzid 11904 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3332, 2syl6eleqr 2861 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀𝑍)
34 fprodefsum.3 . . . . . . . . . . . 12 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
35 efcl 15020 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
3634, 35syl 17 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (exp‘𝐴) ∈ ℂ)
37 eqid 2771 . . . . . . . . . . 11 (𝑘𝑍 ↦ (exp‘𝐴)) = (𝑘𝑍 ↦ (exp‘𝐴))
3836, 37fmptd 6528 . . . . . . . . . 10 (𝜑 → (𝑘𝑍 ↦ (exp‘𝐴)):𝑍⟶ℂ)
3938ffvelrnda 6503 . . . . . . . . 9 ((𝜑𝑀𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
4033, 39sylan2 574 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
41 fveq2 6333 . . . . . . . . 9 (𝑚 = 𝑀 → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4241prodsn 14900 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ) → ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4331, 40, 42syl2anc 567 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4433adantl 467 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → 𝑀𝑍)
45 fvex 6343 . . . . . . . 8 (exp‘𝑀 / 𝑘𝐴) ∈ V
46 nfcv 2913 . . . . . . . . 9 𝑘𝑀
47 nfcv 2913 . . . . . . . . . 10 𝑘exp
48 nfcsb1v 3699 . . . . . . . . . 10 𝑘𝑀 / 𝑘𝐴
4947, 48nffv 6340 . . . . . . . . 9 𝑘(exp‘𝑀 / 𝑘𝐴)
50 csbeq1a 3692 . . . . . . . . . 10 (𝑘 = 𝑀𝐴 = 𝑀 / 𝑘𝐴)
5150fveq2d 6337 . . . . . . . . 9 (𝑘 = 𝑀 → (exp‘𝐴) = (exp‘𝑀 / 𝑘𝐴))
5246, 49, 51, 37fvmptf 6444 . . . . . . . 8 ((𝑀𝑍 ∧ (exp‘𝑀 / 𝑘𝐴) ∈ V) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘𝑀 / 𝑘𝐴))
5344, 45, 52sylancl 568 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘𝑀 / 𝑘𝐴))
5430, 43, 533eqtrd 2809 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘𝑀 / 𝑘𝐴))
5529sumeq1d 14640 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚))
56 eqid 2771 . . . . . . . . . . . 12 (𝑘𝑍𝐴) = (𝑘𝑍𝐴)
5734, 56fmptd 6528 . . . . . . . . . . 11 (𝜑 → (𝑘𝑍𝐴):𝑍⟶ℂ)
5857ffvelrnda 6503 . . . . . . . . . 10 ((𝜑𝑀𝑍) → ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ)
5933, 58sylan2 574 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ)
60 fveq2 6333 . . . . . . . . . 10 (𝑚 = 𝑀 → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6160sumsn 14684 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ) → Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6231, 59, 61syl2anc 567 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6334ralrimiva 3115 . . . . . . . . . 10 (𝜑 → ∀𝑘𝑍 𝐴 ∈ ℂ)
6448nfel1 2928 . . . . . . . . . . . 12 𝑘𝑀 / 𝑘𝐴 ∈ ℂ
6550eleq1d 2835 . . . . . . . . . . . 12 (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝑀 / 𝑘𝐴 ∈ ℂ))
6664, 65rspc 3455 . . . . . . . . . . 11 (𝑀𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → 𝑀 / 𝑘𝐴 ∈ ℂ))
6766impcom 394 . . . . . . . . . 10 ((∀𝑘𝑍 𝐴 ∈ ℂ ∧ 𝑀𝑍) → 𝑀 / 𝑘𝐴 ∈ ℂ)
6863, 33, 67syl2an 577 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘𝐴 ∈ ℂ)
6956fvmpts 6428 . . . . . . . . 9 ((𝑀𝑍𝑀 / 𝑘𝐴 ∈ ℂ) → ((𝑘𝑍𝐴)‘𝑀) = 𝑀 / 𝑘𝐴)
7044, 68, 69syl2anc 567 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍𝐴)‘𝑀) = 𝑀 / 𝑘𝐴)
7155, 62, 703eqtrd 2809 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚) = 𝑀 / 𝑘𝐴)
7271fveq2d 6337 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)) = (exp‘𝑀 / 𝑘𝐴))
7354, 72eqtr4d 2808 . . . . 5 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)))
7473expcom 398 . . . 4 (𝑀 ∈ ℤ → (𝜑 → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚))))
75 simp3 1132 . . . . . . . . . 10 ((𝜑𝑛𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))) → ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)))
762peano2uzs 11945 . . . . . . . . . . . 12 (𝑛𝑍 → (𝑛 + 1) ∈ 𝑍)
77 simpr 471 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
78 nfcsb1v 3699 . . . . . . . . . . . . . . . . . 18 𝑘(𝑛 + 1) / 𝑘𝐴
7978nfel1 2928 . . . . . . . . . . . . . . . . 17 𝑘(𝑛 + 1) / 𝑘𝐴 ∈ ℂ
80 csbeq1a 3692 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛 + 1) → 𝐴 = (𝑛 + 1) / 𝑘𝐴)
8180eleq1d 2835 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8279, 81rspc 3455 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8363, 82mpan9 492 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
84 efcl 15020 . . . . . . . . . . . . . . 15 ((𝑛 + 1) / 𝑘𝐴 ∈ ℂ → (exp‘(𝑛 + 1) / 𝑘𝐴) ∈ ℂ)
8583, 84syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (exp‘(𝑛 + 1) / 𝑘𝐴) ∈ ℂ)
86 nfcv 2913 . . . . . . . . . . . . . . 15 𝑘(𝑛 + 1)
8747, 78nffv 6340 . . . . . . . . . . . . . . 15 𝑘(exp‘(𝑛 + 1) / 𝑘𝐴)
8880fveq2d 6337 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → (exp‘𝐴) = (exp‘(𝑛 + 1) / 𝑘𝐴))
8986, 87, 88, 37fvmptf 6444 . . . . . . . . . . . . . 14 (((𝑛 + 1) ∈ 𝑍 ∧ (exp‘(𝑛 + 1) / 𝑘𝐴) ∈ ℂ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9077, 85, 89syl2anc 567 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9156fvmpts 6428 . . . . . . . . . . . . . . 15 (((𝑛 + 1) ∈ 𝑍(𝑛 + 1) / 𝑘𝐴 ∈ ℂ) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) = (𝑛 + 1) / 𝑘𝐴)
9277, 83, 91syl2anc 567 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) = (𝑛 + 1) / 𝑘𝐴)
9392fveq2d 6337 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9490, 93eqtr4d 2808 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))))
9576, 94sylan2 574 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))))
96953adant3 1126 . . . . . . . . . 10 ((𝜑𝑛𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))))
9775, 96oveq12d 6812 . . . . . . . . 9 ((𝜑𝑛𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))) → (∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) · ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1))) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
98 simpr 471 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → 𝑛𝑍)
9998, 2syl6eleq 2860 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → 𝑛 ∈ (ℤ𝑀))
100 elfzuz 12546 . . . . . . . . . . . . . 14 (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚 ∈ (ℤ𝑀))
101100, 2syl6eleqr 2861 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚𝑍)
10238ffvelrnda 6503 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
103101, 102sylan2 574 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
104103adantlr 688 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
105 fveq2 6333 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)))
10699, 104, 105fprodp1 14907 . . . . . . . . . 10 ((𝜑𝑛𝑍) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) · ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1))))
1071063adant3 1126 . . . . . . . . 9 ((𝜑𝑛𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) · ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1))))
10857ffvelrnda 6503 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
109101, 108sylan2 574 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
110109adantlr 688 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
111 fveq2 6333 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘(𝑛 + 1)))
11299, 110, 111fsump1 14696 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚) = (Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1))))
113112fveq2d 6337 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)) = (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1)))))
114 fzfid 12981 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → (𝑀...𝑛) ∈ Fin)
115 elfzuz 12546 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (𝑀...𝑛) → 𝑚 ∈ (ℤ𝑀))
116115, 2syl6eleqr 2861 . . . . . . . . . . . . . . 15 (𝑚 ∈ (𝑀...𝑛) → 𝑚𝑍)
117116, 108sylan2 574 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (𝑀...𝑛)) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
118117adantlr 688 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...𝑛)) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
119114, 118fsumcl 14673 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
12057ffvelrnda 6503 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) ∈ ℂ)
12176, 120sylan2 574 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) ∈ ℂ)
122 efadd 15031 . . . . . . . . . . . 12 ((Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) ∈ ℂ ∧ ((𝑘𝑍𝐴)‘(𝑛 + 1)) ∈ ℂ) → (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1)))) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
123119, 121, 122syl2anc 567 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1)))) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
124113, 123eqtrd 2805 . . . . . . . . . 10 ((𝜑𝑛𝑍) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
1251243adant3 1126 . . . . . . . . 9 ((𝜑𝑛𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
12697, 107, 1253eqtr4d 2815 . . . . . . . 8 ((𝜑𝑛𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)))
1271263exp 1112 . . . . . . 7 (𝜑 → (𝑛𝑍 → (∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)))))
128127com12 32 . . . . . 6 (𝑛𝑍 → (𝜑 → (∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)))))
129128a2d 29 . . . . 5 (𝑛𝑍 → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))) → (𝜑 → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)))))
1302eqcomi 2780 . . . . 5 (ℤ𝑀) = 𝑍
131129, 130eleq2s 2868 . . . 4 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))) → (𝜑 → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)))))
1329, 15, 21, 27, 74, 131uzind4 11949 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → ∏𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚))))
1333, 132mpcom 38 . 2 (𝜑 → ∏𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚)))
134 fzssuz 12590 . . . . . . . 8 (𝑀...𝑁) ⊆ (ℤ𝑀)
135134, 2sseqtr4i 3788 . . . . . . 7 (𝑀...𝑁) ⊆ 𝑍
136 resmpt 5591 . . . . . . 7 ((𝑀...𝑁) ⊆ 𝑍 → ((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴)))
137135, 136ax-mp 5 . . . . . 6 ((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))
138137fveq1i 6334 . . . . 5 (((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
139 fvres 6349 . . . . 5 (𝑚 ∈ (𝑀...𝑁) → (((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
140138, 139syl5reqr 2820 . . . 4 (𝑚 ∈ (𝑀...𝑁) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚))
141140prodeq2i 14857 . . 3 𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
142 prodfc 14883 . . 3 𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
143141, 142eqtri 2793 . 2 𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
144 resmpt 5591 . . . . . . . 8 ((𝑀...𝑁) ⊆ 𝑍 → ((𝑘𝑍𝐴) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴))
145135, 144ax-mp 5 . . . . . . 7 ((𝑘𝑍𝐴) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)
146145fveq1i 6334 . . . . . 6 (((𝑘𝑍𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
147 fvres 6349 . . . . . 6 (𝑚 ∈ (𝑀...𝑁) → (((𝑘𝑍𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘𝑍𝐴)‘𝑚))
148146, 147syl5reqr 2820 . . . . 5 (𝑚 ∈ (𝑀...𝑁) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚))
149148sumeq2i 14638 . . . 4 Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
150 sumfc 14649 . . . 4 Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
151149, 150eqtri 2793 . . 3 Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
152151fveq2i 6336 . 2 (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴)
153133, 143, 1523eqtr3g 2828 1 (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  csb 3683  wss 3724  {csn 4317  cmpt 4864  cres 5252  cfv 6032  (class class class)co 6794  cc 10137  1c1 10140   + caddc 10142   · cmul 10144  cz 11580  cuz 11889  ...cfz 12534  Σcsu 14625  cprod 14843  expce 14999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-inf2 8703  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216  ax-pre-sup 10217  ax-addf 10218  ax-mulf 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-isom 6041  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-oadd 7718  df-er 7897  df-pm 8013  df-en 8111  df-dom 8112  df-sdom 8113  df-fin 8114  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8966  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-div 10888  df-nn 11224  df-2 11282  df-3 11283  df-n0 11496  df-z 11581  df-uz 11890  df-rp 12037  df-ico 12387  df-fz 12535  df-fzo 12675  df-fl 12802  df-seq 13010  df-exp 13069  df-fac 13266  df-bc 13295  df-hash 13323  df-shft 14016  df-cj 14048  df-re 14049  df-im 14050  df-sqrt 14184  df-abs 14185  df-limsup 14411  df-clim 14428  df-rlim 14429  df-sum 14626  df-prod 14844  df-ef 15005
This theorem is referenced by: (None)
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