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Theorem fprodefsum 14869
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 2740 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
4 oveq2 6698 . . . . . . 7 (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
54prodeq1d 14695 . . . . . 6 (𝑎 = 𝑀 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
64sumeq1d 14475 . . . . . . 7 (𝑎 = 𝑀 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚))
76fveq2d 6233 . . . . . 6 (𝑎 = 𝑀 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)))
85, 7eqeq12d 2666 . . . . 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 6698 . . . . . . 7 (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
1110prodeq1d 14695 . . . . . 6 (𝑎 = 𝑛 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
1210sumeq1d 14475 . . . . . . 7 (𝑎 = 𝑛 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))
1312fveq2d 6233 . . . . . 6 (𝑎 = 𝑛 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)))
1411, 13eqeq12d 2666 . . . . 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 6698 . . . . . . 7 (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
1716prodeq1d 14695 . . . . . 6 (𝑎 = (𝑛 + 1) → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
1816sumeq1d 14475 . . . . . . 7 (𝑎 = (𝑛 + 1) → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚))
1918fveq2d 6233 . . . . . 6 (𝑎 = (𝑛 + 1) → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)))
2017, 19eqeq12d 2666 . . . . 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 6698 . . . . . . 7 (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
2322prodeq1d 14695 . . . . . 6 (𝑎 = 𝑁 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
2422sumeq1d 14475 . . . . . . 7 (𝑎 = 𝑁 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚))
2524fveq2d 6233 . . . . . 6 (𝑎 = 𝑁 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚)))
2623, 25eqeq12d 2666 . . . . 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 12421 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
2928adantl 481 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
3029prodeq1d 14695 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
31 simpr 476 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
32 uzid 11740 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3332, 2syl6eleqr 2741 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀𝑍)
34 fprodefsum.3 . . . . . . . . . . . 12 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
35 efcl 14857 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
3634, 35syl 17 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (exp‘𝐴) ∈ ℂ)
37 eqid 2651 . . . . . . . . . . 11 (𝑘𝑍 ↦ (exp‘𝐴)) = (𝑘𝑍 ↦ (exp‘𝐴))
3836, 37fmptd 6425 . . . . . . . . . 10 (𝜑 → (𝑘𝑍 ↦ (exp‘𝐴)):𝑍⟶ℂ)
3938ffvelrnda 6399 . . . . . . . . 9 ((𝜑𝑀𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
4033, 39sylan2 490 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
41 fveq2 6229 . . . . . . . . 9 (𝑚 = 𝑀 → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4241prodsn 14736 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ) → ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4331, 40, 42syl2anc 694 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4433adantl 481 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → 𝑀𝑍)
45 fvex 6239 . . . . . . . 8 (exp‘𝑀 / 𝑘𝐴) ∈ V
46 nfcv 2793 . . . . . . . . 9 𝑘𝑀
47 nfcv 2793 . . . . . . . . . 10 𝑘exp
48 nfcsb1v 3582 . . . . . . . . . 10 𝑘𝑀 / 𝑘𝐴
4947, 48nffv 6236 . . . . . . . . 9 𝑘(exp‘𝑀 / 𝑘𝐴)
50 csbeq1a 3575 . . . . . . . . . 10 (𝑘 = 𝑀𝐴 = 𝑀 / 𝑘𝐴)
5150fveq2d 6233 . . . . . . . . 9 (𝑘 = 𝑀 → (exp‘𝐴) = (exp‘𝑀 / 𝑘𝐴))
5246, 49, 51, 37fvmptf 6340 . . . . . . . 8 ((𝑀𝑍 ∧ (exp‘𝑀 / 𝑘𝐴) ∈ V) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘𝑀 / 𝑘𝐴))
5344, 45, 52sylancl 695 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘𝑀 / 𝑘𝐴))
5430, 43, 533eqtrd 2689 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘𝑀 / 𝑘𝐴))
5529sumeq1d 14475 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚))
56 eqid 2651 . . . . . . . . . . . 12 (𝑘𝑍𝐴) = (𝑘𝑍𝐴)
5734, 56fmptd 6425 . . . . . . . . . . 11 (𝜑 → (𝑘𝑍𝐴):𝑍⟶ℂ)
5857ffvelrnda 6399 . . . . . . . . . 10 ((𝜑𝑀𝑍) → ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ)
5933, 58sylan2 490 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ)
60 fveq2 6229 . . . . . . . . . 10 (𝑚 = 𝑀 → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6160sumsn 14519 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ) → Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6231, 59, 61syl2anc 694 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6334ralrimiva 2995 . . . . . . . . . 10 (𝜑 → ∀𝑘𝑍 𝐴 ∈ ℂ)
6448nfel1 2808 . . . . . . . . . . . 12 𝑘𝑀 / 𝑘𝐴 ∈ ℂ
6550eleq1d 2715 . . . . . . . . . . . 12 (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝑀 / 𝑘𝐴 ∈ ℂ))
6664, 65rspc 3334 . . . . . . . . . . 11 (𝑀𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → 𝑀 / 𝑘𝐴 ∈ ℂ))
6766impcom 445 . . . . . . . . . 10 ((∀𝑘𝑍 𝐴 ∈ ℂ ∧ 𝑀𝑍) → 𝑀 / 𝑘𝐴 ∈ ℂ)
6863, 33, 67syl2an 493 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘𝐴 ∈ ℂ)
6956fvmpts 6324 . . . . . . . . 9 ((𝑀𝑍𝑀 / 𝑘𝐴 ∈ ℂ) → ((𝑘𝑍𝐴)‘𝑀) = 𝑀 / 𝑘𝐴)
7044, 68, 69syl2anc 694 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍𝐴)‘𝑀) = 𝑀 / 𝑘𝐴)
7155, 62, 703eqtrd 2689 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚) = 𝑀 / 𝑘𝐴)
7271fveq2d 6233 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)) = (exp‘𝑀 / 𝑘𝐴))
7354, 72eqtr4d 2688 . . . . 5 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)))
7473expcom 450 . . . 4 (𝑀 ∈ ℤ → (𝜑 → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚))))
75 simp3 1083 . . . . . . . . . 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 11780 . . . . . . . . . . . 12 (𝑛𝑍 → (𝑛 + 1) ∈ 𝑍)
77 simpr 476 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
78 nfcsb1v 3582 . . . . . . . . . . . . . . . . . 18 𝑘(𝑛 + 1) / 𝑘𝐴
7978nfel1 2808 . . . . . . . . . . . . . . . . 17 𝑘(𝑛 + 1) / 𝑘𝐴 ∈ ℂ
80 csbeq1a 3575 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛 + 1) → 𝐴 = (𝑛 + 1) / 𝑘𝐴)
8180eleq1d 2715 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8279, 81rspc 3334 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8363, 82mpan9 485 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
84 efcl 14857 . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . . 15 𝑘(𝑛 + 1)
8747, 78nffv 6236 . . . . . . . . . . . . . . 15 𝑘(exp‘(𝑛 + 1) / 𝑘𝐴)
8880fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → (exp‘𝐴) = (exp‘(𝑛 + 1) / 𝑘𝐴))
8986, 87, 88, 37fvmptf 6340 . . . . . . . . . . . . . 14 (((𝑛 + 1) ∈ 𝑍 ∧ (exp‘(𝑛 + 1) / 𝑘𝐴) ∈ ℂ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9077, 85, 89syl2anc 694 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9156fvmpts 6324 . . . . . . . . . . . . . . 15 (((𝑛 + 1) ∈ 𝑍(𝑛 + 1) / 𝑘𝐴 ∈ ℂ) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) = (𝑛 + 1) / 𝑘𝐴)
9277, 83, 91syl2anc 694 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) = (𝑛 + 1) / 𝑘𝐴)
9392fveq2d 6233 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9490, 93eqtr4d 2688 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))))
9576, 94sylan2 490 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))))
96953adant3 1101 . . . . . . . . . 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 6708 . . . . . . . . 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 476 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → 𝑛𝑍)
9998, 2syl6eleq 2740 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → 𝑛 ∈ (ℤ𝑀))
100 elfzuz 12376 . . . . . . . . . . . . . 14 (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚 ∈ (ℤ𝑀))
101100, 2syl6eleqr 2741 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚𝑍)
10238ffvelrnda 6399 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
103101, 102sylan2 490 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
104103adantlr 751 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
105 fveq2 6229 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)))
10699, 104, 105fprodp1 14743 . . . . . . . . . 10 ((𝜑𝑛𝑍) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) · ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1))))
1071063adant3 1101 . . . . . . . . 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 6399 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
109101, 108sylan2 490 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
110109adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
111 fveq2 6229 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘(𝑛 + 1)))
11299, 110, 111fsump1 14531 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚) = (Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1))))
113112fveq2d 6233 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)) = (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1)))))
114 fzfid 12812 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → (𝑀...𝑛) ∈ Fin)
115 elfzuz 12376 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (𝑀...𝑛) → 𝑚 ∈ (ℤ𝑀))
116115, 2syl6eleqr 2741 . . . . . . . . . . . . . . 15 (𝑚 ∈ (𝑀...𝑛) → 𝑚𝑍)
117116, 108sylan2 490 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (𝑀...𝑛)) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
118117adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...𝑛)) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
119114, 118fsumcl 14508 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
12057ffvelrnda 6399 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) ∈ ℂ)
12176, 120sylan2 490 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) ∈ ℂ)
122 efadd 14868 . . . . . . . . . . . 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 694 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1)))) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
124113, 123eqtrd 2685 . . . . . . . . . 10 ((𝜑𝑛𝑍) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
1251243adant3 1101 . . . . . . . . 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 2695 . . . . . . . 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 1283 . . . . . . 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 2660 . . . . 5 (ℤ𝑀) = 𝑍
131129, 130eleq2s 2748 . . . 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 11784 . . 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 12420 . . . . . . . 8 (𝑀...𝑁) ⊆ (ℤ𝑀)
135134, 2sseqtr4i 3671 . . . . . . 7 (𝑀...𝑁) ⊆ 𝑍
136 resmpt 5484 . . . . . . 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 6230 . . . . 5 (((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
139 fvres 6245 . . . . 5 (𝑚 ∈ (𝑀...𝑁) → (((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
140138, 139syl5reqr 2700 . . . 4 (𝑚 ∈ (𝑀...𝑁) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚))
141140prodeq2i 14693 . . 3 𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
142 prodfc 14719 . . 3 𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
143141, 142eqtri 2673 . 2 𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
144 resmpt 5484 . . . . . . . 8 ((𝑀...𝑁) ⊆ 𝑍 → ((𝑘𝑍𝐴) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴))
145135, 144ax-mp 5 . . . . . . 7 ((𝑘𝑍𝐴) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)
146145fveq1i 6230 . . . . . 6 (((𝑘𝑍𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
147 fvres 6245 . . . . . 6 (𝑚 ∈ (𝑀...𝑁) → (((𝑘𝑍𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘𝑍𝐴)‘𝑚))
148146, 147syl5reqr 2700 . . . . 5 (𝑚 ∈ (𝑀...𝑁) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚))
149148sumeq2i 14473 . . . 4 Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
150 sumfc 14484 . . . 4 Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
151149, 150eqtri 2673 . . 3 Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
152151fveq2i 6232 . 2 (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴)
153133, 143, 1523eqtr3g 2708 1 (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  csb 3566  wss 3607  {csn 4210  cmpt 4762  cres 5145  cfv 5926  (class class class)co 6690  cc 9972  1c1 9975   + caddc 9977   · cmul 9979  cz 11415  cuz 11725  ...cfz 12364  Σcsu 14460  cprod 14679  expce 14836
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  ax-addf 10053  ax-mulf 10054
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-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  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-ico 12219  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-prod 14680  df-ef 14842
This theorem is referenced by: (None)
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