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Theorem fprodefsum 15052
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 2906 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
4 oveq2 6889 . . . . . . 7 (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
54prodeq1d 14879 . . . . . 6 (𝑎 = 𝑀 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
64sumeq1d 14661 . . . . . . 7 (𝑎 = 𝑀 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚))
76fveq2d 6419 . . . . . 6 (𝑎 = 𝑀 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)))
85, 7eqeq12d 2832 . . . . 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 331 . . . 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 6889 . . . . . . 7 (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
1110prodeq1d 14879 . . . . . 6 (𝑎 = 𝑛 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
1210sumeq1d 14661 . . . . . . 7 (𝑎 = 𝑛 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))
1312fveq2d 6419 . . . . . 6 (𝑎 = 𝑛 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)))
1411, 13eqeq12d 2832 . . . . 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 331 . . . 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 6889 . . . . . . 7 (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
1716prodeq1d 14879 . . . . . 6 (𝑎 = (𝑛 + 1) → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
1816sumeq1d 14661 . . . . . . 7 (𝑎 = (𝑛 + 1) → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚))
1918fveq2d 6419 . . . . . 6 (𝑎 = (𝑛 + 1) → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)))
2017, 19eqeq12d 2832 . . . . 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 331 . . . 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 6889 . . . . . . 7 (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
2322prodeq1d 14879 . . . . . 6 (𝑎 = 𝑁 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
2422sumeq1d 14661 . . . . . . 7 (𝑎 = 𝑁 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚))
2524fveq2d 6419 . . . . . 6 (𝑎 = 𝑁 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚)))
2623, 25eqeq12d 2832 . . . . 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 331 . . . 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 12613 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
2928adantl 469 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
3029prodeq1d 14879 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
31 simpr 473 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
32 uzid 11926 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3332, 2syl6eleqr 2907 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀𝑍)
34 fprodefsum.3 . . . . . . . . . . . 12 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
35 efcl 15040 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
3634, 35syl 17 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (exp‘𝐴) ∈ ℂ)
3736fmpttd 6614 . . . . . . . . . 10 (𝜑 → (𝑘𝑍 ↦ (exp‘𝐴)):𝑍⟶ℂ)
3837ffvelrnda 6588 . . . . . . . . 9 ((𝜑𝑀𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
3933, 38sylan2 582 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
40 fveq2 6415 . . . . . . . . 9 (𝑚 = 𝑀 → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4140prodsn 14920 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ) → ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4231, 39, 41syl2anc 575 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4333adantl 469 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → 𝑀𝑍)
44 fvex 6428 . . . . . . . 8 (exp‘𝑀 / 𝑘𝐴) ∈ V
45 nfcv 2959 . . . . . . . . 9 𝑘𝑀
46 nfcv 2959 . . . . . . . . . 10 𝑘exp
47 nfcsb1v 3755 . . . . . . . . . 10 𝑘𝑀 / 𝑘𝐴
4846, 47nffv 6425 . . . . . . . . 9 𝑘(exp‘𝑀 / 𝑘𝐴)
49 csbeq1a 3748 . . . . . . . . . 10 (𝑘 = 𝑀𝐴 = 𝑀 / 𝑘𝐴)
5049fveq2d 6419 . . . . . . . . 9 (𝑘 = 𝑀 → (exp‘𝐴) = (exp‘𝑀 / 𝑘𝐴))
51 eqid 2817 . . . . . . . . 9 (𝑘𝑍 ↦ (exp‘𝐴)) = (𝑘𝑍 ↦ (exp‘𝐴))
5245, 48, 50, 51fvmptf 6529 . . . . . . . 8 ((𝑀𝑍 ∧ (exp‘𝑀 / 𝑘𝐴) ∈ V) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘𝑀 / 𝑘𝐴))
5343, 44, 52sylancl 576 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘𝑀 / 𝑘𝐴))
5430, 42, 533eqtrd 2855 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘𝑀 / 𝑘𝐴))
5529sumeq1d 14661 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚))
5634fmpttd 6614 . . . . . . . . . . 11 (𝜑 → (𝑘𝑍𝐴):𝑍⟶ℂ)
5756ffvelrnda 6588 . . . . . . . . . 10 ((𝜑𝑀𝑍) → ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ)
5833, 57sylan2 582 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ)
59 fveq2 6415 . . . . . . . . . 10 (𝑚 = 𝑀 → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6059sumsn 14705 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ) → Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6131, 58, 60syl2anc 575 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6234ralrimiva 3165 . . . . . . . . . 10 (𝜑 → ∀𝑘𝑍 𝐴 ∈ ℂ)
6347nfel1 2974 . . . . . . . . . . . 12 𝑘𝑀 / 𝑘𝐴 ∈ ℂ
6449eleq1d 2881 . . . . . . . . . . . 12 (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝑀 / 𝑘𝐴 ∈ ℂ))
6563, 64rspc 3507 . . . . . . . . . . 11 (𝑀𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → 𝑀 / 𝑘𝐴 ∈ ℂ))
6665impcom 396 . . . . . . . . . 10 ((∀𝑘𝑍 𝐴 ∈ ℂ ∧ 𝑀𝑍) → 𝑀 / 𝑘𝐴 ∈ ℂ)
6762, 33, 66syl2an 585 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘𝐴 ∈ ℂ)
68 eqid 2817 . . . . . . . . . 10 (𝑘𝑍𝐴) = (𝑘𝑍𝐴)
6968fvmpts 6513 . . . . . . . . 9 ((𝑀𝑍𝑀 / 𝑘𝐴 ∈ ℂ) → ((𝑘𝑍𝐴)‘𝑀) = 𝑀 / 𝑘𝐴)
7043, 67, 69syl2anc 575 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍𝐴)‘𝑀) = 𝑀 / 𝑘𝐴)
7155, 61, 703eqtrd 2855 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚) = 𝑀 / 𝑘𝐴)
7271fveq2d 6419 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)) = (exp‘𝑀 / 𝑘𝐴))
7354, 72eqtr4d 2854 . . . . 5 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)))
7473expcom 400 . . . 4 (𝑀 ∈ ℤ → (𝜑 → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚))))
75 simp3 1161 . . . . . . . . . 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 11967 . . . . . . . . . . . 12 (𝑛𝑍 → (𝑛 + 1) ∈ 𝑍)
77 simpr 473 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
78 nfcsb1v 3755 . . . . . . . . . . . . . . . . . 18 𝑘(𝑛 + 1) / 𝑘𝐴
7978nfel1 2974 . . . . . . . . . . . . . . . . 17 𝑘(𝑛 + 1) / 𝑘𝐴 ∈ ℂ
80 csbeq1a 3748 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛 + 1) → 𝐴 = (𝑛 + 1) / 𝑘𝐴)
8180eleq1d 2881 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8279, 81rspc 3507 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8362, 82mpan9 498 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
84 efcl 15040 . . . . . . . . . . . . . . 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 2959 . . . . . . . . . . . . . . 15 𝑘(𝑛 + 1)
8746, 78nffv 6425 . . . . . . . . . . . . . . 15 𝑘(exp‘(𝑛 + 1) / 𝑘𝐴)
8880fveq2d 6419 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → (exp‘𝐴) = (exp‘(𝑛 + 1) / 𝑘𝐴))
8986, 87, 88, 51fvmptf 6529 . . . . . . . . . . . . . 14 (((𝑛 + 1) ∈ 𝑍 ∧ (exp‘(𝑛 + 1) / 𝑘𝐴) ∈ ℂ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9077, 85, 89syl2anc 575 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9168fvmpts 6513 . . . . . . . . . . . . . . 15 (((𝑛 + 1) ∈ 𝑍(𝑛 + 1) / 𝑘𝐴 ∈ ℂ) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) = (𝑛 + 1) / 𝑘𝐴)
9277, 83, 91syl2anc 575 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) = (𝑛 + 1) / 𝑘𝐴)
9392fveq2d 6419 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9490, 93eqtr4d 2854 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))))
9576, 94sylan2 582 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))))
96953adant3 1155 . . . . . . . . . 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 6899 . . . . . . . . 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 473 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → 𝑛𝑍)
9998, 2syl6eleq 2906 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → 𝑛 ∈ (ℤ𝑀))
100 elfzuz 12568 . . . . . . . . . . . . . 14 (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚 ∈ (ℤ𝑀))
101100, 2syl6eleqr 2907 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚𝑍)
10237ffvelrnda 6588 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
103101, 102sylan2 582 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
104103adantlr 697 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
105 fveq2 6415 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)))
10699, 104, 105fprodp1 14927 . . . . . . . . . 10 ((𝜑𝑛𝑍) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) · ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1))))
1071063adant3 1155 . . . . . . . . 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))))
10856ffvelrnda 6588 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
109101, 108sylan2 582 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
110109adantlr 697 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
111 fveq2 6415 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘(𝑛 + 1)))
11299, 110, 111fsump1 14717 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚) = (Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1))))
113112fveq2d 6419 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)) = (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1)))))
114 fzfid 13003 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → (𝑀...𝑛) ∈ Fin)
115 elfzuz 12568 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (𝑀...𝑛) → 𝑚 ∈ (ℤ𝑀))
116115, 2syl6eleqr 2907 . . . . . . . . . . . . . . 15 (𝑚 ∈ (𝑀...𝑛) → 𝑚𝑍)
117116, 108sylan2 582 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (𝑀...𝑛)) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
118117adantlr 697 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...𝑛)) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
119114, 118fsumcl 14694 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
12056ffvelrnda 6588 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) ∈ ℂ)
12176, 120sylan2 582 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) ∈ ℂ)
122 efadd 15051 . . . . . . . . . . . 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 575 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1)))) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
124113, 123eqtrd 2851 . . . . . . . . . 10 ((𝜑𝑛𝑍) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
1251243adant3 1155 . . . . . . . . 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 2861 . . . . . . . 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 1141 . . . . . . 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 2826 . . . . 5 (ℤ𝑀) = 𝑍
131129, 130eleq2s 2914 . . . 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 11971 . . 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 12612 . . . . . . . 8 (𝑀...𝑁) ⊆ (ℤ𝑀)
135134, 2sseqtr4i 3846 . . . . . . 7 (𝑀...𝑁) ⊆ 𝑍
136 resmpt 5665 . . . . . . 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 6416 . . . . 5 (((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
139 fvres 6434 . . . . 5 (𝑚 ∈ (𝑀...𝑁) → (((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
140138, 139syl5reqr 2866 . . . 4 (𝑚 ∈ (𝑀...𝑁) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚))
141140prodeq2i 14877 . . 3 𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
142 prodfc 14903 . . 3 𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
143141, 142eqtri 2839 . 2 𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
144 resmpt 5665 . . . . . . . 8 ((𝑀...𝑁) ⊆ 𝑍 → ((𝑘𝑍𝐴) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴))
145135, 144ax-mp 5 . . . . . . 7 ((𝑘𝑍𝐴) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)
146145fveq1i 6416 . . . . . 6 (((𝑘𝑍𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
147 fvres 6434 . . . . . 6 (𝑚 ∈ (𝑀...𝑁) → (((𝑘𝑍𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘𝑍𝐴)‘𝑚))
148146, 147syl5reqr 2866 . . . . 5 (𝑚 ∈ (𝑀...𝑁) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚))
149148sumeq2i 14659 . . . 4 Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
150 sumfc 14670 . . . 4 Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
151149, 150eqtri 2839 . . 3 Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
152151fveq2i 6418 . 2 (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴)
153133, 143, 1523eqtr3g 2874 1 (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2157  wral 3107  Vcvv 3402  csb 3739  wss 3780  {csn 4381  cmpt 4934  cres 5324  cfv 6108  (class class class)co 6881  cc 10226  1c1 10229   + caddc 10231   · cmul 10233  cz 11650  cuz 11911  ...cfz 12556  Σcsu 14646  cprod 14863  expce 15019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4975  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186  ax-inf2 8792  ax-cnex 10284  ax-resscn 10285  ax-1cn 10286  ax-icn 10287  ax-addcl 10288  ax-addrcl 10289  ax-mulcl 10290  ax-mulrcl 10291  ax-mulcom 10292  ax-addass 10293  ax-mulass 10294  ax-distr 10295  ax-i2m1 10296  ax-1ne0 10297  ax-1rid 10298  ax-rnegex 10299  ax-rrecex 10300  ax-cnre 10301  ax-pre-lttri 10302  ax-pre-lttrn 10303  ax-pre-ltadd 10304  ax-pre-mulgt0 10305  ax-pre-sup 10306  ax-addf 10307  ax-mulf 10308
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5230  df-eprel 5235  df-po 5243  df-so 5244  df-fr 5281  df-se 5282  df-we 5283  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-pred 5904  df-ord 5950  df-on 5951  df-lim 5952  df-suc 5953  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-isom 6117  df-riota 6842  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-om 7303  df-1st 7405  df-2nd 7406  df-wrecs 7649  df-recs 7711  df-rdg 7749  df-1o 7803  df-oadd 7807  df-er 7986  df-pm 8102  df-en 8200  df-dom 8201  df-sdom 8202  df-fin 8203  df-sup 8594  df-inf 8595  df-oi 8661  df-card 9055  df-pnf 10368  df-mnf 10369  df-xr 10370  df-ltxr 10371  df-le 10372  df-sub 10560  df-neg 10561  df-div 10977  df-nn 11313  df-2 11371  df-3 11372  df-n0 11567  df-z 11651  df-uz 11912  df-rp 12054  df-ico 12406  df-fz 12557  df-fzo 12697  df-fl 12824  df-seq 13032  df-exp 13091  df-fac 13288  df-bc 13317  df-hash 13345  df-shft 14037  df-cj 14069  df-re 14070  df-im 14071  df-sqrt 14205  df-abs 14206  df-limsup 14432  df-clim 14449  df-rlim 14450  df-sum 14647  df-prod 14864  df-ef 15025
This theorem is referenced by: (None)
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