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Theorem fprodefsum 16071
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, 2eleqtrdi 2839 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
4 oveq2 7428 . . . . . . 7 (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
54prodeq1d 15897 . . . . . 6 (𝑎 = 𝑀 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
64sumeq1d 15679 . . . . . . 7 (𝑎 = 𝑀 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚))
76fveq2d 6901 . . . . . 6 (𝑎 = 𝑀 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)))
85, 7eqeq12d 2744 . . . . 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 340 . . . 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 7428 . . . . . . 7 (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
1110prodeq1d 15897 . . . . . 6 (𝑎 = 𝑛 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
1210sumeq1d 15679 . . . . . . 7 (𝑎 = 𝑛 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚))
1312fveq2d 6901 . . . . . 6 (𝑎 = 𝑛 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)))
1411, 13eqeq12d 2744 . . . . 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 340 . . . 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 7428 . . . . . . 7 (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
1716prodeq1d 15897 . . . . . 6 (𝑎 = (𝑛 + 1) → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
1816sumeq1d 15679 . . . . . . 7 (𝑎 = (𝑛 + 1) → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚))
1918fveq2d 6901 . . . . . 6 (𝑎 = (𝑛 + 1) → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)))
2017, 19eqeq12d 2744 . . . . 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 340 . . . 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 7428 . . . . . . 7 (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
2322prodeq1d 15897 . . . . . 6 (𝑎 = 𝑁 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
2422sumeq1d 15679 . . . . . . 7 (𝑎 = 𝑁 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚))
2524fveq2d 6901 . . . . . 6 (𝑎 = 𝑁 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚)))
2623, 25eqeq12d 2744 . . . . 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 340 . . . 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 13575 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
2928adantl 481 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
3029prodeq1d 15897 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
31 simpr 484 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
32 uzid 12867 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3332, 2eleqtrrdi 2840 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀𝑍)
34 fprodefsum.3 . . . . . . . . . . . 12 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
35 efcl 16058 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
3634, 35syl 17 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (exp‘𝐴) ∈ ℂ)
3736fmpttd 7125 . . . . . . . . . 10 (𝜑 → (𝑘𝑍 ↦ (exp‘𝐴)):𝑍⟶ℂ)
3837ffvelcdmda 7094 . . . . . . . . 9 ((𝜑𝑀𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
3933, 38sylan2 592 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
40 fveq2 6897 . . . . . . . . 9 (𝑚 = 𝑀 → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4140prodsn 15938 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ) → ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4231, 39, 41syl2anc 583 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ {𝑀} ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀))
4333adantl 481 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → 𝑀𝑍)
44 fvex 6910 . . . . . . . 8 (exp‘𝑀 / 𝑘𝐴) ∈ V
45 nfcv 2899 . . . . . . . . 9 𝑘𝑀
46 nfcv 2899 . . . . . . . . . 10 𝑘exp
47 nfcsb1v 3917 . . . . . . . . . 10 𝑘𝑀 / 𝑘𝐴
4846, 47nffv 6907 . . . . . . . . 9 𝑘(exp‘𝑀 / 𝑘𝐴)
49 csbeq1a 3906 . . . . . . . . . 10 (𝑘 = 𝑀𝐴 = 𝑀 / 𝑘𝐴)
5049fveq2d 6901 . . . . . . . . 9 (𝑘 = 𝑀 → (exp‘𝐴) = (exp‘𝑀 / 𝑘𝐴))
51 eqid 2728 . . . . . . . . 9 (𝑘𝑍 ↦ (exp‘𝐴)) = (𝑘𝑍 ↦ (exp‘𝐴))
5245, 48, 50, 51fvmptf 7026 . . . . . . . 8 ((𝑀𝑍 ∧ (exp‘𝑀 / 𝑘𝐴) ∈ V) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘𝑀 / 𝑘𝐴))
5343, 44, 52sylancl 585 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘𝑀 / 𝑘𝐴))
5430, 42, 533eqtrd 2772 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘𝑀 / 𝑘𝐴))
5529sumeq1d 15679 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚))
5634fmpttd 7125 . . . . . . . . . . 11 (𝜑 → (𝑘𝑍𝐴):𝑍⟶ℂ)
5756ffvelcdmda 7094 . . . . . . . . . 10 ((𝜑𝑀𝑍) → ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ)
5833, 57sylan2 592 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ)
59 fveq2 6897 . . . . . . . . . 10 (𝑚 = 𝑀 → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6059sumsn 15724 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ ((𝑘𝑍𝐴)‘𝑀) ∈ ℂ) → Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6131, 58, 60syl2anc 583 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ {𝑀} ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘𝑀))
6234ralrimiva 3143 . . . . . . . . . 10 (𝜑 → ∀𝑘𝑍 𝐴 ∈ ℂ)
6347nfel1 2916 . . . . . . . . . . . 12 𝑘𝑀 / 𝑘𝐴 ∈ ℂ
6449eleq1d 2814 . . . . . . . . . . . 12 (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝑀 / 𝑘𝐴 ∈ ℂ))
6563, 64rspc 3597 . . . . . . . . . . 11 (𝑀𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → 𝑀 / 𝑘𝐴 ∈ ℂ))
6665impcom 407 . . . . . . . . . 10 ((∀𝑘𝑍 𝐴 ∈ ℂ ∧ 𝑀𝑍) → 𝑀 / 𝑘𝐴 ∈ ℂ)
6762, 33, 66syl2an 595 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘𝐴 ∈ ℂ)
68 eqid 2728 . . . . . . . . . 10 (𝑘𝑍𝐴) = (𝑘𝑍𝐴)
6968fvmpts 7008 . . . . . . . . 9 ((𝑀𝑍𝑀 / 𝑘𝐴 ∈ ℂ) → ((𝑘𝑍𝐴)‘𝑀) = 𝑀 / 𝑘𝐴)
7043, 67, 69syl2anc 583 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → ((𝑘𝑍𝐴)‘𝑀) = 𝑀 / 𝑘𝐴)
7155, 61, 703eqtrd 2772 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚) = 𝑀 / 𝑘𝐴)
7271fveq2d 6901 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)) = (exp‘𝑀 / 𝑘𝐴))
7354, 72eqtr4d 2771 . . . . 5 ((𝜑𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚)))
7473expcom 413 . . . 4 (𝑀 ∈ ℤ → (𝜑 → ∏𝑚 ∈ (𝑀...𝑀)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘𝑍𝐴)‘𝑚))))
75 simp3 1136 . . . . . . . . . 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 12916 . . . . . . . . . . . 12 (𝑛𝑍 → (𝑛 + 1) ∈ 𝑍)
77 simpr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
78 nfcsb1v 3917 . . . . . . . . . . . . . . . . . 18 𝑘(𝑛 + 1) / 𝑘𝐴
7978nfel1 2916 . . . . . . . . . . . . . . . . 17 𝑘(𝑛 + 1) / 𝑘𝐴 ∈ ℂ
80 csbeq1a 3906 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛 + 1) → 𝐴 = (𝑛 + 1) / 𝑘𝐴)
8180eleq1d 2814 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8279, 81rspc 3597 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8362, 82mpan9 506 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
84 efcl 16058 . . . . . . . . . . . . . . 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 2899 . . . . . . . . . . . . . . 15 𝑘(𝑛 + 1)
8746, 78nffv 6907 . . . . . . . . . . . . . . 15 𝑘(exp‘(𝑛 + 1) / 𝑘𝐴)
8880fveq2d 6901 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → (exp‘𝐴) = (exp‘(𝑛 + 1) / 𝑘𝐴))
8986, 87, 88, 51fvmptf 7026 . . . . . . . . . . . . . 14 (((𝑛 + 1) ∈ 𝑍 ∧ (exp‘(𝑛 + 1) / 𝑘𝐴) ∈ ℂ) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9077, 85, 89syl2anc 583 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9168fvmpts 7008 . . . . . . . . . . . . . . 15 (((𝑛 + 1) ∈ 𝑍(𝑛 + 1) / 𝑘𝐴 ∈ ℂ) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) = (𝑛 + 1) / 𝑘𝐴)
9277, 83, 91syl2anc 583 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) = (𝑛 + 1) / 𝑘𝐴)
9392fveq2d 6901 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))) = (exp‘(𝑛 + 1) / 𝑘𝐴))
9490, 93eqtr4d 2771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))))
9576, 94sylan2 592 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1))))
96953adant3 1130 . . . . . . . . . 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 7438 . . . . . . . . 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 484 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → 𝑛𝑍)
9998, 2eleqtrdi 2839 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → 𝑛 ∈ (ℤ𝑀))
100 elfzuz 13529 . . . . . . . . . . . . . 14 (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚 ∈ (ℤ𝑀))
101100, 2eleqtrrdi 2840 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚𝑍)
10237ffvelcdmda 7094 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
103101, 102sylan2 592 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
104103adantlr 714 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
105 fveq2 6897 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)))
10699, 104, 105fprodp1 15945 . . . . . . . . . 10 ((𝜑𝑛𝑍) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = (∏𝑚 ∈ (𝑀...𝑛)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) · ((𝑘𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1))))
1071063adant3 1130 . . . . . . . . 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))))
10856ffvelcdmda 7094 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
109101, 108sylan2 592 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
110109adantlr 714 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
111 fveq2 6897 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘𝑍𝐴)‘(𝑛 + 1)))
11299, 110, 111fsump1 15734 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚) = (Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1))))
113112fveq2d 6901 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)) = (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1)))))
114 fzfid 13970 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → (𝑀...𝑛) ∈ Fin)
115 elfzuz 13529 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (𝑀...𝑛) → 𝑚 ∈ (ℤ𝑀))
116115, 2eleqtrrdi 2840 . . . . . . . . . . . . . . 15 (𝑚 ∈ (𝑀...𝑛) → 𝑚𝑍)
117116, 108sylan2 592 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (𝑀...𝑛)) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
118117adantlr 714 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (𝑀...𝑛)) → ((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
119114, 118fsumcl 15711 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) ∈ ℂ)
12056ffvelcdmda 7094 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) ∈ ℂ)
12176, 120sylan2 592 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → ((𝑘𝑍𝐴)‘(𝑛 + 1)) ∈ ℂ)
122 efadd 16070 . . . . . . . . . . . 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 583 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚) + ((𝑘𝑍𝐴)‘(𝑛 + 1)))) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
124113, 123eqtrd 2768 . . . . . . . . . 10 ((𝜑𝑛𝑍) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘𝑍𝐴)‘𝑚)) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘𝑍𝐴)‘𝑚)) · (exp‘((𝑘𝑍𝐴)‘(𝑛 + 1)))))
1251243adant3 1130 . . . . . . . . 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 2778 . . . . . . . 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 1117 . . . . . . 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 2737 . . . . 5 (ℤ𝑀) = 𝑍
131129, 130eleq2s 2847 . . . 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 12920 . . 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 fvres 6916 . . . . 5 (𝑚 ∈ (𝑀...𝑁) → (((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚))
135 fzssuz 13574 . . . . . . . 8 (𝑀...𝑁) ⊆ (ℤ𝑀)
136135, 2sseqtrri 4017 . . . . . . 7 (𝑀...𝑁) ⊆ 𝑍
137 resmpt 6041 . . . . . . 7 ((𝑀...𝑁) ⊆ 𝑍 → ((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴)))
138136, 137ax-mp 5 . . . . . 6 ((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))
139138fveq1i 6898 . . . . 5 (((𝑘𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
140134, 139eqtr3di 2783 . . . 4 (𝑚 ∈ (𝑀...𝑁) → ((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚))
141140prodeq2i 15895 . . 3 𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
142 prodfc 15921 . . 3 𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
143141, 142eqtri 2756 . 2 𝑚 ∈ (𝑀...𝑁)((𝑘𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
144 fvres 6916 . . . . . 6 (𝑚 ∈ (𝑀...𝑁) → (((𝑘𝑍𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘𝑍𝐴)‘𝑚))
145 resmpt 6041 . . . . . . . 8 ((𝑀...𝑁) ⊆ 𝑍 → ((𝑘𝑍𝐴) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴))
146136, 145ax-mp 5 . . . . . . 7 ((𝑘𝑍𝐴) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)
147146fveq1i 6898 . . . . . 6 (((𝑘𝑍𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
148144, 147eqtr3di 2783 . . . . 5 (𝑚 ∈ (𝑀...𝑁) → ((𝑘𝑍𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚))
149148sumeq2i 15677 . . . 4 Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
150 sumfc 15687 . . . 4 Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
151149, 150eqtri 2756 . . 3 Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
152151fveq2i 6900 . 2 (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘𝑍𝐴)‘𝑚)) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴)
153133, 143, 1523eqtr3g 2791 1 (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3058  Vcvv 3471  csb 3892  wss 3947  {csn 4629  cmpt 5231  cres 5680  cfv 6548  (class class class)co 7420  cc 11136  1c1 11139   + caddc 11141   · cmul 11143  cz 12588  cuz 12852  ...cfz 13516  Σcsu 15664  cprod 15881  expce 16037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9664  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-er 8724  df-pm 8847  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-sup 9465  df-inf 9466  df-oi 9533  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-n0 12503  df-z 12589  df-uz 12853  df-rp 13007  df-ico 13362  df-fz 13517  df-fzo 13660  df-fl 13789  df-seq 13999  df-exp 14059  df-fac 14265  df-bc 14294  df-hash 14322  df-shft 15046  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-limsup 15447  df-clim 15464  df-rlim 15465  df-sum 15665  df-prod 15882  df-ef 16043
This theorem is referenced by:  aks4d1p1p1  41534
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