MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prodfrec Structured version   Visualization version   GIF version

Theorem prodfrec 14547
Description: The reciprocal of an infinite product. (Contributed by Scott Fenton, 15-Jan-2018.)
Hypotheses
Ref Expression
prodfn0.1 (𝜑𝑁 ∈ (ℤ𝑀))
prodfn0.2 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℂ)
prodfn0.3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ≠ 0)
prodfrec.4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) = (1 / (𝐹𝑘)))
Assertion
Ref Expression
prodfrec (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))
Distinct variable groups:   𝑘,𝐹   𝜑,𝑘   𝑘,𝑀   𝑘,𝑁   𝑘,𝐺

Proof of Theorem prodfrec
Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prodfn0.1 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 12288 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 6150 . . . . 5 (𝑚 = 𝑀 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑀))
5 fveq2 6150 . . . . . 6 (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀))
65oveq2d 6621 . . . . 5 (𝑚 = 𝑀 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))
74, 6eqeq12d 2641 . . . 4 (𝑚 = 𝑀 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀))))
87imbi2d 330 . . 3 (𝑚 = 𝑀 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))))
9 fveq2 6150 . . . . 5 (𝑚 = 𝑛 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑛))
10 fveq2 6150 . . . . . 6 (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛))
1110oveq2d 6621 . . . . 5 (𝑚 = 𝑛 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))
129, 11eqeq12d 2641 . . . 4 (𝑚 = 𝑛 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))))
1312imbi2d 330 . . 3 (𝑚 = 𝑛 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))))
14 fveq2 6150 . . . . 5 (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘(𝑛 + 1)))
15 fveq2 6150 . . . . . 6 (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
1615oveq2d 6621 . . . . 5 (𝑚 = (𝑛 + 1) → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))
1714, 16eqeq12d 2641 . . . 4 (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1)))))
1817imbi2d 330 . . 3 (𝑚 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
19 fveq2 6150 . . . . 5 (𝑚 = 𝑁 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑁))
20 fveq2 6150 . . . . . 6 (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁))
2120oveq2d 6621 . . . . 5 (𝑚 = 𝑁 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))
2219, 21eqeq12d 2641 . . . 4 (𝑚 = 𝑁 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))))
2322imbi2d 330 . . 3 (𝑚 = 𝑁 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))))
24 eluzfz1 12287 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
251, 24syl 17 . . . . . 6 (𝜑𝑀 ∈ (𝑀...𝑁))
26 fveq2 6150 . . . . . . . . 9 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
27 fveq2 6150 . . . . . . . . . 10 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
2827oveq2d 6621 . . . . . . . . 9 (𝑘 = 𝑀 → (1 / (𝐹𝑘)) = (1 / (𝐹𝑀)))
2926, 28eqeq12d 2641 . . . . . . . 8 (𝑘 = 𝑀 → ((𝐺𝑘) = (1 / (𝐹𝑘)) ↔ (𝐺𝑀) = (1 / (𝐹𝑀))))
3029imbi2d 330 . . . . . . 7 (𝑘 = 𝑀 → ((𝜑 → (𝐺𝑘) = (1 / (𝐹𝑘))) ↔ (𝜑 → (𝐺𝑀) = (1 / (𝐹𝑀)))))
31 prodfrec.4 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) = (1 / (𝐹𝑘)))
3231expcom 451 . . . . . . 7 (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐺𝑘) = (1 / (𝐹𝑘))))
3330, 32vtoclga 3263 . . . . . 6 (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐺𝑀) = (1 / (𝐹𝑀))))
3425, 33mpcom 38 . . . . 5 (𝜑 → (𝐺𝑀) = (1 / (𝐹𝑀)))
35 eluzel2 11636 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
361, 35syl 17 . . . . . 6 (𝜑𝑀 ∈ ℤ)
37 seq1 12751 . . . . . 6 (𝑀 ∈ ℤ → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺𝑀))
3836, 37syl 17 . . . . 5 (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺𝑀))
39 seq1 12751 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹𝑀))
4036, 39syl 17 . . . . . 6 (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹𝑀))
4140oveq2d 6621 . . . . 5 (𝜑 → (1 / (seq𝑀( · , 𝐹)‘𝑀)) = (1 / (𝐹𝑀)))
4234, 38, 413eqtr4d 2670 . . . 4 (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))
4342a1i 11 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀))))
44 oveq1 6612 . . . . . . . . 9 ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))))
45443ad2ant3 1082 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))))
46 fzofzp1 12503 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
47 fveq2 6150 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
48 fveq2 6150 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
4948oveq2d 6621 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → (1 / (𝐹𝑘)) = (1 / (𝐹‘(𝑛 + 1))))
5047, 49eqeq12d 2641 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → ((𝐺𝑘) = (1 / (𝐹𝑘)) ↔ (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
5150imbi2d 330 . . . . . . . . . . . . . 14 (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐺𝑘) = (1 / (𝐹𝑘))) ↔ (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))))
5251, 32vtoclga 3263 . . . . . . . . . . . . 13 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
5346, 52syl 17 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
5453impcom 446 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))
5554oveq2d 6621 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))))
56 1cnd 10001 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
57 elfzouz 12412 . . . . . . . . . . . . . 14 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
5857adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
59 elfzouz2 12422 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝑛))
60 fzss2 12320 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
6159, 60syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝑀..^𝑁) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
6261sselda 3588 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (𝑀...𝑁))
63 prodfn0.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℂ)
6462, 63sylan2 491 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛))) → (𝐹𝑘) ∈ ℂ)
6564anassrs 679 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹𝑘) ∈ ℂ)
66 mulcl 9965 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ)
6766adantl 482 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ)
6858, 65, 67seqcl 12758 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
6948eleq1d 2688 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
7069imbi2d 330 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹𝑘) ∈ ℂ) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ)))
7163expcom 451 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹𝑘) ∈ ℂ))
7270, 71vtoclga 3263 . . . . . . . . . . . . . 14 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
7346, 72syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
7473impcom 446 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
75 prodfn0.3 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ≠ 0)
7662, 75sylan2 491 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛))) → (𝐹𝑘) ≠ 0)
7776anassrs 679 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹𝑘) ≠ 0)
7858, 65, 77prodfn0 14546 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ≠ 0)
7948neeq1d 2855 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ≠ 0 ↔ (𝐹‘(𝑛 + 1)) ≠ 0))
8079imbi2d 330 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹𝑘) ≠ 0) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0)))
8175expcom 451 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹𝑘) ≠ 0))
8280, 81vtoclga 3263 . . . . . . . . . . . . . 14 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0))
8346, 82syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0))
8483impcom 446 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ≠ 0)
8556, 68, 56, 74, 78, 84divmuldivd 10787 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
86 1t1e1 11120 . . . . . . . . . . . 12 (1 · 1) = 1
8786oveq1i 6615 . . . . . . . . . . 11 ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
8885, 87syl6eq 2676 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
8955, 88eqtrd 2660 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
90893adant3 1079 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
9145, 90eqtrd 2660 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
92 seqp1 12753 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
9357, 92syl 17 . . . . . . . 8 (𝑛 ∈ (𝑀..^𝑁) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
94933ad2ant2 1081 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
95 seqp1 12753 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
9657, 95syl 17 . . . . . . . . 9 (𝑛 ∈ (𝑀..^𝑁) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
9796oveq2d 6621 . . . . . . . 8 (𝑛 ∈ (𝑀..^𝑁) → (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
98973ad2ant2 1081 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
9991, 94, 983eqtr4d 2670 . . . . . 6 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))
100993exp 1261 . . . . 5 (𝜑 → (𝑛 ∈ (𝑀..^𝑁) → ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
101100com12 32 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
102101a2d 29 . . 3 (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (𝜑 → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
1038, 13, 18, 23, 43, 102fzind2 12523 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))))
1043, 103mpcom 38 1 (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wss 3560  cfv 5850  (class class class)co 6605  cc 9879  0cc0 9881  1c1 9882   + caddc 9884   · cmul 9886   / cdiv 10629  cz 11322  cuz 11631  ...cfz 12265  ..^cfzo 12403  seqcseq 12738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-seq 12739
This theorem is referenced by:  prodfdiv  14548
  Copyright terms: Public domain W3C validator