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Theorem prodfrec 15230
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 12898 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 6643 . . . . 5 (𝑚 = 𝑀 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑀))
5 fveq2 6643 . . . . . 6 (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀))
65oveq2d 7146 . . . . 5 (𝑚 = 𝑀 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))
74, 6eqeq12d 2837 . . . 4 (𝑚 = 𝑀 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀))))
87imbi2d 344 . . 3 (𝑚 = 𝑀 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))))
9 fveq2 6643 . . . . 5 (𝑚 = 𝑛 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑛))
10 fveq2 6643 . . . . . 6 (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛))
1110oveq2d 7146 . . . . 5 (𝑚 = 𝑛 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))
129, 11eqeq12d 2837 . . . 4 (𝑚 = 𝑛 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))))
1312imbi2d 344 . . 3 (𝑚 = 𝑛 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))))
14 fveq2 6643 . . . . 5 (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘(𝑛 + 1)))
15 fveq2 6643 . . . . . 6 (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
1615oveq2d 7146 . . . . 5 (𝑚 = (𝑛 + 1) → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))
1714, 16eqeq12d 2837 . . . 4 (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1)))))
1817imbi2d 344 . . 3 (𝑚 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
19 fveq2 6643 . . . . 5 (𝑚 = 𝑁 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑁))
20 fveq2 6643 . . . . . 6 (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁))
2120oveq2d 7146 . . . . 5 (𝑚 = 𝑁 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))
2219, 21eqeq12d 2837 . . . 4 (𝑚 = 𝑁 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))))
2322imbi2d 344 . . 3 (𝑚 = 𝑁 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))))
24 eluzfz1 12897 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
251, 24syl 17 . . . . . 6 (𝜑𝑀 ∈ (𝑀...𝑁))
26 fveq2 6643 . . . . . . . . 9 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
27 fveq2 6643 . . . . . . . . . 10 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
2827oveq2d 7146 . . . . . . . . 9 (𝑘 = 𝑀 → (1 / (𝐹𝑘)) = (1 / (𝐹𝑀)))
2926, 28eqeq12d 2837 . . . . . . . 8 (𝑘 = 𝑀 → ((𝐺𝑘) = (1 / (𝐹𝑘)) ↔ (𝐺𝑀) = (1 / (𝐹𝑀))))
3029imbi2d 344 . . . . . . 7 (𝑘 = 𝑀 → ((𝜑 → (𝐺𝑘) = (1 / (𝐹𝑘))) ↔ (𝜑 → (𝐺𝑀) = (1 / (𝐹𝑀)))))
31 prodfrec.4 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) = (1 / (𝐹𝑘)))
3231expcom 417 . . . . . . 7 (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐺𝑘) = (1 / (𝐹𝑘))))
3330, 32vtoclga 3551 . . . . . 6 (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐺𝑀) = (1 / (𝐹𝑀))))
3425, 33mpcom 38 . . . . 5 (𝜑 → (𝐺𝑀) = (1 / (𝐹𝑀)))
35 eluzel2 12226 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
361, 35syl 17 . . . . . 6 (𝜑𝑀 ∈ ℤ)
37 seq1 13365 . . . . . 6 (𝑀 ∈ ℤ → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺𝑀))
3836, 37syl 17 . . . . 5 (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺𝑀))
39 seq1 13365 . . . . . . 7 (𝑀 ∈ ℤ → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹𝑀))
4036, 39syl 17 . . . . . 6 (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹𝑀))
4140oveq2d 7146 . . . . 5 (𝜑 → (1 / (seq𝑀( · , 𝐹)‘𝑀)) = (1 / (𝐹𝑀)))
4234, 38, 413eqtr4d 2866 . . . 4 (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))
4342a1i 11 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀))))
44 oveq1 7137 . . . . . . . . 9 ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))))
45443ad2ant3 1132 . . . . . . . 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 13117 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
47 fveq2 6643 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
48 fveq2 6643 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
4948oveq2d 7146 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → (1 / (𝐹𝑘)) = (1 / (𝐹‘(𝑛 + 1))))
5047, 49eqeq12d 2837 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → ((𝐺𝑘) = (1 / (𝐹𝑘)) ↔ (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
5150imbi2d 344 . . . . . . . . . . . . . 14 (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐺𝑘) = (1 / (𝐹𝑘))) ↔ (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))))
5251, 32vtoclga 3551 . . . . . . . . . . . . 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 411 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))
5554oveq2d 7146 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))))
56 1cnd 10613 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
57 elfzouz 13025 . . . . . . . . . . . . . 14 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
5857adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
59 elfzouz2 13035 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝑛))
60 fzss2 12930 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
6159, 60syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝑀..^𝑁) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
6261sselda 3943 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (𝑀...𝑁))
63 prodfn0.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℂ)
6462, 63sylan2 595 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛))) → (𝐹𝑘) ∈ ℂ)
6564anassrs 471 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹𝑘) ∈ ℂ)
66 mulcl 10598 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ)
6766adantl 485 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ)
6858, 65, 67seqcl 13374 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
6948eleq1d 2896 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
7069imbi2d 344 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹𝑘) ∈ ℂ) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ)))
7163expcom 417 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹𝑘) ∈ ℂ))
7270, 71vtoclga 3551 . . . . . . . . . . . . . 14 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
7346, 72syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
7473impcom 411 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
75 prodfn0.3 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ≠ 0)
7662, 75sylan2 595 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛))) → (𝐹𝑘) ≠ 0)
7776anassrs 471 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹𝑘) ≠ 0)
7858, 65, 77prodfn0 15229 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ≠ 0)
7948neeq1d 3066 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ≠ 0 ↔ (𝐹‘(𝑛 + 1)) ≠ 0))
8079imbi2d 344 . . . . . . . . . . . . . . 15 (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹𝑘) ≠ 0) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0)))
8175expcom 417 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹𝑘) ≠ 0))
8280, 81vtoclga 3551 . . . . . . . . . . . . . 14 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0))
8346, 82syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0))
8483impcom 411 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ≠ 0)
8556, 68, 56, 74, 78, 84divmuldivd 11434 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
86 1t1e1 11777 . . . . . . . . . . . 12 (1 · 1) = 1
8786oveq1i 7140 . . . . . . . . . . 11 ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
8885, 87syl6eq 2872 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
8955, 88eqtrd 2856 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
90893adant3 1129 . . . . . . . 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 2856 . . . . . . 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 13367 . . . . . . . . 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 1131 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
95 seqp1 13367 . . . . . . . . . 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 7146 . . . . . . . 8 (𝑛 ∈ (𝑀..^𝑁) → (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
98973ad2ant2 1131 . . . . . . 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 2866 . . . . . 6 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))
100993exp 1116 . . . . 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 13138 . 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 399  w3a 1084   = wceq 1538  wcel 2115  wne 3007  wss 3910  cfv 6328  (class class class)co 7130  cc 10512  0cc0 10514  1c1 10515   + caddc 10517   · cmul 10519   / cdiv 11274  cz 11959  cuz 12221  ...cfz 12875  ..^cfzo 13016  seqcseq 13352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-n0 11876  df-z 11960  df-uz 12222  df-fz 12876  df-fzo 13017  df-seq 13353
This theorem is referenced by:  prodfdiv  15231
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