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Theorem faclim 31607
Description: An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)
Hypothesis
Ref Expression
faclim.1 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))
Assertion
Ref Expression
faclim (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))
Distinct variable group:   𝐴,𝑛
Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem faclim
Dummy variables 𝑎 𝑏 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 faclim.1 . . 3 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))
2 seqeq3 12789 . . 3 (𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))) → seq1( · , 𝐹) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))))
31, 2ax-mp 5 . 2 seq1( · , 𝐹) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))))
4 oveq2 6643 . . . . . . 7 (𝑎 = 0 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑0))
5 oveq1 6642 . . . . . . . 8 (𝑎 = 0 → (𝑎 / 𝑛) = (0 / 𝑛))
65oveq2d 6651 . . . . . . 7 (𝑎 = 0 → (1 + (𝑎 / 𝑛)) = (1 + (0 / 𝑛)))
74, 6oveq12d 6653 . . . . . 6 (𝑎 = 0 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))
87mpteq2dv 4736 . . . . 5 (𝑎 = 0 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))))
98seqeq3d 12792 . . . 4 (𝑎 = 0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))))
10 fveq2 6178 . . . . 5 (𝑎 = 0 → (!‘𝑎) = (!‘0))
11 fac0 13046 . . . . 5 (!‘0) = 1
1210, 11syl6eq 2670 . . . 4 (𝑎 = 0 → (!‘𝑎) = 1)
139, 12breq12d 4657 . . 3 (𝑎 = 0 → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) ⇝ (!‘𝑎) ↔ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) ⇝ 1))
14 oveq2 6643 . . . . . . 7 (𝑎 = 𝑚 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑𝑚))
15 oveq1 6642 . . . . . . . 8 (𝑎 = 𝑚 → (𝑎 / 𝑛) = (𝑚 / 𝑛))
1615oveq2d 6651 . . . . . . 7 (𝑎 = 𝑚 → (1 + (𝑎 / 𝑛)) = (1 + (𝑚 / 𝑛)))
1714, 16oveq12d 6653 . . . . . 6 (𝑎 = 𝑚 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))
1817mpteq2dv 4736 . . . . 5 (𝑎 = 𝑚 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))
1918seqeq3d 12792 . . . 4 (𝑎 = 𝑚 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))))
20 fveq2 6178 . . . 4 (𝑎 = 𝑚 → (!‘𝑎) = (!‘𝑚))
2119, 20breq12d 4657 . . 3 (𝑎 = 𝑚 → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) ⇝ (!‘𝑎) ↔ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)))
22 oveq2 6643 . . . . . . 7 (𝑎 = (𝑚 + 1) → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑(𝑚 + 1)))
23 oveq1 6642 . . . . . . . 8 (𝑎 = (𝑚 + 1) → (𝑎 / 𝑛) = ((𝑚 + 1) / 𝑛))
2423oveq2d 6651 . . . . . . 7 (𝑎 = (𝑚 + 1) → (1 + (𝑎 / 𝑛)) = (1 + ((𝑚 + 1) / 𝑛)))
2522, 24oveq12d 6653 . . . . . 6 (𝑎 = (𝑚 + 1) → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))
2625mpteq2dv 4736 . . . . 5 (𝑎 = (𝑚 + 1) → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛)))))
2726seqeq3d 12792 . . . 4 (𝑎 = (𝑚 + 1) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))))
28 fveq2 6178 . . . 4 (𝑎 = (𝑚 + 1) → (!‘𝑎) = (!‘(𝑚 + 1)))
2927, 28breq12d 4657 . . 3 (𝑎 = (𝑚 + 1) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) ⇝ (!‘𝑎) ↔ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (!‘(𝑚 + 1))))
30 oveq2 6643 . . . . . . 7 (𝑎 = 𝐴 → ((1 + (1 / 𝑛))↑𝑎) = ((1 + (1 / 𝑛))↑𝐴))
31 oveq1 6642 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎 / 𝑛) = (𝐴 / 𝑛))
3231oveq2d 6651 . . . . . . 7 (𝑎 = 𝐴 → (1 + (𝑎 / 𝑛)) = (1 + (𝐴 / 𝑛)))
3330, 32oveq12d 6653 . . . . . 6 (𝑎 = 𝐴 → (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))) = (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))
3433mpteq2dv 4736 . . . . 5 (𝑎 = 𝐴 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))))
3534seqeq3d 12792 . . . 4 (𝑎 = 𝐴 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) = seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))))
36 fveq2 6178 . . . 4 (𝑎 = 𝐴 → (!‘𝑎) = (!‘𝐴))
3735, 36breq12d 4657 . . 3 (𝑎 = 𝐴 → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑎) / (1 + (𝑎 / 𝑛))))) ⇝ (!‘𝑎) ↔ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))) ⇝ (!‘𝐴)))
38 1red 10040 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 1 ∈ ℝ)
39 nnrecre 11042 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
4038, 39readdcld 10054 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1 + (1 / 𝑛)) ∈ ℝ)
4140recnd 10053 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 + (1 / 𝑛)) ∈ ℂ)
4241exp0d 12985 . . . . . . . . 9 (𝑛 ∈ ℕ → ((1 + (1 / 𝑛))↑0) = 1)
43 nncn 11013 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
44 nnne0 11038 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
4543, 44div0d 10785 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (0 / 𝑛) = 0)
4645oveq2d 6651 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 + (0 / 𝑛)) = (1 + 0))
47 1p0e1 11118 . . . . . . . . . 10 (1 + 0) = 1
4846, 47syl6eq 2670 . . . . . . . . 9 (𝑛 ∈ ℕ → (1 + (0 / 𝑛)) = 1)
4942, 48oveq12d 6653 . . . . . . . 8 (𝑛 ∈ ℕ → (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))) = (1 / 1))
50 1div1e1 10702 . . . . . . . 8 (1 / 1) = 1
5149, 50syl6eq 2670 . . . . . . 7 (𝑛 ∈ ℕ → (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))) = 1)
5251mpteq2ia 4731 . . . . . 6 (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))) = (𝑛 ∈ ℕ ↦ 1)
53 fconstmpt 5153 . . . . . 6 (ℕ × {1}) = (𝑛 ∈ ℕ ↦ 1)
5452, 53eqtr4i 2645 . . . . 5 (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))) = (ℕ × {1})
55 seqeq3 12789 . . . . 5 ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛)))) = (ℕ × {1}) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) = seq1( · , (ℕ × {1})))
5654, 55ax-mp 5 . . . 4 seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) = seq1( · , (ℕ × {1}))
57 nnuz 11708 . . . . . 6 ℕ = (ℤ‘1)
58 1zzd 11393 . . . . . 6 (⊤ → 1 ∈ ℤ)
5957, 58climprod1 14676 . . . . 5 (⊤ → seq1( · , (ℕ × {1})) ⇝ 1)
6059trud 1491 . . . 4 seq1( · , (ℕ × {1})) ⇝ 1
6156, 60eqbrtri 4665 . . 3 seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑0) / (1 + (0 / 𝑛))))) ⇝ 1
62 1zzd 11393 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → 1 ∈ ℤ)
63 simpr 477 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚))
64 seqex 12786 . . . . . . 7 seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ∈ V
6564a1i 11 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ∈ V)
66 faclimlem2 31605 . . . . . . 7 (𝑚 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (𝑚 + 1))
6766adantr 481 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (𝑚 + 1))
68 elnnuz 11709 . . . . . . . . . 10 (𝑎 ∈ ℕ ↔ 𝑎 ∈ (ℤ‘1))
6968biimpi 206 . . . . . . . . 9 (𝑎 ∈ ℕ → 𝑎 ∈ (ℤ‘1))
7069adantl 482 . . . . . . . 8 ((𝑚 ∈ ℕ0𝑎 ∈ ℕ) → 𝑎 ∈ (ℤ‘1))
71 1rp 11821 . . . . . . . . . . . . . . . 16 1 ∈ ℝ+
7271a1i 11 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → 1 ∈ ℝ+)
73 nnrp 11827 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
7473rpreccld 11867 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ+)
7574adantl 482 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ+)
7672, 75rpaddcld 11872 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (1 + (1 / 𝑛)) ∈ ℝ+)
77 nn0z 11385 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
7877adantr 481 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
7976, 78rpexpcld 13015 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → ((1 + (1 / 𝑛))↑𝑚) ∈ ℝ+)
80 1cnd 10041 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → 1 ∈ ℂ)
81 nn0nndivcl 11347 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℝ)
8281recnd 10053 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (𝑚 / 𝑛) ∈ ℂ)
8380, 82addcomd 10223 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (1 + (𝑚 / 𝑛)) = ((𝑚 / 𝑛) + 1))
84 nn0ge0div 11431 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → 0 ≤ (𝑚 / 𝑛))
8581, 84ge0p1rpd 11887 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → ((𝑚 / 𝑛) + 1) ∈ ℝ+)
8683, 85eqeltrd 2699 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (1 + (𝑚 / 𝑛)) ∈ ℝ+)
8779, 86rpdivcld 11874 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) ∈ ℝ+)
8887rpcnd 11859 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) ∈ ℂ)
89 eqid 2620 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))
9088, 89fmptd 6371 . . . . . . . . . 10 (𝑚 ∈ ℕ0 → (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))):ℕ⟶ℂ)
91 elfznn 12355 . . . . . . . . . 10 (𝑏 ∈ (1...𝑎) → 𝑏 ∈ ℕ)
92 ffvelrn 6343 . . . . . . . . . 10 (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))):ℕ⟶ℂ ∧ 𝑏 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
9390, 91, 92syl2an 494 . . . . . . . . 9 ((𝑚 ∈ ℕ0𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
9493adantlr 750 . . . . . . . 8 (((𝑚 ∈ ℕ0𝑎 ∈ ℕ) ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) ∈ ℂ)
95 mulcl 10005 . . . . . . . . 9 ((𝑏 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑏 · 𝑥) ∈ ℂ)
9695adantl 482 . . . . . . . 8 (((𝑚 ∈ ℕ0𝑎 ∈ ℕ) ∧ (𝑏 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑏 · 𝑥) ∈ ℂ)
9770, 94, 96seqcl 12804 . . . . . . 7 ((𝑚 ∈ ℕ0𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) ∈ ℂ)
9897adantlr 750 . . . . . 6 (((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) ∈ ℂ)
9986, 76rpmulcld 11873 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → ((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) ∈ ℝ+)
100 nn0p1nn 11317 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
101100nnrpd 11855 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ+)
102 rpdivcl 11841 . . . . . . . . . . . . . . 15 (((𝑚 + 1) ∈ ℝ+𝑛 ∈ ℝ+) → ((𝑚 + 1) / 𝑛) ∈ ℝ+)
103101, 73, 102syl2an 494 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → ((𝑚 + 1) / 𝑛) ∈ ℝ+)
10472, 103rpaddcld 11872 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (1 + ((𝑚 + 1) / 𝑛)) ∈ ℝ+)
10599, 104rpdivcld 11874 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) ∈ ℝ+)
106105rpcnd 11859 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0𝑛 ∈ ℕ) → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) ∈ ℂ)
107 eqid 2620 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))
108106, 107fmptd 6371 . . . . . . . . . 10 (𝑚 ∈ ℕ0 → (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))):ℕ⟶ℂ)
109 ffvelrn 6343 . . . . . . . . . 10 (((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))):ℕ⟶ℂ ∧ 𝑏 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) ∈ ℂ)
110108, 91, 109syl2an 494 . . . . . . . . 9 ((𝑚 ∈ ℕ0𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) ∈ ℂ)
111110adantlr 750 . . . . . . . 8 (((𝑚 ∈ ℕ0𝑎 ∈ ℕ) ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) ∈ ℂ)
11270, 111, 96seqcl 12804 . . . . . . 7 ((𝑚 ∈ ℕ0𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) ∈ ℂ)
113112adantlr 750 . . . . . 6 (((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) ∈ ℂ)
114 faclimlem3 31606 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0𝑏 ∈ ℕ) → (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))) = ((((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) · (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏)))))
115 oveq2 6643 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑏 → (1 / 𝑛) = (1 / 𝑏))
116115oveq2d 6651 . . . . . . . . . . . . . . 15 (𝑛 = 𝑏 → (1 + (1 / 𝑛)) = (1 + (1 / 𝑏)))
117116oveq1d 6650 . . . . . . . . . . . . . 14 (𝑛 = 𝑏 → ((1 + (1 / 𝑛))↑(𝑚 + 1)) = ((1 + (1 / 𝑏))↑(𝑚 + 1)))
118 oveq2 6643 . . . . . . . . . . . . . . 15 (𝑛 = 𝑏 → ((𝑚 + 1) / 𝑛) = ((𝑚 + 1) / 𝑏))
119118oveq2d 6651 . . . . . . . . . . . . . 14 (𝑛 = 𝑏 → (1 + ((𝑚 + 1) / 𝑛)) = (1 + ((𝑚 + 1) / 𝑏)))
120117, 119oveq12d 6653 . . . . . . . . . . . . 13 (𝑛 = 𝑏 → (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))) = (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))))
121 eqid 2620 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))
122 ovex 6663 . . . . . . . . . . . . 13 (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))) ∈ V
123120, 121, 122fvmpt 6269 . . . . . . . . . . . 12 (𝑏 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))))
124123adantl 482 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0𝑏 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((1 + (1 / 𝑏))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑏))))
125116oveq1d 6650 . . . . . . . . . . . . . . 15 (𝑛 = 𝑏 → ((1 + (1 / 𝑛))↑𝑚) = ((1 + (1 / 𝑏))↑𝑚))
126 oveq2 6643 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑏 → (𝑚 / 𝑛) = (𝑚 / 𝑏))
127126oveq2d 6651 . . . . . . . . . . . . . . 15 (𝑛 = 𝑏 → (1 + (𝑚 / 𝑛)) = (1 + (𝑚 / 𝑏)))
128125, 127oveq12d 6653 . . . . . . . . . . . . . 14 (𝑛 = 𝑏 → (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))) = (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))))
129 ovex 6663 . . . . . . . . . . . . . 14 (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) ∈ V
130128, 89, 129fvmpt 6269 . . . . . . . . . . . . 13 (𝑏 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) = (((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))))
131127, 116oveq12d 6653 . . . . . . . . . . . . . . 15 (𝑛 = 𝑏 → ((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) = ((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))))
132131, 119oveq12d 6653 . . . . . . . . . . . . . 14 (𝑛 = 𝑏 → (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))) = (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏))))
133 ovex 6663 . . . . . . . . . . . . . 14 (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏))) ∈ V
134132, 107, 133fvmpt 6269 . . . . . . . . . . . . 13 (𝑏 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏))))
135130, 134oveq12d 6653 . . . . . . . . . . . 12 (𝑏 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) · ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏)) = ((((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) · (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏)))))
136135adantl 482 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0𝑏 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) · ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏)) = ((((1 + (1 / 𝑏))↑𝑚) / (1 + (𝑚 / 𝑏))) · (((1 + (𝑚 / 𝑏)) · (1 + (1 / 𝑏))) / (1 + ((𝑚 + 1) / 𝑏)))))
137114, 124, 1363eqtr4d 2664 . . . . . . . . . 10 ((𝑚 ∈ ℕ0𝑏 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) · ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏)))
13891, 137sylan2 491 . . . . . . . . 9 ((𝑚 ∈ ℕ0𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) · ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏)))
139138adantlr 750 . . . . . . . 8 (((𝑚 ∈ ℕ0𝑎 ∈ ℕ) ∧ 𝑏 ∈ (1...𝑎)) → ((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏) = (((𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))‘𝑏) · ((𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛))))‘𝑏)))
14070, 94, 111, 139prodfmul 14603 . . . . . . 7 ((𝑚 ∈ ℕ0𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) = ((seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) · (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎)))
141140adantlr 750 . . . . . 6 (((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) ∧ 𝑎 ∈ ℕ) → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎) = ((seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛)))))‘𝑎) · (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑚 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑚 + 1) / 𝑛)))))‘𝑎)))
14257, 62, 63, 65, 67, 98, 113, 141climmul 14344 . . . . 5 ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ ((!‘𝑚) · (𝑚 + 1)))
143 facp1 13048 . . . . . 6 (𝑚 ∈ ℕ0 → (!‘(𝑚 + 1)) = ((!‘𝑚) · (𝑚 + 1)))
144143adantr 481 . . . . 5 ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → (!‘(𝑚 + 1)) = ((!‘𝑚) · (𝑚 + 1)))
145142, 144breqtrrd 4672 . . . 4 ((𝑚 ∈ ℕ0 ∧ seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚)) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (!‘(𝑚 + 1)))
146145ex 450 . . 3 (𝑚 ∈ ℕ0 → (seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝑚) / (1 + (𝑚 / 𝑛))))) ⇝ (!‘𝑚) → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑(𝑚 + 1)) / (1 + ((𝑚 + 1) / 𝑛))))) ⇝ (!‘(𝑚 + 1))))
14713, 21, 29, 37, 61, 146nn0ind 11457 . 2 (𝐴 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))) ⇝ (!‘𝐴))
1483, 147syl5eqbr 4679 1 (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wtru 1482  wcel 1988  Vcvv 3195  {csn 4168   class class class wbr 4644  cmpt 4720   × cxp 5102  wf 5872  cfv 5876  (class class class)co 6635  cc 9919  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926   / cdiv 10669  cn 11005  0cn0 11277  cz 11362  cuz 11672  +crp 11817  ...cfz 12311  seqcseq 12784  cexp 12843  !cfa 13043  cli 14196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-sup 8333  df-inf 8334  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-rp 11818  df-fz 12312  df-fzo 12450  df-fl 12576  df-seq 12785  df-exp 12844  df-fac 13044  df-shft 13788  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-rlim 14201
This theorem is referenced by:  iprodfac  31608
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