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Theorem efrlim 26319
Description: The limit of the sequence (1 + 𝐴 / 𝑘)↑𝑘 is the exponential function. This is often taken as an alternate definition of the exponential function (see also dfef2 26320). (Contributed by Mario Carneiro, 1-Mar-2015.)
Hypothesis
Ref Expression
efrlim.1 𝑆 = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))
Assertion
Ref Expression
efrlim (𝐴 ∈ ℂ → (𝑘 ∈ ℝ+ ↦ ((1 + (𝐴 / 𝑘))↑𝑐𝑘)) ⇝𝑟 (exp‘𝐴))
Distinct variable group:   𝐴,𝑘
Allowed substitution hint:   𝑆(𝑘)

Proof of Theorem efrlim
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rge0ssre 13373 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
2 ax-resscn 11108 . . . . . . . 8 ℝ ⊆ ℂ
31, 2sstri 3953 . . . . . . 7 (0[,)+∞) ⊆ ℂ
43sseli 3940 . . . . . 6 (𝑥 ∈ (0[,)+∞) → 𝑥 ∈ ℂ)
5 simpll 765 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝐴 ∈ ℂ)
6 1cnd 11150 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 1 ∈ ℂ)
7 simplr 767 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
8 ax-1ne0 11120 . . . . . . . . . . . 12 1 ≠ 0
98a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 1 ≠ 0)
10 simpr 485 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ¬ 𝑥 = 0)
1110neqned 2950 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ≠ 0)
125, 6, 7, 9, 11divdiv2d 11963 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = ((𝐴 · 𝑥) / 1))
13 mulcl 11135 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴 · 𝑥) ∈ ℂ)
1413adantr 481 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) ∈ ℂ)
1514div1d 11923 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((𝐴 · 𝑥) / 1) = (𝐴 · 𝑥))
1612, 15eqtrd 2776 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = (𝐴 · 𝑥))
1716oveq2d 7373 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 / (1 / 𝑥))) = (1 + (𝐴 · 𝑥)))
1817oveq1d 7372 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥)) = ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))
1918ifeq2da 4518 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥))) = if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))))
204, 19sylan2 593 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ (0[,)+∞)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥))) = if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))))
2120mpteq2dva 5205 . . . 4 (𝐴 ∈ ℂ → (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥)))) = (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))))
22 resmpt 5991 . . . . 5 ((0[,)+∞) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ↾ (0[,)+∞)) = (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))))
233, 22ax-mp 5 . . . 4 ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ↾ (0[,)+∞)) = (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))))
2421, 23eqtr4di 2794 . . 3 (𝐴 ∈ ℂ → (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥)))) = ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ↾ (0[,)+∞)))
25 0e0icopnf 13375 . . . . 5 0 ∈ (0[,)+∞)
2625a1i 11 . . . 4 (𝐴 ∈ ℂ → 0 ∈ (0[,)+∞))
27 eqeq2 2748 . . . . . . . . 9 ((exp‘(𝐴 · 1)) = if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) → (if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)) ↔ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))))
28 eqeq2 2748 . . . . . . . . 9 ((exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) = if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) → (if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) ↔ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))))
29 efrlim.1 . . . . . . . . . . . . . 14 𝑆 = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))
30 cnxmet 24136 . . . . . . . . . . . . . . 15 (abs ∘ − ) ∈ (∞Met‘ℂ)
31 0cnd 11148 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → 0 ∈ ℂ)
32 abscl 15163 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
33 peano2re 11328 . . . . . . . . . . . . . . . . . . 19 ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
3432, 33syl 17 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ)
35 0red 11158 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℂ → 0 ∈ ℝ)
36 absge0 15172 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
3732ltp1d 12085 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℂ → (abs‘𝐴) < ((abs‘𝐴) + 1))
3835, 32, 34, 36, 37lelttrd 11313 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℂ → 0 < ((abs‘𝐴) + 1))
3934, 38elrpd 12954 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ+)
4039rpreccld 12967 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ+)
4140rpxrd 12958 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ*)
42 blssm 23771 . . . . . . . . . . . . . . 15 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ*) → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ⊆ ℂ)
4330, 31, 41, 42mp3an2i 1466 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ⊆ ℂ)
4429, 43eqsstrid 3992 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → 𝑆 ⊆ ℂ)
4544sselda 3944 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 𝑥 ∈ ℂ)
46 mul0or 11795 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
4745, 46syldan 591 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
4847biimpa 477 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 · 𝑥) = 0) → (𝐴 = 0 ∨ 𝑥 = 0))
497, 11reccld 11924 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
5045, 49syldanl 602 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
5150adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
52511cxpd 26062 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1↑𝑐(1 / 𝑥)) = 1)
53 simplr 767 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝐴 = 0)
5453oveq1d 7372 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = (0 · 𝑥))
5545ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
5655mul02d 11353 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (0 · 𝑥) = 0)
5754, 56eqtrd 2776 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = 0)
5857oveq2d 7373 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = (1 + 0))
59 1p0e1 12277 . . . . . . . . . . . . . . . . 17 (1 + 0) = 1
6058, 59eqtrdi 2792 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = 1)
6160oveq1d 7372 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) = (1↑𝑐(1 / 𝑥)))
6253fveq2d 6846 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = (exp‘0))
63 ef0 15973 . . . . . . . . . . . . . . . 16 (exp‘0) = 1
6462, 63eqtrdi 2792 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = 1)
6552, 61, 643eqtr4d 2786 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) = (exp‘𝐴))
6665ifeq2da 4518 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = if(𝑥 = 0, (exp‘𝐴), (exp‘𝐴)))
67 ifid 4526 . . . . . . . . . . . . 13 if(𝑥 = 0, (exp‘𝐴), (exp‘𝐴)) = (exp‘𝐴)
6866, 67eqtrdi 2792 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
69 iftrue 4492 . . . . . . . . . . . . 13 (𝑥 = 0 → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
7069adantl 482 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝑥 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
7168, 70jaodan 956 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
72 mulid1 11153 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
7372ad2antrr 724 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (𝐴 · 1) = 𝐴)
7473fveq2d 6846 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (exp‘(𝐴 · 1)) = (exp‘𝐴))
7571, 74eqtr4d 2779 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)))
7648, 75syldan 591 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 · 𝑥) = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)))
77 mulne0b 11796 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
7845, 77syldan 591 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
79 df-ne 2944 . . . . . . . . . . . 12 ((𝐴 · 𝑥) ≠ 0 ↔ ¬ (𝐴 · 𝑥) = 0)
8078, 79bitrdi 286 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ ¬ (𝐴 · 𝑥) = 0))
81 simprr 771 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ≠ 0)
8281neneqd 2948 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ¬ 𝑥 = 0)
8382iffalsed 4497 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))
84 ax-1cn 11109 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
8545, 13syldan 591 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝐴 · 𝑥) ∈ ℂ)
86 addcl 11133 . . . . . . . . . . . . . . . 16 ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
8784, 85, 86sylancr 587 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
8887adantr 481 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
89 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
9089dvlog2lem 26007 . . . . . . . . . . . . . . . . . 18 (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
91 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
9291logdmss 25997 . . . . . . . . . . . . . . . . . 18 (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
9390, 92sstri 3953 . . . . . . . . . . . . . . . . 17 (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
94 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (abs ∘ − ) = (abs ∘ − )
9594cnmetdval 24134 . . . . . . . . . . . . . . . . . . . . 21 (((1 + (𝐴 · 𝑥)) ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
9687, 84, 95sylancl 586 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
97 pncan2 11408 . . . . . . . . . . . . . . . . . . . . . 22 ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
9884, 85, 97sylancr 587 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
9998fveq2d 6846 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘((1 + (𝐴 · 𝑥)) − 1)) = (abs‘(𝐴 · 𝑥)))
10096, 99eqtrd 2776 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘(𝐴 · 𝑥)))
10185abscld 15321 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) ∈ ℝ)
10234adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((abs‘𝐴) + 1) ∈ ℝ)
10345abscld 15321 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝑥) ∈ ℝ)
104102, 103remulcld 11185 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) ∈ ℝ)
105 1red 11156 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 1 ∈ ℝ)
106 absmul 15179 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
10745, 106syldan 591 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
10832adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝐴) ∈ ℝ)
109108, 33syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((abs‘𝐴) + 1) ∈ ℝ)
11045absge0d 15329 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 0 ≤ (abs‘𝑥))
111108lep1d 12086 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
112108, 109, 103, 110, 111lemul1ad 12094 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((abs‘𝐴) · (abs‘𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
113107, 112eqbrtrd 5127 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
114 0cn 11147 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℂ
11594cnmetdval 24134 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
11645, 114, 115sylancl 586 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
11745subid1d 11501 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥 − 0) = 𝑥)
118117fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝑥 − 0)) = (abs‘𝑥))
119116, 118eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥(abs ∘ − )0) = (abs‘𝑥))
120 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 𝑥𝑆)
121120, 29eleqtrdi 2848 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
12230a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs ∘ − ) ∈ (∞Met‘ℂ))
12341adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 / ((abs‘𝐴) + 1)) ∈ ℝ*)
124 0cnd 11148 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 0 ∈ ℂ)
125 elbl3 23745 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ*) ∧ (0 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ↔ (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1))))
126122, 123, 124, 45, 125syl22anc 837 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ↔ (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1))))
127121, 126mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1)))
128119, 127eqbrtrrd 5129 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝑥) < (1 / ((abs‘𝐴) + 1)))
12938adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 0 < ((abs‘𝐴) + 1))
130 ltmuldiv2 12029 . . . . . . . . . . . . . . . . . . . . . 22 (((abs‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (((abs‘𝐴) + 1) ∈ ℝ ∧ 0 < ((abs‘𝐴) + 1))) → ((((abs‘𝐴) + 1) · (abs‘𝑥)) < 1 ↔ (abs‘𝑥) < (1 / ((abs‘𝐴) + 1))))
131103, 105, 109, 129, 130syl112anc 1374 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((((abs‘𝐴) + 1) · (abs‘𝑥)) < 1 ↔ (abs‘𝑥) < (1 / ((abs‘𝐴) + 1))))
132128, 131mpbird 256 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) < 1)
133101, 104, 105, 113, 132lelttrd 11313 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) < 1)
134100, 133eqbrtrd 5127 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1)
135 1rp 12919 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
136 rpxr 12924 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ ℝ+ → 1 ∈ ℝ*)
137135, 136mp1i 13 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 1 ∈ ℝ*)
138 1cnd 11150 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 1 ∈ ℂ)
139 elbl3 23745 . . . . . . . . . . . . . . . . . . 19 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ*) ∧ (1 ∈ ℂ ∧ (1 + (𝐴 · 𝑥)) ∈ ℂ)) → ((1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1))
140122, 137, 138, 87, 139syl22anc 837 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1))
141134, 140mpbird 256 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1))
14293, 141sselid 3942 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}))
143 eldifsni 4750 . . . . . . . . . . . . . . . 16 ((1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}) → (1 + (𝐴 · 𝑥)) ≠ 0)
144142, 143syl 17 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ≠ 0)
145144adantr 481 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 + (𝐴 · 𝑥)) ≠ 0)
14645adantr 481 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ∈ ℂ)
147146, 81reccld 11924 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ ℂ)
14888, 145, 147cxpefd 26067 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) = (exp‘((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥))))))
14987, 144logcld 25926 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
150149adantr 481 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
151147, 150mulcomd 11176 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)))
152 simpll 765 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ∈ ℂ)
153 simprl 769 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ≠ 0)
154152, 153dividd 11929 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 / 𝐴) = 1)
155154oveq1d 7372 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (1 / 𝑥))
156152, 152, 146, 153, 81divdiv1d 11962 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
157155, 156eqtr3d 2778 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
158157oveq2d 7373 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)) = ((log‘(1 + (𝐴 · 𝑥))) · (𝐴 / (𝐴 · 𝑥))))
15985adantr 481 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 · 𝑥) ∈ ℂ)
16078biimpa 477 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 · 𝑥) ≠ 0)
161150, 152, 159, 160div12d 11967 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((log‘(1 + (𝐴 · 𝑥))) · (𝐴 / (𝐴 · 𝑥))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
162151, 158, 1613eqtrd 2780 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
163162fveq2d 6846 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (exp‘((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥))))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
16483, 148, 1633eqtrd 2780 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
165164ex 413 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))))
16680, 165sylbird 259 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (¬ (𝐴 · 𝑥) = 0 → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))))
167166imp 407 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
16827, 28, 76, 167ifbothda 4524 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))))
169168mpteq2dva 5205 . . . . . . 7 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) = (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))))
17044resmptd 5994 . . . . . . 7 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ↾ 𝑆) = (𝑥𝑆 ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))))
171 1cnd 11150 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 · 𝑥) = 0) → 1 ∈ ℂ)
172149adantr 481 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
17385adantr 481 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ∈ ℂ)
174 simpr 485 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ¬ (𝐴 · 𝑥) = 0)
175174neqned 2950 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ≠ 0)
176172, 173, 175divcld 11931 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)) ∈ ℂ)
177171, 176ifclda 4521 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) ∈ ℂ)
178 eqidd 2737 . . . . . . . 8 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) = (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
179 eqidd 2737 . . . . . . . 8 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))))
180 oveq2 7365 . . . . . . . . . 10 (𝑦 = if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) → (𝐴 · 𝑦) = (𝐴 · if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
181180fveq2d 6846 . . . . . . . . 9 (𝑦 = if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))))
182 oveq2 7365 . . . . . . . . . . 11 (if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) = 1 → (𝐴 · if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) = (𝐴 · 1))
183182fveq2d 6846 . . . . . . . . . 10 (if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) = 1 → (exp‘(𝐴 · if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) = (exp‘(𝐴 · 1)))
184 oveq2 7365 . . . . . . . . . . 11 (if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) = ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)) → (𝐴 · if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
185184fveq2d 6846 . . . . . . . . . 10 (if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) = ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)) → (exp‘(𝐴 · if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
186183, 185ifsb 4499 . . . . . . . . 9 (exp‘(𝐴 · if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) = if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
187181, 186eqtrdi 2792 . . . . . . . 8 (𝑦 = if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) → (exp‘(𝐴 · 𝑦)) = if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))))
188177, 178, 179, 187fmptco 7075 . . . . . . 7 (𝐴 ∈ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∘ (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) = (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))))
189169, 170, 1883eqtr4d 2786 . . . . . 6 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∘ (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))))
190 eqidd 2737 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))))
191 eqidd 2737 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))))
192 eqeq1 2740 . . . . . . . . . . 11 (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 = 1 ↔ (1 + (𝐴 · 𝑥)) = 1))
193 fveq2 6842 . . . . . . . . . . . 12 (𝑦 = (1 + (𝐴 · 𝑥)) → (log‘𝑦) = (log‘(1 + (𝐴 · 𝑥))))
194 oveq1 7364 . . . . . . . . . . . 12 (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 − 1) = ((1 + (𝐴 · 𝑥)) − 1))
195193, 194oveq12d 7375 . . . . . . . . . . 11 (𝑦 = (1 + (𝐴 · 𝑥)) → ((log‘𝑦) / (𝑦 − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)))
196192, 195ifbieq2d 4512 . . . . . . . . . 10 (𝑦 = (1 + (𝐴 · 𝑥)) → if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1))) = if((1 + (𝐴 · 𝑥)) = 1, 1, ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1))))
197141, 190, 191, 196fmptco 7075 . . . . . . . . 9 (𝐴 ∈ ℂ → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∘ (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))) = (𝑥𝑆 ↦ if((1 + (𝐴 · 𝑥)) = 1, 1, ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)))))
19859eqeq2i 2749 . . . . . . . . . . . 12 ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (1 + (𝐴 · 𝑥)) = 1)
199138, 85, 124addcand 11358 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (𝐴 · 𝑥) = 0))
200198, 199bitr3id 284 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) = 1 ↔ (𝐴 · 𝑥) = 0))
20198oveq2d 7373 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))
202200, 201ifbieq2d 4512 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → if((1 + (𝐴 · 𝑥)) = 1, 1, ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1))) = if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
203202mpteq2dva 5205 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ if((1 + (𝐴 · 𝑥)) = 1, 1, ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)))) = (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
204197, 203eqtrd 2776 . . . . . . . 8 (𝐴 ∈ ℂ → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∘ (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))) = (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
205 eqid 2736 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
206 eqid 2736 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
207206cnfldtopon 24146 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
208207a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
209 1cnd 11150 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → 1 ∈ ℂ)
210208, 208, 209cnmptc 23013 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
211 id 22 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
212208, 208, 211cnmptc 23013 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
213208cnmptid 23012 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝑥) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
214206mulcn 24230 . . . . . . . . . . . . . . 15 · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
215214a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
216208, 212, 213, 215cnmpt12f 23017 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
217206addcn 24228 . . . . . . . . . . . . . 14 + ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
218217a1i 11 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → + ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
219208, 210, 216, 218cnmpt12f 23017 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (1 + (𝐴 · 𝑥))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
220205, 208, 44, 219cnmpt1res 23027 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
221141fmpttd 7063 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))):𝑆⟶(1(ball‘(abs ∘ − ))1))
222221frnd 6676 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ran (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ⊆ (1(ball‘(abs ∘ − ))1))
223 difss 4091 . . . . . . . . . . . . . 14 (ℂ ∖ {0}) ⊆ ℂ
22493, 223sstri 3953 . . . . . . . . . . . . 13 (1(ball‘(abs ∘ − ))1) ⊆ ℂ
225224a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (1(ball‘(abs ∘ − ))1) ⊆ ℂ)
226 cnrest2 22637 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ ran (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ⊆ (1(ball‘(abs ∘ − ))1) ∧ (1(ball‘(abs ∘ − ))1) ⊆ ℂ) → ((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)))))
227207, 222, 225, 226mp3an2i 1466 . . . . . . . . . . 11 (𝐴 ∈ ℂ → ((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)))))
228220, 227mpbid 231 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1))))
229 blcntr 23766 . . . . . . . . . . . . 13 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ+) → 0 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
23030, 31, 40, 229mp3an2i 1466 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → 0 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
231230, 29eleqtrrdi 2849 . . . . . . . . . . 11 (𝐴 ∈ ℂ → 0 ∈ 𝑆)
232 resttopon 22512 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
233207, 44, 232sylancr 587 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
234 toponuni 22263 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
235233, 234syl 17 . . . . . . . . . . 11 (𝐴 ∈ ℂ → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
236231, 235eleqtrd 2840 . . . . . . . . . 10 (𝐴 ∈ ℂ → 0 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
237 eqid 2736 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
238237cncnpi 22629 . . . . . . . . . 10 (((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1))) ∧ 0 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)))‘0))
239228, 236, 238syl2anc 584 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)))‘0))
240 cnelprrecn 11144 . . . . . . . . . . 11 ℂ ∈ {ℝ, ℂ}
241 logf1o 25920 . . . . . . . . . . . . . 14 log:(ℂ ∖ {0})–1-1-onto→ran log
242 f1of 6784 . . . . . . . . . . . . . 14 (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
243241, 242ax-mp 5 . . . . . . . . . . . . 13 log:(ℂ ∖ {0})⟶ran log
244 logrncn 25918 . . . . . . . . . . . . . 14 (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
245244ssriv 3948 . . . . . . . . . . . . 13 ran log ⊆ ℂ
246 fss 6685 . . . . . . . . . . . . 13 ((log:(ℂ ∖ {0})⟶ran log ∧ ran log ⊆ ℂ) → log:(ℂ ∖ {0})⟶ℂ)
247243, 245, 246mp2an 690 . . . . . . . . . . . 12 log:(ℂ ∖ {0})⟶ℂ
248 fssres 6708 . . . . . . . . . . . 12 ((log:(ℂ ∖ {0})⟶ℂ ∧ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})) → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ)
249247, 93, 248mp2an 690 . . . . . . . . . . 11 (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ
250 blcntr 23766 . . . . . . . . . . . . . 14 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ+) → 1 ∈ (1(ball‘(abs ∘ − ))1))
25130, 84, 135, 250mp3an 1461 . . . . . . . . . . . . 13 1 ∈ (1(ball‘(abs ∘ − ))1)
252 ovex 7390 . . . . . . . . . . . . . 14 (1 / 𝑦) ∈ V
25389dvlog2 26008 . . . . . . . . . . . . . 14 (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
254252, 253dmmpti 6645 . . . . . . . . . . . . 13 dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (1(ball‘(abs ∘ − ))1)
255251, 254eleqtrri 2837 . . . . . . . . . . . 12 1 ∈ dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))
256 eqid 2736 . . . . . . . . . . . . 13 ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)) = ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1))
257253fveq1i 6843 . . . . . . . . . . . . . . . . 17 ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1) = ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1)
258 oveq2 7365 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 1 → (1 / 𝑦) = (1 / 1))
259 1div1e1 11845 . . . . . . . . . . . . . . . . . . . 20 (1 / 1) = 1
260258, 259eqtrdi 2792 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 1 → (1 / 𝑦) = 1)
261 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦)) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
262 1ex 11151 . . . . . . . . . . . . . . . . . . 19 1 ∈ V
263260, 261, 262fvmpt 6948 . . . . . . . . . . . . . . . . . 18 (1 ∈ (1(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1) = 1)
264251, 263ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1) = 1
265257, 264eqtr2i 2765 . . . . . . . . . . . . . . . 16 1 = ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1)
266265a1i 11 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → 1 = ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1))
267 fvres 6861 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) = (log‘𝑦))
268 fvres 6861 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = (log‘1))
269251, 268mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = (log‘1))
270 log1 25941 . . . . . . . . . . . . . . . . . . 19 (log‘1) = 0
271269, 270eqtrdi 2792 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = 0)
272267, 271oveq12d 7375 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) = ((log‘𝑦) − 0))
27393sseli 3940 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → 𝑦 ∈ (ℂ ∖ {0}))
274 eldifsn 4747 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
275273, 274sylib 217 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
276 logcl 25924 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (log‘𝑦) ∈ ℂ)
277275, 276syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑦) ∈ ℂ)
278277subid1d 11501 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log‘𝑦) − 0) = (log‘𝑦))
279272, 278eqtr2d 2777 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑦) = (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)))
280279oveq1d 7372 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log‘𝑦) / (𝑦 − 1)) = ((((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) / (𝑦 − 1)))
281266, 280ifeq12d 4507 . . . . . . . . . . . . . 14 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1))) = if(𝑦 = 1, ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1), ((((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) / (𝑦 − 1))))
282281mpteq2ia 5208 . . . . . . . . . . . . 13 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1), ((((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) / (𝑦 − 1))))
283256, 206, 282dvcnp 25283 . . . . . . . . . . . 12 (((ℂ ∈ {ℝ, ℂ} ∧ (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ ∧ (1(ball‘(abs ∘ − ))1) ⊆ ℂ) ∧ 1 ∈ dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))) → (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∈ ((((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂfld))‘1))
284255, 283mpan2 689 . . . . . . . . . . 11 ((ℂ ∈ {ℝ, ℂ} ∧ (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ ∧ (1(ball‘(abs ∘ − ))1) ⊆ ℂ) → (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∈ ((((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂfld))‘1))
285240, 249, 224, 284mp3an 1461 . . . . . . . . . 10 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∈ ((((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂfld))‘1)
286 oveq2 7365 . . . . . . . . . . . . . . 15 (𝑥 = 0 → (𝐴 · 𝑥) = (𝐴 · 0))
287286oveq2d 7373 . . . . . . . . . . . . . 14 (𝑥 = 0 → (1 + (𝐴 · 𝑥)) = (1 + (𝐴 · 0)))
288 eqid 2736 . . . . . . . . . . . . . 14 (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))
289 ovex 7390 . . . . . . . . . . . . . 14 (1 + (𝐴 · 0)) ∈ V
290287, 288, 289fvmpt 6948 . . . . . . . . . . . . 13 (0 ∈ 𝑆 → ((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = (1 + (𝐴 · 0)))
291231, 290syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = (1 + (𝐴 · 0)))
292 mul01 11334 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
293292oveq2d 7373 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = (1 + 0))
294293, 59eqtrdi 2792 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = 1)
295291, 294eqtrd 2776 . . . . . . . . . . 11 (𝐴 ∈ ℂ → ((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = 1)
296295fveq2d 6846 . . . . . . . . . 10 (𝐴 ∈ ℂ → ((((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂfld))‘((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0)) = ((((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂfld))‘1))
297285, 296eleqtrrid 2845 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∈ ((((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂfld))‘((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0)))
298 cnpco 22618 . . . . . . . . 9 (((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)))‘0) ∧ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∈ ((((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂfld))‘((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0))) → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∘ (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0))
299239, 297, 298syl2anc 584 . . . . . . . 8 (𝐴 ∈ ℂ → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∘ (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0))
300204, 299eqeltrrd 2839 . . . . . . 7 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0))
301208, 208, 211cnmptc 23013 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
302208cnmptid 23012 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝑦) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
303208, 301, 302, 215cnmpt12f 23017 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
304 efcn 25802 . . . . . . . . . . 11 exp ∈ (ℂ–cn→ℂ)
305206cncfcn1 24274 . . . . . . . . . . 11 (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))
306304, 305eleqtri 2836 . . . . . . . . . 10 exp ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))
307306a1i 11 . . . . . . . . 9 (𝐴 ∈ ℂ → exp ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
308208, 303, 307cnmpt11f 23015 . . . . . . . 8 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
309177fmpttd 7063 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))):𝑆⟶ℂ)
310309, 231ffvelcdmd 7036 . . . . . . . 8 (𝐴 ∈ ℂ → ((𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0) ∈ ℂ)
311 unicntop 24149 . . . . . . . . 9 ℂ = (TopOpen‘ℂfld)
312311cncnpi 22629 . . . . . . . 8 (((𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) ∧ ((𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0) ∈ ℂ) → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ (((TopOpen‘ℂfld) CnP (TopOpen‘ℂfld))‘((𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0)))
313308, 310, 312syl2anc 584 . . . . . . 7 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ (((TopOpen‘ℂfld) CnP (TopOpen‘ℂfld))‘((𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0)))
314 cnpco 22618 . . . . . . 7 (((𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0) ∧ (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ (((TopOpen‘ℂfld) CnP (TopOpen‘ℂfld))‘((𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0))) → ((𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∘ (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0))
315300, 313, 314syl2anc 584 . . . . . 6 (𝐴 ∈ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∘ (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0))
316189, 315eqeltrd 2838 . . . . 5 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ↾ 𝑆) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0))
317206cnfldtop 24147 . . . . . . 7 (TopOpen‘ℂfld) ∈ Top
318317a1i 11 . . . . . 6 (𝐴 ∈ ℂ → (TopOpen‘ℂfld) ∈ Top)
319206cnfldtopn 24145 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
320319blopn 23856 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ*) → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂfld))
32130, 31, 41, 320mp3an2i 1466 . . . . . . . . 9 (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂfld))
32229, 321eqeltrid 2842 . . . . . . . 8 (𝐴 ∈ ℂ → 𝑆 ∈ (TopOpen‘ℂfld))
323 isopn3i 22433 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆)
324317, 322, 323sylancr 587 . . . . . . 7 (𝐴 ∈ ℂ → ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆)
325231, 324eleqtrrd 2841 . . . . . 6 (𝐴 ∈ ℂ → 0 ∈ ((int‘(TopOpen‘ℂfld))‘𝑆))
326 efcl 15965 . . . . . . . . 9 (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
327326ad2antrr 724 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ 𝑥 = 0) → (exp‘𝐴) ∈ ℂ)
32884, 14, 86sylancr 587 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
329328, 49cxpcld 26063 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) ∈ ℂ)
330327, 329ifclda 4521 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) ∈ ℂ)
331330fmpttd 7063 . . . . . 6 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))):ℂ⟶ℂ)
332311, 311cnprest 22640 . . . . . 6 ((((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) ∧ (0 ∈ ((int‘(TopOpen‘ℂfld))‘𝑆) ∧ (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))):ℂ⟶ℂ)) → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ∈ (((TopOpen‘ℂfld) CnP (TopOpen‘ℂfld))‘0) ↔ ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ↾ 𝑆) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0)))
333318, 44, 325, 331, 332syl22anc 837 . . . . 5 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ∈ (((TopOpen‘ℂfld) CnP (TopOpen‘ℂfld))‘0) ↔ ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ↾ 𝑆) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0)))
334316, 333mpbird 256 . . . 4 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ∈ (((TopOpen‘ℂfld) CnP (TopOpen‘ℂfld))‘0))
335311cnpresti 22639 . . . 4 (((0[,)+∞) ⊆ ℂ ∧ 0 ∈ (0[,)+∞) ∧ (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ∈ (((TopOpen‘ℂfld) CnP (TopOpen‘ℂfld))‘0)) → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ↾ (0[,)+∞)) ∈ ((((TopOpen‘ℂfld) ↾t (0[,)+∞)) CnP (TopOpen‘ℂfld))‘0))
3363, 26, 334, 335mp3an2i 1466 . . 3 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ↾ (0[,)+∞)) ∈ ((((TopOpen‘ℂfld) ↾t (0[,)+∞)) CnP (TopOpen‘ℂfld))‘0))
33724, 336eqeltrd 2838 . 2 (𝐴 ∈ ℂ → (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥)))) ∈ ((((TopOpen‘ℂfld) ↾t (0[,)+∞)) CnP (TopOpen‘ℂfld))‘0))
338 simpl 483 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → 𝐴 ∈ ℂ)
339 rpcn 12925 . . . . . . 7 (𝑘 ∈ ℝ+𝑘 ∈ ℂ)
340339adantl 482 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → 𝑘 ∈ ℂ)
341 rpne0 12931 . . . . . . 7 (𝑘 ∈ ℝ+𝑘 ≠ 0)
342341adantl 482 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → 𝑘 ≠ 0)
343338, 340, 342divcld 11931 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → (𝐴 / 𝑘) ∈ ℂ)
344 addcl 11133 . . . . 5 ((1 ∈ ℂ ∧ (𝐴 / 𝑘) ∈ ℂ) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
34584, 343, 344sylancr 587 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
346345, 340cxpcld 26063 . . 3 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → ((1 + (𝐴 / 𝑘))↑𝑐𝑘) ∈ ℂ)
347 oveq2 7365 . . . . 5 (𝑘 = (1 / 𝑥) → (𝐴 / 𝑘) = (𝐴 / (1 / 𝑥)))
348347oveq2d 7373 . . . 4 (𝑘 = (1 / 𝑥) → (1 + (𝐴 / 𝑘)) = (1 + (𝐴 / (1 / 𝑥))))
349 id 22 . . . 4 (𝑘 = (1 / 𝑥) → 𝑘 = (1 / 𝑥))
350348, 349oveq12d 7375 . . 3 (𝑘 = (1 / 𝑥) → ((1 + (𝐴 / 𝑘))↑𝑐𝑘) = ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥)))
351 eqid 2736 . . 3 ((TopOpen‘ℂfld) ↾t (0[,)+∞)) = ((TopOpen‘ℂfld) ↾t (0[,)+∞))
352326, 346, 350, 206, 351rlimcnp3 26317 . 2 (𝐴 ∈ ℂ → ((𝑘 ∈ ℝ+ ↦ ((1 + (𝐴 / 𝑘))↑𝑐𝑘)) ⇝𝑟 (exp‘𝐴) ↔ (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥)))) ∈ ((((TopOpen‘ℂfld) ↾t (0[,)+∞)) CnP (TopOpen‘ℂfld))‘0)))
353337, 352mpbird 256 1 (𝐴 ∈ ℂ → (𝑘 ∈ ℝ+ ↦ ((1 + (𝐴 / 𝑘))↑𝑐𝑘)) ⇝𝑟 (exp‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  cdif 3907  wss 3910  ifcif 4486  {csn 4586  {cpr 4588   cuni 4865   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634  cres 5635  ccom 5637  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  +∞cpnf 11186  -∞cmnf 11187  *cxr 11188   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  +crp 12915  (,]cioc 13265  [,)cico 13266  abscabs 15119  𝑟 crli 15367  expce 15944  t crest 17302  TopOpenctopn 17303  ∞Metcxmet 20781  ballcbl 20783  fldccnfld 20796  Topctop 22242  TopOnctopon 22259  intcnt 22368   Cn ccn 22575   CnP ccnp 22576   ×t ctx 22911  cnccncf 24239   D cdv 25227  logclog 25910  𝑐ccxp 25911
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-tan 15954  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231  df-log 25912  df-cxp 25913
This theorem is referenced by:  dfef2  26320
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