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Theorem efrlim 26816
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 26818). (Contributed by Mario Carneiro, 1-Mar-2015.) Avoid ax-mulf 11196. (Revised by GG, 19-Apr-2025.)
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 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rge0ssre 13440 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
2 ax-resscn 11173 . . . . . . . 8 ℝ ⊆ ℂ
31, 2sstri 3991 . . . . . . 7 (0[,)+∞) ⊆ ℂ
43sseli 3978 . . . . . 6 (𝑥 ∈ (0[,)+∞) → 𝑥 ∈ ℂ)
5 simpll 764 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝐴 ∈ ℂ)
6 1cnd 11216 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 1 ∈ ℂ)
7 simplr 766 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
8 ax-1ne0 11185 . . . . . . . . . . . 12 1 ≠ 0
98a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 1 ≠ 0)
10 simpr 484 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ¬ 𝑥 = 0)
1110neqned 2946 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ≠ 0)
125, 6, 7, 9, 11divdiv2d 12029 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = ((𝐴 · 𝑥) / 1))
13 mulcl 11200 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴 · 𝑥) ∈ ℂ)
1413adantr 480 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) ∈ ℂ)
1514div1d 11989 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((𝐴 · 𝑥) / 1) = (𝐴 · 𝑥))
1612, 15eqtrd 2771 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = (𝐴 · 𝑥))
1716oveq2d 7428 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 / (1 / 𝑥))) = (1 + (𝐴 · 𝑥)))
1817oveq1d 7427 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥)) = ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))
1918ifeq2da 4560 . . . . . 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 592 . . . . 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 5248 . . . 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 6037 . . . . 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 2789 . . 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 13442 . . . . 5 0 ∈ (0[,)+∞)
2625a1i 11 . . . 4 (𝐴 ∈ ℂ → 0 ∈ (0[,)+∞))
27 eqeq2 2743 . . . . . . . . 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 2743 . . . . . . . . 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 . . . . . . . . . . . . 13 𝑆 = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))
30 cnxmet 24610 . . . . . . . . . . . . . 14 (abs ∘ − ) ∈ (∞Met‘ℂ)
31 0cnd 11214 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → 0 ∈ ℂ)
32 abscl 15232 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
33 peano2re 11394 . . . . . . . . . . . . . . . . . 18 ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
3432, 33syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ)
35 0red 11224 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℂ → 0 ∈ ℝ)
36 absge0 15241 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
3732ltp1d 12151 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℂ → (abs‘𝐴) < ((abs‘𝐴) + 1))
3835, 32, 34, 36, 37lelttrd 11379 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℂ → 0 < ((abs‘𝐴) + 1))
3934, 38elrpd 13020 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ+)
4039rpreccld 13033 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ+)
4140rpxrd 13024 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ*)
42 blssm 24245 . . . . . . . . . . . . . 14 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ*) → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ⊆ ℂ)
4330, 31, 41, 42mp3an2i 1465 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ⊆ ℂ)
4429, 43eqsstrid 4030 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → 𝑆 ⊆ ℂ)
4544sselda 3982 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 𝑥 ∈ ℂ)
46 mul0or 11861 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
4745, 46syldan 590 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
487, 11reccld 11990 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
4945, 48syldanl 601 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
5049adantlr 712 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
51501cxpd 26556 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1↑𝑐(1 / 𝑥)) = 1)
52 simplr 766 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝐴 = 0)
5352oveq1d 7427 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = (0 · 𝑥))
5445ad2antrr 723 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
5554mul02d 11419 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (0 · 𝑥) = 0)
5653, 55eqtrd 2771 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = 0)
5756oveq2d 7428 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = (1 + 0))
58 1p0e1 12343 . . . . . . . . . . . . . . . . 17 (1 + 0) = 1
5957, 58eqtrdi 2787 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = 1)
6059oveq1d 7427 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) = (1↑𝑐(1 / 𝑥)))
6152fveq2d 6895 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = (exp‘0))
62 ef0 16041 . . . . . . . . . . . . . . . 16 (exp‘0) = 1
6361, 62eqtrdi 2787 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = 1)
6451, 60, 633eqtr4d 2781 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) = (exp‘𝐴))
6564ifeq2da 4560 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = if(𝑥 = 0, (exp‘𝐴), (exp‘𝐴)))
66 ifid 4568 . . . . . . . . . . . . 13 if(𝑥 = 0, (exp‘𝐴), (exp‘𝐴)) = (exp‘𝐴)
6765, 66eqtrdi 2787 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
68 iftrue 4534 . . . . . . . . . . . . 13 (𝑥 = 0 → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
6968adantl 481 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝑥 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
7067, 69jaodan 955 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
71 mulrid 11219 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
7271ad2antrr 723 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (𝐴 · 1) = 𝐴)
7372fveq2d 6895 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (exp‘(𝐴 · 1)) = (exp‘𝐴))
7470, 73eqtr4d 2774 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)))
7547, 74sylbida 591 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 · 𝑥) = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)))
76 mulne0b 11862 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
7745, 76syldan 590 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
78 df-ne 2940 . . . . . . . . . . . 12 ((𝐴 · 𝑥) ≠ 0 ↔ ¬ (𝐴 · 𝑥) = 0)
7977, 78bitrdi 287 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ ¬ (𝐴 · 𝑥) = 0))
80 simprr 770 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ≠ 0)
8180neneqd 2944 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ¬ 𝑥 = 0)
8281iffalsed 4539 . . . . . . . . . . . . 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)))
83 ax-1cn 11174 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
8445, 13syldan 590 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝐴 · 𝑥) ∈ ℂ)
85 addcl 11198 . . . . . . . . . . . . . . . 16 ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
8683, 84, 85sylancr 586 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
8786adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
88 eqid 2731 . . . . . . . . . . . . . . . . . . 19 (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
8988dvlog2lem 26501 . . . . . . . . . . . . . . . . . 18 (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
90 eqid 2731 . . . . . . . . . . . . . . . . . . 19 (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
9190logdmss 26491 . . . . . . . . . . . . . . . . . 18 (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
9289, 91sstri 3991 . . . . . . . . . . . . . . . . 17 (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
93 eqid 2731 . . . . . . . . . . . . . . . . . . . . . 22 (abs ∘ − ) = (abs ∘ − )
9493cnmetdval 24608 . . . . . . . . . . . . . . . . . . . . 21 (((1 + (𝐴 · 𝑥)) ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
9586, 83, 94sylancl 585 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
96 pncan2 11474 . . . . . . . . . . . . . . . . . . . . . 22 ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
9783, 84, 96sylancr 586 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
9897fveq2d 6895 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘((1 + (𝐴 · 𝑥)) − 1)) = (abs‘(𝐴 · 𝑥)))
9995, 98eqtrd 2771 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘(𝐴 · 𝑥)))
10084abscld 15390 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) ∈ ℝ)
10134adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((abs‘𝐴) + 1) ∈ ℝ)
10245abscld 15390 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝑥) ∈ ℝ)
103101, 102remulcld 11251 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) ∈ ℝ)
104 1red 11222 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 1 ∈ ℝ)
105 absmul 15248 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
10645, 105syldan 590 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
10732adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝐴) ∈ ℝ)
108107, 33syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((abs‘𝐴) + 1) ∈ ℝ)
10945absge0d 15398 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 0 ≤ (abs‘𝑥))
110107lep1d 12152 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
111107, 108, 102, 109, 110lemul1ad 12160 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((abs‘𝐴) · (abs‘𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
112106, 111eqbrtrd 5170 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
113 0cn 11213 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℂ
11493cnmetdval 24608 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
11545, 113, 114sylancl 585 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
11645subid1d 11567 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥 − 0) = 𝑥)
117116fveq2d 6895 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝑥 − 0)) = (abs‘𝑥))
118115, 117eqtrd 2771 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥(abs ∘ − )0) = (abs‘𝑥))
119 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 𝑥𝑆)
120119, 29eleqtrdi 2842 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
12130a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs ∘ − ) ∈ (∞Met‘ℂ))
12241adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 / ((abs‘𝐴) + 1)) ∈ ℝ*)
123 0cnd 11214 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 0 ∈ ℂ)
124 elbl3 24219 . . . . . . . . . . . . . . . . . . . . . . . 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))))
125121, 122, 123, 45, 124syl22anc 836 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ↔ (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1))))
126120, 125mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1)))
127118, 126eqbrtrrd 5172 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝑥) < (1 / ((abs‘𝐴) + 1)))
12838adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 0 < ((abs‘𝐴) + 1))
129 ltmuldiv2 12095 . . . . . . . . . . . . . . . . . . . . . 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))))
130102, 104, 108, 128, 129syl112anc 1373 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((((abs‘𝐴) + 1) · (abs‘𝑥)) < 1 ↔ (abs‘𝑥) < (1 / ((abs‘𝐴) + 1))))
131127, 130mpbird 257 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) < 1)
132100, 103, 104, 112, 131lelttrd 11379 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) < 1)
13399, 132eqbrtrd 5170 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1)
134 1rp 12985 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
135 rpxr 12990 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ ℝ+ → 1 ∈ ℝ*)
136134, 135mp1i 13 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 1 ∈ ℝ*)
137 1cnd 11216 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 1 ∈ ℂ)
138 elbl3 24219 . . . . . . . . . . . . . . . . . . 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))
139121, 136, 137, 86, 138syl22anc 836 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1))
140133, 139mpbird 257 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1))
14192, 140sselid 3980 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}))
142 eldifsni 4793 . . . . . . . . . . . . . . . 16 ((1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}) → (1 + (𝐴 · 𝑥)) ≠ 0)
143141, 142syl 17 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ≠ 0)
144143adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 + (𝐴 · 𝑥)) ≠ 0)
14545adantr 480 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ∈ ℂ)
146145, 80reccld 11990 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ ℂ)
14787, 144, 146cxpefd 26561 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) = (exp‘((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥))))))
14886, 143logcld 26420 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
149148adantr 480 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
150146, 149mulcomd 11242 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)))
151 simpll 764 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ∈ ℂ)
152 simprl 768 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ≠ 0)
153151, 152dividd 11995 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 / 𝐴) = 1)
154153oveq1d 7427 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (1 / 𝑥))
155151, 151, 145, 152, 80divdiv1d 12028 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
156154, 155eqtr3d 2773 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
157156oveq2d 7428 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)) = ((log‘(1 + (𝐴 · 𝑥))) · (𝐴 / (𝐴 · 𝑥))))
15884adantr 480 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 · 𝑥) ∈ ℂ)
15977biimpa 476 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 · 𝑥) ≠ 0)
160149, 151, 158, 159div12d 12033 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((log‘(1 + (𝐴 · 𝑥))) · (𝐴 / (𝐴 · 𝑥))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
161150, 157, 1603eqtrd 2775 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
162161fveq2d 6895 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (exp‘((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥))))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
16382, 147, 1623eqtrd 2775 . . . . . . . . . . . 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)))))
164163ex 412 . . . . . . . . . . 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))))))
16579, 164sylbird 260 . . . . . . . . . 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))))))
166165imp 406 . . . . . . . . 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)))))
16727, 28, 75, 166ifbothda 4566 . . . . . . . 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))))))
168167mpteq2dva 5248 . . . . . . 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)))))))
16944resmptd 6040 . . . . . . 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)))))
170 1cnd 11216 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 · 𝑥) = 0) → 1 ∈ ℂ)
171148adantr 480 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
17284adantr 480 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ∈ ℂ)
173 simpr 484 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ¬ (𝐴 · 𝑥) = 0)
174173neqned 2946 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ≠ 0)
175171, 172, 174divcld 11997 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)) ∈ ℂ)
176170, 175ifclda 4563 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) ∈ ℂ)
177 eqidd 2732 . . . . . . . 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)))))
178 eqidd 2732 . . . . . . . 8 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))))
179 oveq2 7420 . . . . . . . . . 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)))))
180179fveq2d 6895 . . . . . . . . 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))))))
181 oveq2 7420 . . . . . . . . . . 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))
182181fveq2d 6895 . . . . . . . . . 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)))
183 oveq2 7420 . . . . . . . . . . 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))))
184183fveq2d 6895 . . . . . . . . . 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)))))
185182, 184ifsb 4541 . . . . . . . . 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)))))
186180, 185eqtrdi 2787 . . . . . . . 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))))))
187176, 177, 178, 186fmptco 7129 . . . . . . 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)))))))
188168, 169, 1873eqtr4d 2781 . . . . . 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))))))
189 eqidd 2732 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))))
190 eqidd 2732 . . . . . . . . . 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)))))
191 eqeq1 2735 . . . . . . . . . . 11 (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 = 1 ↔ (1 + (𝐴 · 𝑥)) = 1))
192 fveq2 6891 . . . . . . . . . . . 12 (𝑦 = (1 + (𝐴 · 𝑥)) → (log‘𝑦) = (log‘(1 + (𝐴 · 𝑥))))
193 oveq1 7419 . . . . . . . . . . . 12 (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 − 1) = ((1 + (𝐴 · 𝑥)) − 1))
194192, 193oveq12d 7430 . . . . . . . . . . 11 (𝑦 = (1 + (𝐴 · 𝑥)) → ((log‘𝑦) / (𝑦 − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)))
195191, 194ifbieq2d 4554 . . . . . . . . . 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))))
196140, 189, 190, 195fmptco 7129 . . . . . . . . 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)))))
19758eqeq2i 2744 . . . . . . . . . . . 12 ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (1 + (𝐴 · 𝑥)) = 1)
198137, 84, 123addcand 11424 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (𝐴 · 𝑥) = 0))
199197, 198bitr3id 285 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) = 1 ↔ (𝐴 · 𝑥) = 0))
20097oveq2d 7428 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))
201199, 200ifbieq2d 4554 . . . . . . . . . 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))))
202201mpteq2dva 5248 . . . . . . . . 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)))))
203196, 202eqtrd 2771 . . . . . . . 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)))))
204 eqid 2731 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
205 eqid 2731 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
206205cnfldtopon 24620 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
207206a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
208 1cnd 11216 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → 1 ∈ ℂ)
209207, 207, 208cnmptc 23487 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
210 id 22 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
211207, 207, 210cnmptc 23487 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
212207cnmptid 23486 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝑥) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
213205mpomulcn 24706 . . . . . . . . . . . . . . 15 (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
214213a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
215 oveq12 7421 . . . . . . . . . . . . . 14 ((𝑢 = 𝐴𝑣 = 𝑥) → (𝑢 · 𝑣) = (𝐴 · 𝑥))
216207, 211, 212, 207, 207, 214, 215cnmpt12 23492 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
217205addcn 24702 . . . . . . . . . . . . . 14 + ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
218217a1i 11 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → + ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
219207, 209, 216, 218cnmpt12f 23491 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (1 + (𝐴 · 𝑥))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
220204, 207, 44, 219cnmpt1res 23501 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
221140fmpttd 7116 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))):𝑆⟶(1(ball‘(abs ∘ − ))1))
222221frnd 6725 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ran (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ⊆ (1(ball‘(abs ∘ − ))1))
223 difss 4131 . . . . . . . . . . . . . 14 (ℂ ∖ {0}) ⊆ ℂ
22492, 223sstri 3991 . . . . . . . . . . . . 13 (1(ball‘(abs ∘ − ))1) ⊆ ℂ
225224a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (1(ball‘(abs ∘ − ))1) ⊆ ℂ)
226 cnrest2 23111 . . . . . . . . . . . 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)))))
227206, 222, 225, 226mp3an2i 1465 . . . . . . . . . . 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 24240 . . . . . . . . . . . . 13 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ+) → 0 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
23030, 31, 40, 229mp3an2i 1465 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → 0 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
231230, 29eleqtrrdi 2843 . . . . . . . . . . 11 (𝐴 ∈ ℂ → 0 ∈ 𝑆)
232 resttopon 22986 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
233206, 44, 232sylancr 586 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
234 toponuni 22737 . . . . . . . . . . . 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 2834 . . . . . . . . . 10 (𝐴 ∈ ℂ → 0 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
237 eqid 2731 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
238237cncnpi 23103 . . . . . . . . . 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 583 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)))‘0))
240 cnelprrecn 11209 . . . . . . . . . . 11 ℂ ∈ {ℝ, ℂ}
241 logf1o 26414 . . . . . . . . . . . . . 14 log:(ℂ ∖ {0})–1-1-onto→ran log
242 f1of 6833 . . . . . . . . . . . . . 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 26412 . . . . . . . . . . . . . 14 (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
245244ssriv 3986 . . . . . . . . . . . . 13 ran log ⊆ ℂ
246 fss 6734 . . . . . . . . . . . . 13 ((log:(ℂ ∖ {0})⟶ran log ∧ ran log ⊆ ℂ) → log:(ℂ ∖ {0})⟶ℂ)
247243, 245, 246mp2an 689 . . . . . . . . . . . 12 log:(ℂ ∖ {0})⟶ℂ
248 fssres 6757 . . . . . . . . . . . 12 ((log:(ℂ ∖ {0})⟶ℂ ∧ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})) → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ)
249247, 92, 248mp2an 689 . . . . . . . . . . 11 (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ
250 blcntr 24240 . . . . . . . . . . . . . 14 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ+) → 1 ∈ (1(ball‘(abs ∘ − ))1))
25130, 83, 134, 250mp3an 1460 . . . . . . . . . . . . 13 1 ∈ (1(ball‘(abs ∘ − ))1)
252 ovex 7445 . . . . . . . . . . . . . 14 (1 / 𝑦) ∈ V
25388dvlog2 26502 . . . . . . . . . . . . . 14 (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
254252, 253dmmpti 6694 . . . . . . . . . . . . 13 dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (1(ball‘(abs ∘ − ))1)
255251, 254eleqtrri 2831 . . . . . . . . . . . 12 1 ∈ dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))
256 eqid 2731 . . . . . . . . . . . . 13 ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)) = ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1))
257253fveq1i 6892 . . . . . . . . . . . . . . . . 17 ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1) = ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1)
258 oveq2 7420 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 1 → (1 / 𝑦) = (1 / 1))
259 1div1e1 11911 . . . . . . . . . . . . . . . . . . . 20 (1 / 1) = 1
260258, 259eqtrdi 2787 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 1 → (1 / 𝑦) = 1)
261 eqid 2731 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦)) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
262 1ex 11217 . . . . . . . . . . . . . . . . . . 19 1 ∈ V
263260, 261, 262fvmpt 6998 . . . . . . . . . . . . . . . . . 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 2760 . . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) = (log‘𝑦))
268 fvres 6910 . . . . . . . . . . . . . . . . . . . 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 26435 . . . . . . . . . . . . . . . . . . 19 (log‘1) = 0
271269, 270eqtrdi 2787 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = 0)
272267, 271oveq12d 7430 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) = ((log‘𝑦) − 0))
27392sseli 3978 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → 𝑦 ∈ (ℂ ∖ {0}))
274 eldifsn 4790 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
275273, 274sylib 217 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
276 logcl 26418 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (log‘𝑦) ∈ ℂ)
277275, 276syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑦) ∈ ℂ)
278277subid1d 11567 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log‘𝑦) − 0) = (log‘𝑦))
279272, 278eqtr2d 2772 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑦) = (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)))
280279oveq1d 7427 . . . . . . . . . . . . . . 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 4549 . . . . . . . . . . . . . 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 5251 . . . . . . . . . . . . 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, 205, 282dvcnp 25769 . . . . . . . . . . . 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 688 . . . . . . . . . . 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 1460 . . . . . . . . . 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 7420 . . . . . . . . . . . . . . 15 (𝑥 = 0 → (𝐴 · 𝑥) = (𝐴 · 0))
287286oveq2d 7428 . . . . . . . . . . . . . 14 (𝑥 = 0 → (1 + (𝐴 · 𝑥)) = (1 + (𝐴 · 0)))
288 eqid 2731 . . . . . . . . . . . . . 14 (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))
289 ovex 7445 . . . . . . . . . . . . . 14 (1 + (𝐴 · 0)) ∈ V
290287, 288, 289fvmpt 6998 . . . . . . . . . . . . 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 11400 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
293292oveq2d 7428 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = (1 + 0))
294293, 58eqtrdi 2787 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = 1)
295291, 294eqtrd 2771 . . . . . . . . . . 11 (𝐴 ∈ ℂ → ((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = 1)
296295fveq2d 6895 . . . . . . . . . 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 2839 . . . . . . . . 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 23092 . . . . . . . . 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 583 . . . . . . . 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))
300203, 299eqeltrrd 2833 . . . . . . 7 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0))
301207, 207, 210cnmptc 23487 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
302207cnmptid 23486 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝑦) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
303 oveq12 7421 . . . . . . . . . 10 ((𝑢 = 𝐴𝑣 = 𝑦) → (𝑢 · 𝑣) = (𝐴 · 𝑦))
304207, 301, 302, 207, 207, 214, 303cnmpt12 23492 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
305 efcn 26296 . . . . . . . . . . 11 exp ∈ (ℂ–cn→ℂ)
306205cncfcn1 24752 . . . . . . . . . . 11 (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))
307305, 306eleqtri 2830 . . . . . . . . . 10 exp ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))
308307a1i 11 . . . . . . . . 9 (𝐴 ∈ ℂ → exp ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
309207, 304, 308cnmpt11f 23489 . . . . . . . 8 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
310176fmpttd 7116 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))):𝑆⟶ℂ)
311310, 231ffvelcdmd 7087 . . . . . . . 8 (𝐴 ∈ ℂ → ((𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0) ∈ ℂ)
312 unicntop 24623 . . . . . . . . 9 ℂ = (TopOpen‘ℂfld)
313312cncnpi 23103 . . . . . . . 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)))
314309, 311, 313syl2anc 583 . . . . . . 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)))
315 cnpco 23092 . . . . . . 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))
316300, 314, 315syl2anc 583 . . . . . 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))
317188, 316eqeltrd 2832 . . . . 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))
318205cnfldtop 24621 . . . . . . 7 (TopOpen‘ℂfld) ∈ Top
319318a1i 11 . . . . . 6 (𝐴 ∈ ℂ → (TopOpen‘ℂfld) ∈ Top)
320205cnfldtopn 24619 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
321320blopn 24330 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ*) → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂfld))
32230, 31, 41, 321mp3an2i 1465 . . . . . . . . 9 (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂfld))
32329, 322eqeltrid 2836 . . . . . . . 8 (𝐴 ∈ ℂ → 𝑆 ∈ (TopOpen‘ℂfld))
324 isopn3i 22907 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆)
325318, 323, 324sylancr 586 . . . . . . 7 (𝐴 ∈ ℂ → ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆)
326231, 325eleqtrrd 2835 . . . . . 6 (𝐴 ∈ ℂ → 0 ∈ ((int‘(TopOpen‘ℂfld))‘𝑆))
327 efcl 16033 . . . . . . . . 9 (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
328327ad2antrr 723 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ 𝑥 = 0) → (exp‘𝐴) ∈ ℂ)
32983, 14, 85sylancr 586 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
330329, 48cxpcld 26557 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) ∈ ℂ)
331328, 330ifclda 4563 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) ∈ ℂ)
332331fmpttd 7116 . . . . . 6 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))):ℂ⟶ℂ)
333312, 312cnprest 23114 . . . . . 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)))
334319, 44, 326, 332, 333syl22anc 836 . . . . 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)))
335317, 334mpbird 257 . . . 4 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ∈ (((TopOpen‘ℂfld) CnP (TopOpen‘ℂfld))‘0))
336312cnpresti 23113 . . . 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))
3373, 26, 335, 336mp3an2i 1465 . . 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))
33824, 337eqeltrd 2832 . 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))
339 simpl 482 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → 𝐴 ∈ ℂ)
340 rpcn 12991 . . . . . . 7 (𝑘 ∈ ℝ+𝑘 ∈ ℂ)
341340adantl 481 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → 𝑘 ∈ ℂ)
342 rpne0 12997 . . . . . . 7 (𝑘 ∈ ℝ+𝑘 ≠ 0)
343342adantl 481 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → 𝑘 ≠ 0)
344339, 341, 343divcld 11997 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → (𝐴 / 𝑘) ∈ ℂ)
345 addcl 11198 . . . . 5 ((1 ∈ ℂ ∧ (𝐴 / 𝑘) ∈ ℂ) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
34683, 344, 345sylancr 586 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
347346, 341cxpcld 26557 . . 3 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → ((1 + (𝐴 / 𝑘))↑𝑐𝑘) ∈ ℂ)
348 oveq2 7420 . . . . 5 (𝑘 = (1 / 𝑥) → (𝐴 / 𝑘) = (𝐴 / (1 / 𝑥)))
349348oveq2d 7428 . . . 4 (𝑘 = (1 / 𝑥) → (1 + (𝐴 / 𝑘)) = (1 + (𝐴 / (1 / 𝑥))))
350 id 22 . . . 4 (𝑘 = (1 / 𝑥) → 𝑘 = (1 / 𝑥))
351349, 350oveq12d 7430 . . 3 (𝑘 = (1 / 𝑥) → ((1 + (𝐴 / 𝑘))↑𝑐𝑘) = ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥)))
352 eqid 2731 . . 3 ((TopOpen‘ℂfld) ↾t (0[,)+∞)) = ((TopOpen‘ℂfld) ↾t (0[,)+∞))
353327, 347, 351, 205, 352rlimcnp3 26814 . 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)))
354338, 353mpbird 257 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 395  wo 844  w3a 1086   = wceq 1540  wcel 2105  wne 2939  cdif 3945  wss 3948  ifcif 4528  {csn 4628  {cpr 4630   cuni 4908   class class class wbr 5148  cmpt 5231  dom cdm 5676  ran crn 5677  cres 5678  ccom 5680  wf 6539  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7412  cmpo 7414  cc 11114  cr 11115  0cc0 11116  1c1 11117   + caddc 11119   · cmul 11121  +∞cpnf 11252  -∞cmnf 11253  *cxr 11254   < clt 11255  cle 11256  cmin 11451   / cdiv 11878  +crp 12981  (,]cioc 13332  [,)cico 13333  abscabs 15188  𝑟 crli 15436  expce 16012  t crest 17373  TopOpenctopn 17374  ∞Metcxmet 21219  ballcbl 21221  fldccnfld 21234  Topctop 22716  TopOnctopon 22733  intcnt 22842   Cn ccn 23049   CnP ccnp 23050   ×t ctx 23385  cnccncf 24717   D cdv 25713  logclog 26404  𝑐ccxp 26405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194  ax-addf 11195
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-fi 9412  df-sup 9443  df-inf 9444  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-q 12940  df-rp 12982  df-xneg 13099  df-xadd 13100  df-xmul 13101  df-ioo 13335  df-ioc 13336  df-ico 13337  df-icc 13338  df-fz 13492  df-fzo 13635  df-fl 13764  df-mod 13842  df-seq 13974  df-exp 14035  df-fac 14241  df-bc 14270  df-hash 14298  df-shft 15021  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-limsup 15422  df-clim 15439  df-rlim 15440  df-sum 15640  df-ef 16018  df-sin 16020  df-cos 16021  df-tan 16022  df-pi 16023  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-starv 17219  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-unif 17227  df-hom 17228  df-cco 17229  df-rest 17375  df-topn 17376  df-0g 17394  df-gsum 17395  df-topgen 17396  df-pt 17397  df-prds 17400  df-xrs 17455  df-qtop 17460  df-imas 17461  df-xps 17463  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-submnd 18712  df-mulg 18994  df-cntz 19229  df-cmn 19698  df-psmet 21226  df-xmet 21227  df-met 21228  df-bl 21229  df-mopn 21230  df-fbas 21231  df-fg 21232  df-cnfld 21235  df-top 22717  df-topon 22734  df-topsp 22756  df-bases 22770  df-cld 22844  df-ntr 22845  df-cls 22846  df-nei 22923  df-lp 22961  df-perf 22962  df-cn 23052  df-cnp 23053  df-haus 23140  df-cmp 23212  df-tx 23387  df-hmeo 23580  df-fil 23671  df-fm 23763  df-flim 23764  df-flf 23765  df-xms 24147  df-ms 24148  df-tms 24149  df-cncf 24719  df-limc 25716  df-dv 25717  df-log 26406  df-cxp 26407
This theorem is referenced by:  dfef2  26818
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