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Theorem efrlim 24741
 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 24742). (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 12318 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
2 ax-resscn 10031 . . . . . . . 8 ℝ ⊆ ℂ
31, 2sstri 3645 . . . . . . 7 (0[,)+∞) ⊆ ℂ
43sseli 3632 . . . . . 6 (𝑥 ∈ (0[,)+∞) → 𝑥 ∈ ℂ)
5 simpll 805 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝐴 ∈ ℂ)
6 1cnd 10094 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 1 ∈ ℂ)
7 simplr 807 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
8 ax-1ne0 10043 . . . . . . . . . . . 12 1 ≠ 0
98a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 1 ≠ 0)
10 simpr 476 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ¬ 𝑥 = 0)
1110neqned 2830 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ≠ 0)
125, 6, 7, 9, 11divdiv2d 10871 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = ((𝐴 · 𝑥) / 1))
13 mulcl 10058 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴 · 𝑥) ∈ ℂ)
1413adantr 480 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) ∈ ℂ)
1514div1d 10831 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((𝐴 · 𝑥) / 1) = (𝐴 · 𝑥))
1612, 15eqtrd 2685 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = (𝐴 · 𝑥))
1716oveq2d 6706 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 / (1 / 𝑥))) = (1 + (𝐴 · 𝑥)))
1817oveq1d 6705 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥)) = ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))
1918ifeq2da 4150 . . . . . 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 490 . . . . 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 4777 . . . 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 5484 . . . . 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, 23syl6eqr 2703 . . 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[,)+∞)))
253a1i 11 . . . 4 (𝐴 ∈ ℂ → (0[,)+∞) ⊆ ℂ)
26 0e0icopnf 12320 . . . . 5 0 ∈ (0[,)+∞)
2726a1i 11 . . . 4 (𝐴 ∈ ℂ → 0 ∈ (0[,)+∞))
28 eqeq2 2662 . . . . . . . . 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)))))))
29 eqeq2 2662 . . . . . . . . 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)))))))
30 efrlim.1 . . . . . . . . . . . . . 14 𝑆 = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))
31 cnxmet 22623 . . . . . . . . . . . . . . . 16 (abs ∘ − ) ∈ (∞Met‘ℂ)
3231a1i 11 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → (abs ∘ − ) ∈ (∞Met‘ℂ))
33 0cnd 10071 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → 0 ∈ ℂ)
34 abscl 14062 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
35 peano2re 10247 . . . . . . . . . . . . . . . . . . 19 ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
3634, 35syl 17 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ)
37 0red 10079 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℂ → 0 ∈ ℝ)
38 absge0 14071 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
3934ltp1d 10992 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℂ → (abs‘𝐴) < ((abs‘𝐴) + 1))
4037, 34, 36, 38, 39lelttrd 10233 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℂ → 0 < ((abs‘𝐴) + 1))
4136, 40elrpd 11907 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ+)
4241rpreccld 11920 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ+)
4342rpxrd 11911 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ*)
44 blssm 22270 . . . . . . . . . . . . . . 15 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ*) → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ⊆ ℂ)
4532, 33, 43, 44syl3anc 1366 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ⊆ ℂ)
4630, 45syl5eqss 3682 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → 𝑆 ⊆ ℂ)
4746sselda 3636 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 𝑥 ∈ ℂ)
48 mul0or 10705 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
4947, 48syldan 486 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
5049biimpa 500 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 · 𝑥) = 0) → (𝐴 = 0 ∨ 𝑥 = 0))
51 simpl 472 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 𝐴 ∈ ℂ)
5251, 47jca 553 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ))
537, 11reccld 10832 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
5452, 53sylan 487 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
5554adantlr 751 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
56551cxpd 24498 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1↑𝑐(1 / 𝑥)) = 1)
57 simplr 807 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝐴 = 0)
5857oveq1d 6705 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = (0 · 𝑥))
5947ad2antrr 762 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
6059mul02d 10272 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (0 · 𝑥) = 0)
6158, 60eqtrd 2685 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = 0)
6261oveq2d 6706 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = (1 + 0))
63 1p0e1 11171 . . . . . . . . . . . . . . . . 17 (1 + 0) = 1
6462, 63syl6eq 2701 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = 1)
6564oveq1d 6705 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) = (1↑𝑐(1 / 𝑥)))
6657fveq2d 6233 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = (exp‘0))
67 ef0 14865 . . . . . . . . . . . . . . . 16 (exp‘0) = 1
6866, 67syl6eq 2701 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = 1)
6956, 65, 683eqtr4d 2695 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) = (exp‘𝐴))
7069ifeq2da 4150 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = if(𝑥 = 0, (exp‘𝐴), (exp‘𝐴)))
71 ifid 4158 . . . . . . . . . . . . 13 if(𝑥 = 0, (exp‘𝐴), (exp‘𝐴)) = (exp‘𝐴)
7270, 71syl6eq 2701 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝐴 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
73 iftrue 4125 . . . . . . . . . . . . 13 (𝑥 = 0 → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
7473adantl 481 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ 𝑥 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
7572, 74jaodan 843 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘𝐴))
76 mulid1 10075 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
7776ad2antrr 762 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (𝐴 · 1) = 𝐴)
7877fveq2d 6233 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (exp‘(𝐴 · 1)) = (exp‘𝐴))
7975, 78eqtr4d 2688 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)))
8050, 79syldan 486 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 · 𝑥) = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)))
81 mulne0b 10706 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
8247, 81syldan 486 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
83 df-ne 2824 . . . . . . . . . . . 12 ((𝐴 · 𝑥) ≠ 0 ↔ ¬ (𝐴 · 𝑥) = 0)
8482, 83syl6bb 276 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ ¬ (𝐴 · 𝑥) = 0))
85 simprr 811 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ≠ 0)
8685neneqd 2828 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ¬ 𝑥 = 0)
8786iffalsed 4130 . . . . . . . . . . . . 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)))
88 ax-1cn 10032 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
8947, 13syldan 486 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝐴 · 𝑥) ∈ ℂ)
90 addcl 10056 . . . . . . . . . . . . . . . 16 ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
9188, 89, 90sylancr 696 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
9291adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
93 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
9493dvlog2lem 24443 . . . . . . . . . . . . . . . . . 18 (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
95 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
9695logdmss 24433 . . . . . . . . . . . . . . . . . 18 (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
9794, 96sstri 3645 . . . . . . . . . . . . . . . . 17 (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
98 eqid 2651 . . . . . . . . . . . . . . . . . . . . . 22 (abs ∘ − ) = (abs ∘ − )
9998cnmetdval 22621 . . . . . . . . . . . . . . . . . . . . 21 (((1 + (𝐴 · 𝑥)) ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
10091, 88, 99sylancl 695 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
101 pncan2 10326 . . . . . . . . . . . . . . . . . . . . . 22 ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
10288, 89, 101sylancr 696 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
103102fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘((1 + (𝐴 · 𝑥)) − 1)) = (abs‘(𝐴 · 𝑥)))
104100, 103eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘(𝐴 · 𝑥)))
10589abscld 14219 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) ∈ ℝ)
10636adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((abs‘𝐴) + 1) ∈ ℝ)
10747abscld 14219 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝑥) ∈ ℝ)
108106, 107remulcld 10108 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) ∈ ℝ)
109 1red 10093 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 1 ∈ ℝ)
110 absmul 14078 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
11147, 110syldan 486 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
11234adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝐴) ∈ ℝ)
113112, 35syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((abs‘𝐴) + 1) ∈ ℝ)
11447absge0d 14227 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 0 ≤ (abs‘𝑥))
115112lep1d 10993 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
116112, 113, 107, 114, 115lemul1ad 11001 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((abs‘𝐴) · (abs‘𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
117111, 116eqbrtrd 4707 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
118 0cn 10070 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℂ
11998cnmetdval 22621 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
12047, 118, 119sylancl 695 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
12147subid1d 10419 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥 − 0) = 𝑥)
122121fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝑥 − 0)) = (abs‘𝑥))
123120, 122eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥(abs ∘ − )0) = (abs‘𝑥))
124 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 𝑥𝑆)
125124, 30syl6eleq 2740 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
12631a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs ∘ − ) ∈ (∞Met‘ℂ))
12743adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 / ((abs‘𝐴) + 1)) ∈ ℝ*)
128 0cnd 10071 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 0 ∈ ℂ)
129 elbl3 22244 . . . . . . . . . . . . . . . . . . . . . . . 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))))
130126, 127, 128, 47, 129syl22anc 1367 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ↔ (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1))))
131125, 130mpbid 222 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1)))
132123, 131eqbrtrrd 4709 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘𝑥) < (1 / ((abs‘𝐴) + 1)))
13340adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 0 < ((abs‘𝐴) + 1))
134 ltmuldiv2 10935 . . . . . . . . . . . . . . . . . . . . . 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))))
135107, 109, 113, 133, 134syl112anc 1370 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((((abs‘𝐴) + 1) · (abs‘𝑥)) < 1 ↔ (abs‘𝑥) < (1 / ((abs‘𝐴) + 1))))
136132, 135mpbird 247 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) < 1)
137105, 108, 109, 117, 136lelttrd 10233 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (abs‘(𝐴 · 𝑥)) < 1)
138104, 137eqbrtrd 4707 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1)
139 1rp 11874 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
140 rpxr 11878 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ ℝ+ → 1 ∈ ℝ*)
141139, 140mp1i 13 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 1 ∈ ℝ*)
142 1cnd 10094 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → 1 ∈ ℂ)
143 elbl3 22244 . . . . . . . . . . . . . . . . . . 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))
144126, 141, 142, 91, 143syl22anc 1367 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1))
145138, 144mpbird 247 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1))
14697, 145sseldi 3634 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}))
147 eldifsni 4353 . . . . . . . . . . . . . . . 16 ((1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}) → (1 + (𝐴 · 𝑥)) ≠ 0)
148146, 147syl 17 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (1 + (𝐴 · 𝑥)) ≠ 0)
149148adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 + (𝐴 · 𝑥)) ≠ 0)
15047adantr 480 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ∈ ℂ)
151150, 85reccld 10832 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ ℂ)
15292, 149, 151cxpefd 24503 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) = (exp‘((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥))))))
15391, 148logcld 24362 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
154153adantr 480 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
155151, 154mulcomd 10099 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)))
156 simpll 805 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ∈ ℂ)
157 simprl 809 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ≠ 0)
158156, 157dividd 10837 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 / 𝐴) = 1)
159158oveq1d 6705 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (1 / 𝑥))
160156, 156, 150, 157, 85divdiv1d 10870 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
161159, 160eqtr3d 2687 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
162161oveq2d 6706 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)) = ((log‘(1 + (𝐴 · 𝑥))) · (𝐴 / (𝐴 · 𝑥))))
16389adantr 480 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 · 𝑥) ∈ ℂ)
16482biimpa 500 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 · 𝑥) ≠ 0)
165154, 156, 163, 164div12d 10875 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((log‘(1 + (𝐴 · 𝑥))) · (𝐴 / (𝐴 · 𝑥))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
166155, 162, 1653eqtrd 2689 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
167166fveq2d 6233 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (exp‘((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥))))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
16887, 152, 1673eqtrd 2689 . . . . . . . . . . . 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)))))
169168ex 449 . . . . . . . . . . 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))))))
17084, 169sylbird 250 . . . . . . . . . 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))))))
171170imp 444 . . . . . . . . 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)))))
17228, 29, 80, 171ifbothda 4156 . . . . . . . 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))))))
173172mpteq2dva 4777 . . . . . . 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)))))))
17446resmptd 5487 . . . . . . 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)))))
175 1cnd 10094 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ (𝐴 · 𝑥) = 0) → 1 ∈ ℂ)
176153adantr 480 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
17789adantr 480 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ∈ ℂ)
178 simpr 476 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ¬ (𝐴 · 𝑥) = 0)
179178neqned 2830 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ≠ 0)
180176, 177, 179divcld 10839 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)) ∈ ℂ)
181175, 180ifclda 4153 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) ∈ ℂ)
182 eqidd 2652 . . . . . . . 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)))))
183 eqidd 2652 . . . . . . . 8 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))))
184 oveq2 6698 . . . . . . . . . 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)))))
185184fveq2d 6233 . . . . . . . . 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))))))
186 oveq2 6698 . . . . . . . . . . 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))
187186fveq2d 6233 . . . . . . . . . 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)))
188 oveq2 6698 . . . . . . . . . . 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))))
189188fveq2d 6233 . . . . . . . . . 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)))))
190187, 189ifsb 4132 . . . . . . . . 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)))))
191185, 190syl6eq 2701 . . . . . . . 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))))))
192181, 182, 183, 191fmptco 6436 . . . . . . 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)))))))
193173, 174, 1923eqtr4d 2695 . . . . . 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))))))
194 eqidd 2652 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))))
195 eqidd 2652 . . . . . . . . . 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)))))
196 eqeq1 2655 . . . . . . . . . . 11 (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 = 1 ↔ (1 + (𝐴 · 𝑥)) = 1))
197 fveq2 6229 . . . . . . . . . . . 12 (𝑦 = (1 + (𝐴 · 𝑥)) → (log‘𝑦) = (log‘(1 + (𝐴 · 𝑥))))
198 oveq1 6697 . . . . . . . . . . . 12 (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 − 1) = ((1 + (𝐴 · 𝑥)) − 1))
199197, 198oveq12d 6708 . . . . . . . . . . 11 (𝑦 = (1 + (𝐴 · 𝑥)) → ((log‘𝑦) / (𝑦 − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)))
200196, 199ifbieq2d 4144 . . . . . . . . . 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))))
201145, 194, 195, 200fmptco 6436 . . . . . . . . 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)))))
20263eqeq2i 2663 . . . . . . . . . . . 12 ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (1 + (𝐴 · 𝑥)) = 1)
203142, 89, 128addcand 10277 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (𝐴 · 𝑥) = 0))
204202, 203syl5bbr 274 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((1 + (𝐴 · 𝑥)) = 1 ↔ (𝐴 · 𝑥) = 0))
205102oveq2d 6706 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥𝑆) → ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))
206204, 205ifbieq2d 4144 . . . . . . . . . 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))))
207206mpteq2dva 4777 . . . . . . . . 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)))))
208201, 207eqtrd 2685 . . . . . . . 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)))))
209 eqid 2651 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
210 eqid 2651 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
211210cnfldtopon 22633 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
212211a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
213 1cnd 10094 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → 1 ∈ ℂ)
214212, 212, 213cnmptc 21513 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
215 id 22 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
216212, 212, 215cnmptc 21513 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
217212cnmptid 21512 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝑥) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
218210mulcn 22717 . . . . . . . . . . . . . . 15 · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
219218a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
220212, 216, 217, 219cnmpt12f 21517 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
221210addcn 22715 . . . . . . . . . . . . . 14 + ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
222221a1i 11 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → + ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
223212, 214, 220, 222cnmpt12f 21517 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (1 + (𝐴 · 𝑥))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
224209, 212, 46, 223cnmpt1res 21527 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
225 eqid 2651 . . . . . . . . . . . . . 14 (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))
226145, 225fmptd 6425 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))):𝑆⟶(1(ball‘(abs ∘ − ))1))
227 frn 6091 . . . . . . . . . . . . 13 ((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))):𝑆⟶(1(ball‘(abs ∘ − ))1) → ran (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ⊆ (1(ball‘(abs ∘ − ))1))
228226, 227syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ran (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ⊆ (1(ball‘(abs ∘ − ))1))
229 difss 3770 . . . . . . . . . . . . . 14 (ℂ ∖ {0}) ⊆ ℂ
23097, 229sstri 3645 . . . . . . . . . . . . 13 (1(ball‘(abs ∘ − ))1) ⊆ ℂ
231230a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (1(ball‘(abs ∘ − ))1) ⊆ ℂ)
232 cnrest2 21138 . . . . . . . . . . . 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)))))
233212, 228, 231, 232syl3anc 1366 . . . . . . . . . . 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)))))
234224, 233mpbid 222 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1))))
235 blcntr 22265 . . . . . . . . . . . . 13 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ+) → 0 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
23632, 33, 42, 235syl3anc 1366 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → 0 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
237236, 30syl6eleqr 2741 . . . . . . . . . . 11 (𝐴 ∈ ℂ → 0 ∈ 𝑆)
238 resttopon 21013 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
239211, 46, 238sylancr 696 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
240 toponuni 20767 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
241239, 240syl 17 . . . . . . . . . . 11 (𝐴 ∈ ℂ → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
242237, 241eleqtrd 2732 . . . . . . . . . 10 (𝐴 ∈ ℂ → 0 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
243 eqid 2651 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
244243cncnpi 21130 . . . . . . . . . 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))
245234, 242, 244syl2anc 694 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)))‘0))
246 cnelprrecn 10067 . . . . . . . . . . 11 ℂ ∈ {ℝ, ℂ}
247 logf1o 24356 . . . . . . . . . . . . . 14 log:(ℂ ∖ {0})–1-1-onto→ran log
248 f1of 6175 . . . . . . . . . . . . . 14 (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
249247, 248ax-mp 5 . . . . . . . . . . . . 13 log:(ℂ ∖ {0})⟶ran log
250 logrncn 24354 . . . . . . . . . . . . . 14 (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
251250ssriv 3640 . . . . . . . . . . . . 13 ran log ⊆ ℂ
252 fss 6094 . . . . . . . . . . . . 13 ((log:(ℂ ∖ {0})⟶ran log ∧ ran log ⊆ ℂ) → log:(ℂ ∖ {0})⟶ℂ)
253249, 251, 252mp2an 708 . . . . . . . . . . . 12 log:(ℂ ∖ {0})⟶ℂ
254 fssres 6108 . . . . . . . . . . . 12 ((log:(ℂ ∖ {0})⟶ℂ ∧ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})) → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ)
255253, 97, 254mp2an 708 . . . . . . . . . . 11 (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ
256 blcntr 22265 . . . . . . . . . . . . . 14 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ+) → 1 ∈ (1(ball‘(abs ∘ − ))1))
25731, 88, 139, 256mp3an 1464 . . . . . . . . . . . . 13 1 ∈ (1(ball‘(abs ∘ − ))1)
258 ovex 6718 . . . . . . . . . . . . . 14 (1 / 𝑦) ∈ V
25993dvlog2 24444 . . . . . . . . . . . . . 14 (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
260258, 259dmmpti 6061 . . . . . . . . . . . . 13 dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (1(ball‘(abs ∘ − ))1)
261257, 260eleqtrri 2729 . . . . . . . . . . . 12 1 ∈ dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))
262 eqid 2651 . . . . . . . . . . . . 13 ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1)) = ((TopOpen‘ℂfld) ↾t (1(ball‘(abs ∘ − ))1))
263259fveq1i 6230 . . . . . . . . . . . . . . . . 17 ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1) = ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1)
264 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 1 → (1 / 𝑦) = (1 / 1))
265 1div1e1 10755 . . . . . . . . . . . . . . . . . . . 20 (1 / 1) = 1
266264, 265syl6eq 2701 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 1 → (1 / 𝑦) = 1)
267 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦)) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
268 1ex 10073 . . . . . . . . . . . . . . . . . . 19 1 ∈ V
269266, 267, 268fvmpt 6321 . . . . . . . . . . . . . . . . . 18 (1 ∈ (1(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1) = 1)
270257, 269ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1) = 1
271263, 270eqtr2i 2674 . . . . . . . . . . . . . . . 16 1 = ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1)
272271a1i 11 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → 1 = ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1))
273 fvres 6245 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) = (log‘𝑦))
274 fvres 6245 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = (log‘1))
275257, 274mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = (log‘1))
276 log1 24377 . . . . . . . . . . . . . . . . . . 19 (log‘1) = 0
277275, 276syl6eq 2701 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = 0)
278273, 277oveq12d 6708 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) = ((log‘𝑦) − 0))
27997sseli 3632 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → 𝑦 ∈ (ℂ ∖ {0}))
280 eldifsn 4350 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
281279, 280sylib 208 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
282 logcl 24360 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (log‘𝑦) ∈ ℂ)
283281, 282syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑦) ∈ ℂ)
284283subid1d 10419 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log‘𝑦) − 0) = (log‘𝑦))
285278, 284eqtr2d 2686 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑦) = (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)))
286285oveq1d 6705 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log‘𝑦) / (𝑦 − 1)) = ((((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) / (𝑦 − 1)))
287272, 286ifeq12d 4139 . . . . . . . . . . . . . 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))))
288287mpteq2ia 4773 . . . . . . . . . . . . 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))))
289262, 210, 288dvcnp 23727 . . . . . . . . . . . 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))
290261, 289mpan2 707 . . . . . . . . . . 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))
291246, 255, 230, 290mp3an 1464 . . . . . . . . . 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)
292 oveq2 6698 . . . . . . . . . . . . . . 15 (𝑥 = 0 → (𝐴 · 𝑥) = (𝐴 · 0))
293292oveq2d 6706 . . . . . . . . . . . . . 14 (𝑥 = 0 → (1 + (𝐴 · 𝑥)) = (1 + (𝐴 · 0)))
294 ovex 6718 . . . . . . . . . . . . . 14 (1 + (𝐴 · 0)) ∈ V
295293, 225, 294fvmpt 6321 . . . . . . . . . . . . 13 (0 ∈ 𝑆 → ((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = (1 + (𝐴 · 0)))
296237, 295syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = (1 + (𝐴 · 0)))
297 mul01 10253 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
298297oveq2d 6706 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = (1 + 0))
299298, 63syl6eq 2701 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = 1)
300296, 299eqtrd 2685 . . . . . . . . . . 11 (𝐴 ∈ ℂ → ((𝑥𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = 1)
301300fveq2d 6233 . . . . . . . . . 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))
302291, 301syl5eleqr 2737 . . . . . . . . 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)))
303 cnpco 21119 . . . . . . . . 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))
304245, 302, 303syl2anc 694 . . . . . . . 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))
305208, 304eqeltrrd 2731 . . . . . . 7 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘0))
306212, 212, 215cnmptc 21513 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
307212cnmptid 21512 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝑦) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
308212, 306, 307, 219cnmpt12f 21517 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
309 efcn 24242 . . . . . . . . . . 11 exp ∈ (ℂ–cn→ℂ)
310210cncfcn1 22760 . . . . . . . . . . 11 (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))
311309, 310eleqtri 2728 . . . . . . . . . 10 exp ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))
312311a1i 11 . . . . . . . . 9 (𝐴 ∈ ℂ → exp ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
313212, 308, 312cnmpt11f 21515 . . . . . . . 8 (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)))
314 eqid 2651 . . . . . . . . . 10 (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) = (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
315181, 314fmptd 6425 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))):𝑆⟶ℂ)
316315, 237ffvelrnd 6400 . . . . . . . 8 (𝐴 ∈ ℂ → ((𝑥𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0) ∈ ℂ)
317211toponunii 20769 . . . . . . . . 9 ℂ = (TopOpen‘ℂfld)
318317cncnpi 21130 . . . . . . . 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)))
319313, 316, 318syl2anc 694 . . . . . . 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)))
320 cnpco 21119 . . . . . . 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))
321305, 319, 320syl2anc 694 . . . . . 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))
322193, 321eqeltrd 2730 . . . . 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))
323210cnfldtop 22634 . . . . . . 7 (TopOpen‘ℂfld) ∈ Top
324323a1i 11 . . . . . 6 (𝐴 ∈ ℂ → (TopOpen‘ℂfld) ∈ Top)
325210cnfldtopn 22632 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
326325blopn 22352 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ*) → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂfld))
32732, 33, 43, 326syl3anc 1366 . . . . . . . . 9 (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂfld))
32830, 327syl5eqel 2734 . . . . . . . 8 (𝐴 ∈ ℂ → 𝑆 ∈ (TopOpen‘ℂfld))
329 isopn3i 20934 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆)
330323, 328, 329sylancr 696 . . . . . . 7 (𝐴 ∈ ℂ → ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆)
331237, 330eleqtrrd 2733 . . . . . 6 (𝐴 ∈ ℂ → 0 ∈ ((int‘(TopOpen‘ℂfld))‘𝑆))
332 efcl 14857 . . . . . . . . 9 (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
333332ad2antrr 762 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ 𝑥 = 0) → (exp‘𝐴) ∈ ℂ)
33488, 14, 90sylancr 696 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
335334, 53cxpcld 24499 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)) ∈ ℂ)
336333, 335ifclda 4153 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))) ∈ ℂ)
337 eqid 2651 . . . . . . 7 (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) = (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥))))
338336, 337fmptd 6425 . . . . . 6 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))):ℂ⟶ℂ)
339317, 317cnprest 21141 . . . . . 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)))
340324, 46, 331, 338, 339syl22anc 1367 . . . . 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)))
341322, 340mpbird 247 . . . 4 (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑𝑐(1 / 𝑥)))) ∈ (((TopOpen‘ℂfld) CnP (TopOpen‘ℂfld))‘0))
342317cnpresti 21140 . . . 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))
34325, 27, 341, 342syl3anc 1366 . . 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))
34424, 343eqeltrd 2730 . 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))
345 simpl 472 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → 𝐴 ∈ ℂ)
346 rpcn 11879 . . . . . . 7 (𝑘 ∈ ℝ+𝑘 ∈ ℂ)
347346adantl 481 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → 𝑘 ∈ ℂ)
348 rpne0 11886 . . . . . . 7 (𝑘 ∈ ℝ+𝑘 ≠ 0)
349348adantl 481 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → 𝑘 ≠ 0)
350345, 347, 349divcld 10839 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → (𝐴 / 𝑘) ∈ ℂ)
351 addcl 10056 . . . . 5 ((1 ∈ ℂ ∧ (𝐴 / 𝑘) ∈ ℂ) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
35288, 350, 351sylancr 696 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
353352, 347cxpcld 24499 . . 3 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ+) → ((1 + (𝐴 / 𝑘))↑𝑐𝑘) ∈ ℂ)
354 oveq2 6698 . . . . 5 (𝑘 = (1 / 𝑥) → (𝐴 / 𝑘) = (𝐴 / (1 / 𝑥)))
355354oveq2d 6706 . . . 4 (𝑘 = (1 / 𝑥) → (1 + (𝐴 / 𝑘)) = (1 + (𝐴 / (1 / 𝑥))))
356 id 22 . . . 4 (𝑘 = (1 / 𝑥) → 𝑘 = (1 / 𝑥))
357355, 356oveq12d 6708 . . 3 (𝑘 = (1 / 𝑥) → ((1 + (𝐴 / 𝑘))↑𝑐𝑘) = ((1 + (𝐴 / (1 / 𝑥)))↑𝑐(1 / 𝑥)))
358 eqid 2651 . . 3 ((TopOpen‘ℂfld) ↾t (0[,)+∞)) = ((TopOpen‘ℂfld) ↾t (0[,)+∞))
359332, 353, 357, 210, 358rlimcnp3 24739 . 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)))
360344, 359mpbird 247 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 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823   ∖ cdif 3604   ⊆ wss 3607  ifcif 4119  {csn 4210  {cpr 4212  ∪ cuni 4468   class class class wbr 4685   ↦ cmpt 4762  dom cdm 5143  ran crn 5144   ↾ cres 5145   ∘ ccom 5147  ⟶wf 5922  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109  -∞cmnf 10110  ℝ*cxr 10111   < clt 10112   ≤ cle 10113   − cmin 10304   / cdiv 10722  ℝ+crp 11870  (,]cioc 12214  [,)cico 12215  abscabs 14018   ⇝𝑟 crli 14260  expce 14836   ↾t crest 16128  TopOpenctopn 16129  ∞Metcxmt 19779  ballcbl 19781  ℂfldccnfld 19794  Topctop 20746  TopOnctopon 20763  intcnt 20869   Cn ccn 21076   CnP ccnp 21077   ×t ctx 21411  –cn→ccncf 22726   D cdv 23672  logclog 24346  ↑𝑐ccxp 24347 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-tan 14846  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-cxp 24349 This theorem is referenced by:  dfef2  24742
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