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Theorem efrlim 26877

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 26879). (Contributed by Mario Carneiro, 1-Mar-2015.) Avoid ax-mulf 11089. (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.

Step	Hyp	Ref	Expression
1		rge0ssre 13359	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
2		ax-resscn 11066	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
3	1, 2	sstri 3945	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
4	3	sseli 3931	. . . . . 6 ⊢ (𝑥 ∈ (0[,)+∞) → 𝑥 ∈ ℂ)
5		simpll 766	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝐴 ∈ ℂ)
6		1cnd 11110	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 1 ∈ ℂ)
7		simplr 768	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
8		ax-1ne0 11078	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 1 ≠ 0)
10		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ¬ 𝑥 = 0)
11	10	neqned 2932	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ≠ 0)
12	5, 6, 7, 9, 11	divdiv2d 11932	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = ((𝐴 · 𝑥) / 1))
13		mulcl 11093	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴 · 𝑥) ∈ ℂ)
14	13	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) ∈ ℂ)
15	14	div1d 11892	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((𝐴 · 𝑥) / 1) = (𝐴 · 𝑥))
16	12, 15	eqtrd 2764	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = (𝐴 · 𝑥))
17	16	oveq2d 7365	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 / (1 / 𝑥))) = (1 + (𝐴 · 𝑥)))
18	17	oveq1d 7364	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 / (1 / 𝑥)))↑_𝑐(1 / 𝑥)) = ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))
19	18	ifeq2da 4509	. . . . . 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))))
20	4, 19	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ (0[,)+∞)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 / (1 / 𝑥)))↑_𝑐(1 / 𝑥))) = if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))))
21	20	mpteq2dva 5185	. . . 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 5988	. . . . 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)))))
23	3, 22	ax-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))))
24	21, 23	eqtr4di 2782	. . 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 13361	. . . . 5 ⊢ 0 ∈ (0[,)+∞)
26	25	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℂ → 0 ∈ (0[,)+∞))
27		eqeq2 2741	. . . . . . . . 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 2741	. . . . . . . . 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 24658	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
31		0cnd 11108	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ)
32		abscl 15185	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
33		peano2re 11289	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ)
35		0red 11118	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℝ)
36		absge0 15194	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
37	32	ltp1d 12055	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) < ((abs‘𝐴) + 1))
38	35, 32, 34, 36, 37	lelttrd 11274	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → 0 < ((abs‘𝐴) + 1))
39	34, 38	elrpd 12934	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ⁺)
40	39	rpreccld 12947	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ⁺)
41	40	rpxrd 12938	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ^*)
42		blssm 24304	. . . . . . . . . . . . . 14 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ^*) → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ⊆ ℂ)
43	30, 31, 41, 42	mp3an2i 1468	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ⊆ ℂ)
44	29, 43	eqsstrid 3974	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → 𝑆 ⊆ ℂ)
45	44	sselda 3935	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
46		mul0or 11760	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
47	45, 46	syldan 591	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
48	7, 11	reccld 11893	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
49	45, 48	syldanl 602	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
50	49	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
51	50	1cxpd 26614	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1↑_𝑐(1 / 𝑥)) = 1)
52		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝐴 = 0)
53	52	oveq1d 7364	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = (0 · 𝑥))
54	45	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
55	54	mul02d 11314	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (0 · 𝑥) = 0)
56	53, 55	eqtrd 2764	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = 0)
57	56	oveq2d 7365	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = (1 + 0))
58		1p0e1 12247	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 0) = 1
59	57, 58	eqtrdi 2780	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = 1)
60	59	oveq1d 7364	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) = (1↑_𝑐(1 / 𝑥)))
61	52	fveq2d 6826	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = (exp‘0))
62		ef0 15998	. . . . . . . . . . . . . . . 16 ⊢ (exp‘0) = 1
63	61, 62	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = 1)
64	51, 60, 63	3eqtr4d 2774	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) = (exp‘𝐴))
65	64	ifeq2da 4509	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = if(𝑥 = 0, (exp‘𝐴), (exp‘𝐴)))
66		ifid 4517	. . . . . . . . . . . . 13 ⊢ if(𝑥 = 0, (exp‘𝐴), (exp‘𝐴)) = (exp‘𝐴)
67	65, 66	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘𝐴))
68		iftrue 4482	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘𝐴))
69	68	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝑥 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘𝐴))
70	67, 69	jaodan 959	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘𝐴))
71		mulrid 11113	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
72	71	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (𝐴 · 1) = 𝐴)
73	72	fveq2d 6826	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (exp‘(𝐴 · 1)) = (exp‘𝐴))
74	70, 73	eqtr4d 2767	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)))
75	47, 74	sylbida 592	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 · 𝑥) = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)))
76		mulne0b 11761	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
77	45, 76	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
78		df-ne 2926	. . . . . . . . . . . 12 ⊢ ((𝐴 · 𝑥) ≠ 0 ↔ ¬ (𝐴 · 𝑥) = 0)
79	77, 78	bitrdi 287	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ ¬ (𝐴 · 𝑥) = 0))
80		simprr 772	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ≠ 0)
81	80	neneqd 2930	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ¬ 𝑥 = 0)
82	81	iffalsed 4487	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))
83		ax-1cn 11067	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
84	45, 13	syldan 591	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝐴 · 𝑥) ∈ ℂ)
85		addcl 11091	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
86	83, 84, 85	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
87	86	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
88		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
89	88	dvlog2lem 26559	. . . . . . . . . . . . . . . . . 18 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
90		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
91	90	logdmss 26549	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
92	89, 91	sstri 3945	. . . . . . . . . . . . . . . . 17 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
93		eqid 2729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (abs ∘ − ) = (abs ∘ − )
94	93	cnmetdval 24656	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1 + (𝐴 · 𝑥)) ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
95	86, 83, 94	sylancl 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
96		pncan2 11370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
97	83, 84, 96	sylancr 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
98	97	fveq2d 6826	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘((1 + (𝐴 · 𝑥)) − 1)) = (abs‘(𝐴 · 𝑥)))
99	95, 98	eqtrd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘(𝐴 · 𝑥)))
100	84	abscld 15346	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝐴 · 𝑥)) ∈ ℝ)
101	34	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((abs‘𝐴) + 1) ∈ ℝ)
102	45	abscld 15346	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘𝑥) ∈ ℝ)
103	101, 102	remulcld 11145	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) ∈ ℝ)
104		1red 11116	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 1 ∈ ℝ)
105		absmul 15201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
106	45, 105	syldan 591	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
107	32	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘𝐴) ∈ ℝ)
108	107, 33	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((abs‘𝐴) + 1) ∈ ℝ)
109	45	absge0d 15354	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 0 ≤ (abs‘𝑥))
110	107	lep1d 12056	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
111	107, 108, 102, 109, 110	lemul1ad 12064	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((abs‘𝐴) · (abs‘𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
112	106, 111	eqbrtrd 5114	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝐴 · 𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
113		0cn 11107	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ ℂ
114	93	cnmetdval 24656	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
115	45, 113, 114	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
116	45	subid1d 11464	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥 − 0) = 𝑥)
117	116	fveq2d 6826	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝑥 − 0)) = (abs‘𝑥))
118	115, 117	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥(abs ∘ − )0) = (abs‘𝑥))
119		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
120	119, 29	eleqtrdi 2838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
121	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs ∘ − ) ∈ (∞Met‘ℂ))
122	41	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 / ((abs‘𝐴) + 1)) ∈ ℝ^*)
123		0cnd 11108	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 0 ∈ ℂ)
124		elbl3 24278	. . . . . . . . . . . . . . . . . . . . . . . 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))))
125	121, 122, 123, 45, 124	syl22anc 838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ↔ (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1))))
126	120, 125	mpbid 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1)))
127	118, 126	eqbrtrrd 5116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘𝑥) < (1 / ((abs‘𝐴) + 1)))
128	38	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 0 < ((abs‘𝐴) + 1))
129		ltmuldiv2 11999	. . . . . . . . . . . . . . . . . . . . . 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))))
130	102, 104, 108, 128, 129	syl112anc 1376	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((((abs‘𝐴) + 1) · (abs‘𝑥)) < 1 ↔ (abs‘𝑥) < (1 / ((abs‘𝐴) + 1))))
131	127, 130	mpbird 257	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) < 1)
132	100, 103, 104, 112, 131	lelttrd 11274	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝐴 · 𝑥)) < 1)
133	99, 132	eqbrtrd 5114	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1)
134		1rp 12897	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℝ⁺
135		rpxr 12903	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
136	134, 135	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 1 ∈ ℝ^*)
137		1cnd 11110	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 1 ∈ ℂ)
138		elbl3 24278	. . . . . . . . . . . . . . . . . . 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))
139	121, 136, 137, 86, 138	syl22anc 838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1))
140	133, 139	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1))
141	92, 140	sselid 3933	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}))
142		eldifsni 4741	. . . . . . . . . . . . . . . 16 ⊢ ((1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}) → (1 + (𝐴 · 𝑥)) ≠ 0)
143	141, 142	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 + (𝐴 · 𝑥)) ≠ 0)
144	143	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 + (𝐴 · 𝑥)) ≠ 0)
145	45	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ∈ ℂ)
146	145, 80	reccld 11893	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ ℂ)
147	87, 144, 146	cxpefd 26619	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) = (exp‘((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥))))))
148	86, 143	logcld 26477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
149	148	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
150	146, 149	mulcomd 11136	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)))
151		simpll 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ∈ ℂ)
152		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ≠ 0)
153	151, 152	dividd 11898	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 / 𝐴) = 1)
154	153	oveq1d 7364	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (1 / 𝑥))
155	151, 151, 145, 152, 80	divdiv1d 11931	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
156	154, 155	eqtr3d 2766	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
157	156	oveq2d 7365	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)) = ((log‘(1 + (𝐴 · 𝑥))) · (𝐴 / (𝐴 · 𝑥))))
158	84	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 · 𝑥) ∈ ℂ)
159	77	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 · 𝑥) ≠ 0)
160	149, 151, 158, 159	div12d 11936	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((log‘(1 + (𝐴 · 𝑥))) · (𝐴 / (𝐴 · 𝑥))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
161	150, 157, 160	3eqtrd 2768	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
162	161	fveq2d 6826	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (exp‘((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥))))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
163	82, 147, 162	3eqtrd 2768	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
164	163	ex 412	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))))
165	79, 164	sylbird 260	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (¬ (𝐴 · 𝑥) = 0 → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))))
166	165	imp 406	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
167	27, 28, 75, 166	ifbothda 4515	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 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))))))
168	167	mpteq2dva 5185	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) = (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))))
169	44	resmptd 5991	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))))
170		1cnd 11110	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 · 𝑥) = 0) → 1 ∈ ℂ)
171	148	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
172	84	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ∈ ℂ)
173		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ¬ (𝐴 · 𝑥) = 0)
174	173	neqned 2932	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ≠ 0)
175	171, 172, 174	divcld 11900	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)) ∈ ℂ)
176	170, 175	ifclda 4512	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) ∈ ℂ)
177		eqidd 2730	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) = (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
178		eqidd 2730	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))))
179		oveq2 7357	. . . . . . . . . 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)))))
180	179	fveq2d 6826	. . . . . . . . 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 7357	. . . . . . . . . . 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))
182	181	fveq2d 6826	. . . . . . . . . 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 7357	. . . . . . . . . . 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))))
184	183	fveq2d 6826	. . . . . . . . . 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)))))
185	182, 184	ifsb 4490	. . . . . . . . 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)))))
186	180, 185	eqtrdi 2780	. . . . . . . 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))))))
187	176, 177, 178, 186	fmptco 7063	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∘ (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) = (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, (exp‘(𝐴 · 1)), (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))))
188	168, 169, 187	3eqtr4d 2774	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∘ (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))))
189		eqidd 2730	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))))
190		eqidd 2730	. . . . . . . . . 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 2733	. . . . . . . . . . 11 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 = 1 ↔ (1 + (𝐴 · 𝑥)) = 1))
192		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → (log‘𝑦) = (log‘(1 + (𝐴 · 𝑥))))
193		oveq1 7356	. . . . . . . . . . . 12 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 − 1) = ((1 + (𝐴 · 𝑥)) − 1))
194	192, 193	oveq12d 7367	. . . . . . . . . . 11 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → ((log‘𝑦) / (𝑦 − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)))
195	191, 194	ifbieq2d 4503	. . . . . . . . . 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))))
196	140, 189, 190, 195	fmptco 7063	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∘ (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))) = (𝑥 ∈ 𝑆 ↦ if((1 + (𝐴 · 𝑥)) = 1, 1, ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)))))
197	58	eqeq2i 2742	. . . . . . . . . . . 12 ⊢ ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (1 + (𝐴 · 𝑥)) = 1)
198	137, 84, 123	addcand 11319	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (𝐴 · 𝑥) = 0))
199	197, 198	bitr3id 285	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥)) = 1 ↔ (𝐴 · 𝑥) = 0))
200	97	oveq2d 7365	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))
201	199, 200	ifbieq2d 4503	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 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))))
202	201	mpteq2dva 5185	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ if((1 + (𝐴 · 𝑥)) = 1, 1, ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)))) = (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
203	196, 202	eqtrd 2764	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∘ (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))) = (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
204		eqid 2729	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
205		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
206	205	cnfldtopon 24668	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
207	206	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
208		1cnd 11110	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ)
209	207, 207, 208	cnmptc 23547	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
210		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
211	207, 207, 210	cnmptc 23547	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
212	207	cnmptid 23546	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝑥) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
213	205	mpomulcn 24756	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
214	213	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
215		oveq12 7358	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝑥) → (𝑢 · 𝑣) = (𝐴 · 𝑥))
216	207, 211, 212, 207, 207, 214, 215	cnmpt12 23552	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
217	205	addcn 24752	. . . . . . . . . . . . . 14 ⊢ + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
218	217	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
219	207, 209, 216, 218	cnmpt12f 23551	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (1 + (𝐴 · 𝑥))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
220	204, 207, 44, 219	cnmpt1res 23561	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn (TopOpen‘ℂ_fld)))
221	140	fmpttd 7049	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))):𝑆⟶(1(ball‘(abs ∘ − ))1))
222	221	frnd 6660	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ran (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ⊆ (1(ball‘(abs ∘ − ))1))
223		difss 4087	. . . . . . . . . . . . . 14 ⊢ (ℂ ∖ {0}) ⊆ ℂ
224	92, 223	sstri 3945	. . . . . . . . . . . . 13 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ ℂ
225	224	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (1(ball‘(abs ∘ − ))1) ⊆ ℂ)
226		cnrest2 23171	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ ran (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ⊆ (1(ball‘(abs ∘ − ))1) ∧ (1(ball‘(abs ∘ − ))1) ⊆ ℂ) → ((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn (TopOpen‘ℂ_fld)) ↔ (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)))))
227	206, 222, 225, 226	mp3an2i 1468	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn (TopOpen‘ℂ_fld)) ↔ (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)))))
228	220, 227	mpbid 232	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1))))
229		blcntr 24299	. . . . . . . . . . . . 13 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ⁺) → 0 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
230	30, 31, 40, 229	mp3an2i 1468	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → 0 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
231	230, 29	eleqtrrdi 2839	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → 0 ∈ 𝑆)
232		resttopon 23046	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
233	206, 44, 232	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
234		toponuni 22799	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
235	233, 234	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
236	231, 235	eleqtrd 2830	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → 0 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
237		eqid 2729	. . . . . . . . . . 11 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)
238	237	cncnpi 23163	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1))) ∧ 0 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)) → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)))‘0))
239	228, 236, 238	syl2anc 584	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)))‘0))
240		cnelprrecn 11102	. . . . . . . . . . 11 ⊢ ℂ ∈ {ℝ, ℂ}
241		logf1o 26471	. . . . . . . . . . . . . 14 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
242		f1of 6764	. . . . . . . . . . . . . 14 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
243	241, 242	ax-mp 5	. . . . . . . . . . . . 13 ⊢ log:(ℂ ∖ {0})⟶ran log
244		logrncn 26469	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
245	244	ssriv 3939	. . . . . . . . . . . . 13 ⊢ ran log ⊆ ℂ
246		fss 6668	. . . . . . . . . . . . 13 ⊢ ((log:(ℂ ∖ {0})⟶ran log ∧ ran log ⊆ ℂ) → log:(ℂ ∖ {0})⟶ℂ)
247	243, 245, 246	mp2an 692	. . . . . . . . . . . 12 ⊢ log:(ℂ ∖ {0})⟶ℂ
248		fssres 6690	. . . . . . . . . . . 12 ⊢ ((log:(ℂ ∖ {0})⟶ℂ ∧ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})) → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ)
249	247, 92, 248	mp2an 692	. . . . . . . . . . 11 ⊢ (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ
250		blcntr 24299	. . . . . . . . . . . . . 14 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ⁺) → 1 ∈ (1(ball‘(abs ∘ − ))1))
251	30, 83, 134, 250	mp3an 1463	. . . . . . . . . . . . 13 ⊢ 1 ∈ (1(ball‘(abs ∘ − ))1)
252		ovex 7382	. . . . . . . . . . . . . 14 ⊢ (1 / 𝑦) ∈ V
253	88	dvlog2 26560	. . . . . . . . . . . . . 14 ⊢ (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
254	252, 253	dmmpti 6626	. . . . . . . . . . . . 13 ⊢ dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (1(ball‘(abs ∘ − ))1)
255	251, 254	eleqtrri 2827	. . . . . . . . . . . 12 ⊢ 1 ∈ dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))
256		eqid 2729	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)) = ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1))
257	253	fveq1i 6823	. . . . . . . . . . . . . . . . 17 ⊢ ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1) = ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1)
258		oveq2 7357	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 1 → (1 / 𝑦) = (1 / 1))
259		1div1e1 11815	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 / 1) = 1
260	258, 259	eqtrdi 2780	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 1 → (1 / 𝑦) = 1)
261		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦)) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
262		1ex 11111	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ V
263	260, 261, 262	fvmpt 6930	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ (1(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1) = 1)
264	251, 263	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1) = 1
265	257, 264	eqtr2i 2753	. . . . . . . . . . . . . . . 16 ⊢ 1 = ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1)
266	265	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → 1 = ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1))
267		fvres 6841	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) = (log‘𝑦))
268		fvres 6841	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = (log‘1))
269	251, 268	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = (log‘1))
270		log1 26492	. . . . . . . . . . . . . . . . . . 19 ⊢ (log‘1) = 0
271	269, 270	eqtrdi 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = 0)
272	267, 271	oveq12d 7367	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) = ((log‘𝑦) − 0))
273	92	sseli 3931	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → 𝑦 ∈ (ℂ ∖ {0}))
274		eldifsn 4737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
275	273, 274	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
276		logcl 26475	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (log‘𝑦) ∈ ℂ)
277	275, 276	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑦) ∈ ℂ)
278	277	subid1d 11464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log‘𝑦) − 0) = (log‘𝑦))
279	272, 278	eqtr2d 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑦) = (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)))
280	279	oveq1d 7364	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log‘𝑦) / (𝑦 − 1)) = ((((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) / (𝑦 − 1)))
281	266, 280	ifeq12d 4498	. . . . . . . . . . . . . 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))))
282	281	mpteq2ia 5187	. . . . . . . . . . . . 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))))
283	256, 205, 282	dvcnp 25818	. . . . . . . . . . . 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‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂ_fld))‘1))
284	255, 283	mpan2 691	. . . . . . . . . . 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‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂ_fld))‘1))
285	240, 249, 224, 284	mp3an 1463	. . . . . . . . . 10 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂ_fld))‘1)
286		oveq2 7357	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0 → (𝐴 · 𝑥) = (𝐴 · 0))
287	286	oveq2d 7365	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → (1 + (𝐴 · 𝑥)) = (1 + (𝐴 · 0)))
288		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))
289		ovex 7382	. . . . . . . . . . . . . 14 ⊢ (1 + (𝐴 · 0)) ∈ V
290	287, 288, 289	fvmpt 6930	. . . . . . . . . . . . 13 ⊢ (0 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = (1 + (𝐴 · 0)))
291	231, 290	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = (1 + (𝐴 · 0)))
292		mul01 11295	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
293	292	oveq2d 7365	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = (1 + 0))
294	293, 58	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = 1)
295	291, 294	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = 1)
296	295	fveq2d 6826	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂ_fld))‘((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0)) = ((((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂ_fld))‘1))
297	285, 296	eleqtrrid 2835	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂ_fld))‘((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0)))
298		cnpco 23152	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)))‘0) ∧ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)) CnP (TopOpen‘ℂ_fld))‘((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0))) → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∘ (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘0))
299	239, 297, 298	syl2anc 584	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ if(𝑦 = 1, 1, ((log‘𝑦) / (𝑦 − 1)))) ∘ (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘0))
300	203, 299	eqeltrrd 2829	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘0))
301	207, 207, 210	cnmptc 23547	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
302	207	cnmptid 23546	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝑦) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
303		oveq12 7358	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝑦) → (𝑢 · 𝑣) = (𝐴 · 𝑦))
304	207, 301, 302, 207, 207, 214, 303	cnmpt12 23552	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
305		efcn 26351	. . . . . . . . . . 11 ⊢ exp ∈ (ℂ–cn→ℂ)
306	205	cncfcn1 24802	. . . . . . . . . . 11 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
307	305, 306	eleqtri 2826	. . . . . . . . . 10 ⊢ exp ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
308	307	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → exp ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
309	207, 304, 308	cnmpt11f 23549	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
310	176	fmpttd 7049	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))):𝑆⟶ℂ)
311	310, 231	ffvelcdmd 7019	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0) ∈ ℂ)
312		unicntop 24671	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
313	312	cncnpi 23163	. . . . . . . 8 ⊢ (((𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)) ∧ ((𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0) ∈ ℂ) → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ (((TopOpen‘ℂ_fld) CnP (TopOpen‘ℂ_fld))‘((𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0)))
314	309, 311, 313	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ (((TopOpen‘ℂ_fld) CnP (TopOpen‘ℂ_fld))‘((𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0)))
315		cnpco 23152	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘0) ∧ (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ (((TopOpen‘ℂ_fld) CnP (TopOpen‘ℂ_fld))‘((𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0))) → ((𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∘ (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘0))
316	300, 314, 315	syl2anc 584	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∘ (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘0))
317	188, 316	eqeltrd 2828	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) ↾ 𝑆) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘0))
318	205	cnfldtop 24669	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ Top
319	318	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (TopOpen‘ℂ_fld) ∈ Top)
320	205	cnfldtopn 24667	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
321	320	blopn 24386	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ^*) → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂ_fld))
322	30, 31, 41, 321	mp3an2i 1468	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂ_fld))
323	29, 322	eqeltrid 2832	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → 𝑆 ∈ (TopOpen‘ℂ_fld))
324		isopn3i 22967	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝑆 ∈ (TopOpen‘ℂ_fld)) → ((int‘(TopOpen‘ℂ_fld))‘𝑆) = 𝑆)
325	318, 323, 324	sylancr 587	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((int‘(TopOpen‘ℂ_fld))‘𝑆) = 𝑆)
326	231, 325	eleqtrrd 2831	. . . . . 6 ⊢ (𝐴 ∈ ℂ → 0 ∈ ((int‘(TopOpen‘ℂ_fld))‘𝑆))
327		efcl 15989	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
328	327	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ 𝑥 = 0) → (exp‘𝐴) ∈ ℂ)
329	83, 14, 85	sylancr 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
330	329, 48	cxpcld 26615	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) ∈ ℂ)
331	328, 330	ifclda 4512	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) ∈ ℂ)
332	331	fmpttd 7049	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))):ℂ⟶ℂ)
333	312, 312	cnprest 23174	. . . . . 6 ⊢ ((((TopOpen‘ℂ_fld) ∈ Top ∧ 𝑆 ⊆ ℂ) ∧ (0 ∈ ((int‘(TopOpen‘ℂ_fld))‘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‘ℂ_fld) CnP (TopOpen‘ℂ_fld))‘0) ↔ ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) ↾ 𝑆) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘0)))
334	319, 44, 326, 332, 333	syl22anc 838	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) ∈ (((TopOpen‘ℂ_fld) CnP (TopOpen‘ℂ_fld))‘0) ↔ ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) ↾ 𝑆) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘0)))
335	317, 334	mpbird 257	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) ∈ (((TopOpen‘ℂ_fld) CnP (TopOpen‘ℂ_fld))‘0))
336	312	cnpresti 23173	. . . 4 ⊢ (((0[,)+∞) ⊆ ℂ ∧ 0 ∈ (0[,)+∞) ∧ (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) ∈ (((TopOpen‘ℂ_fld) CnP (TopOpen‘ℂ_fld))‘0)) → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) ↾ (0[,)+∞)) ∈ ((((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) CnP (TopOpen‘ℂ_fld))‘0))
337	3, 26, 335, 336	mp3an2i 1468	. . 3 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))) ↾ (0[,)+∞)) ∈ ((((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) CnP (TopOpen‘ℂ_fld))‘0))
338	24, 337	eqeltrd 2828	. 2 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 / (1 / 𝑥)))↑_𝑐(1 / 𝑥)))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) CnP (TopOpen‘ℂ_fld))‘0))
339		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
340		rpcn 12904	. . . . . . 7 ⊢ (𝑘 ∈ ℝ⁺ → 𝑘 ∈ ℂ)
341	340	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → 𝑘 ∈ ℂ)
342		rpne0 12910	. . . . . . 7 ⊢ (𝑘 ∈ ℝ⁺ → 𝑘 ≠ 0)
343	342	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → 𝑘 ≠ 0)
344	339, 341, 343	divcld 11900	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → (𝐴 / 𝑘) ∈ ℂ)
345		addcl 11091	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (𝐴 / 𝑘) ∈ ℂ) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
346	83, 344, 345	sylancr 587	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
347	346, 341	cxpcld 26615	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → ((1 + (𝐴 / 𝑘))↑_𝑐𝑘) ∈ ℂ)
348		oveq2 7357	. . . . 5 ⊢ (𝑘 = (1 / 𝑥) → (𝐴 / 𝑘) = (𝐴 / (1 / 𝑥)))
349	348	oveq2d 7365	. . . 4 ⊢ (𝑘 = (1 / 𝑥) → (1 + (𝐴 / 𝑘)) = (1 + (𝐴 / (1 / 𝑥))))
350		id 22	. . . 4 ⊢ (𝑘 = (1 / 𝑥) → 𝑘 = (1 / 𝑥))
351	349, 350	oveq12d 7367	. . 3 ⊢ (𝑘 = (1 / 𝑥) → ((1 + (𝐴 / 𝑘))↑_𝑐𝑘) = ((1 + (𝐴 / (1 / 𝑥)))↑_𝑐(1 / 𝑥)))
352		eqid 2729	. . 3 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) = ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞))
353	327, 347, 351, 205, 352	rlimcnp3 26875	. 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‘ℂ_fld) ↾_t (0[,)+∞)) CnP (TopOpen‘ℂ_fld))‘0)))
354	338, 353	mpbird 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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3900 ⊆ wss 3903 ifcif 4476 {csn 4577 {cpr 4579 ∪ cuni 4858 class class class wbr 5092 ↦ cmpt 5173 dom cdm 5619 ran crn 5620 ↾ cres 5621 ∘ ccom 5623 ⟶wf 6478 –1-1-onto→wf1o 6481 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 +∞cpnf 11146 -∞cmnf 11147 ℝ^*cxr 11148 < clt 11149 ≤ cle 11150 − cmin 11347 / cdiv 11777 ℝ⁺crp 12893 (,]cioc 13249 [,)cico 13250 abscabs 15141 ⇝_𝑟 crli 15392 expce 15968 ↾_t crest 17324 TopOpenctopn 17325 ∞Metcxmet 21246 ballcbl 21248 ℂ_fldccnfld 21261 Topctop 22778 TopOnctopon 22795 intcnt 22902 Cn ccn 23109 CnP ccnp 23110 ×_t ctx 23445 –cn→ccncf 24767 D cdv 25762 logclog 26461 ↑_𝑐ccxp 26462

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-tan 15978 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-log 26463 df-cxp 26464

This theorem is referenced by: dfef2 26879

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