efrlimOLD - Metamath Proof Explorer

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Theorem efrlimOLD 26880

Description: Obsolete version of efrlim 26879 as of 19-Apr-2025. (Contributed by Mario Carneiro, 1-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
efrlimOLD.1	⊢ 𝑆 = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))

**Assertion**
Ref	Expression
efrlimOLD	⊢ (𝐴 ∈ ℂ → (𝑘 ∈ ℝ⁺ ↦ ((1 + (𝐴 / 𝑘))↑_𝑐𝑘)) ⇝_𝑟 (exp‘𝐴))

Distinct variable group: 𝐴,𝑘 Allowed substitution hint: 𝑆(𝑘)

Proof of Theorem efrlimOLD Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rge0ssre 13417	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
2		ax-resscn 11125	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
3	1, 2	sstri 3956	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
4	3	sseli 3942	. . . . . 6 ⊢ (𝑥 ∈ (0[,)+∞) → 𝑥 ∈ ℂ)
5		simpll 766	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝐴 ∈ ℂ)
6		1cnd 11169	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 1 ∈ ℂ)
7		simplr 768	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
8		ax-1ne0 11137	. . . . . . . . . . . 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 11990	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = ((𝐴 · 𝑥) / 1))
13		mulcl 11152	. . . . . . . . . . . 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 11950	. . . . . . . . . 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 7403	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 / (1 / 𝑥))) = (1 + (𝐴 · 𝑥)))
18	17	oveq1d 7402	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 / (1 / 𝑥)))↑_𝑐(1 / 𝑥)) = ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))
19	18	ifeq2da 4521	. . . . . 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 5200	. . . 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 6008	. . . . 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 13419	. . . . 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		efrlimOLD.1	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))
30		cnxmet 24660	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
31		0cnd 11167	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ)
32		abscl 15244	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
33		peano2re 11347	. . . . . . . . . . . . . . . . . . 19 ⊢ ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ)
35		0red 11177	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℝ)
36		absge0 15253	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
37	32	ltp1d 12113	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) < ((abs‘𝐴) + 1))
38	35, 32, 34, 36, 37	lelttrd 11332	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → 0 < ((abs‘𝐴) + 1))
39	34, 38	elrpd 12992	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ⁺)
40	39	rpreccld 13005	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ⁺)
41	40	rpxrd 12996	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ^*)
42		blssm 24306	. . . . . . . . . . . . . . 15 ⊢ (((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	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ⊆ ℂ)
44	29, 43	eqsstrid 3985	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → 𝑆 ⊆ ℂ)
45	44	sselda 3946	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
46		mul0or 11818	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
47	45, 46	syldan 591	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
48	47	biimpa 476	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 · 𝑥) = 0) → (𝐴 = 0 ∨ 𝑥 = 0))
49	7, 11	reccld 11951	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
50	45, 49	syldanl 602	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
51	50	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
52	51	1cxpd 26616	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1↑_𝑐(1 / 𝑥)) = 1)
53		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝐴 = 0)
54	53	oveq1d 7402	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = (0 · 𝑥))
55	45	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
56	55	mul02d 11372	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (0 · 𝑥) = 0)
57	54, 56	eqtrd 2764	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = 0)
58	57	oveq2d 7403	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = (1 + 0))
59		1p0e1 12305	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 0) = 1
60	58, 59	eqtrdi 2780	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = 1)
61	60	oveq1d 7402	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) = (1↑_𝑐(1 / 𝑥)))
62	53	fveq2d 6862	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = (exp‘0))
63		ef0 16057	. . . . . . . . . . . . . . . 16 ⊢ (exp‘0) = 1
64	62, 63	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = 1)
65	52, 61, 64	3eqtr4d 2774	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) = (exp‘𝐴))
66	65	ifeq2da 4521	. . . . . . . . . . . . 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)))
67		ifid 4529	. . . . . . . . . . . . 13 ⊢ if(𝑥 = 0, (exp‘𝐴), (exp‘𝐴)) = (exp‘𝐴)
68	66, 67	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘𝐴))
69		iftrue 4494	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘𝐴))
70	69	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝑥 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘𝐴))
71	68, 70	jaodan 959	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘𝐴))
72		mulrid 11172	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
73	72	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (𝐴 · 1) = 𝐴)
74	73	fveq2d 6862	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (exp‘(𝐴 · 1)) = (exp‘𝐴))
75	71, 74	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)))
76	48, 75	syldan 591	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 · 𝑥) = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)))
77		mulne0b 11819	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
78	45, 77	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
79		df-ne 2926	. . . . . . . . . . . 12 ⊢ ((𝐴 · 𝑥) ≠ 0 ↔ ¬ (𝐴 · 𝑥) = 0)
80	78, 79	bitrdi 287	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ ¬ (𝐴 · 𝑥) = 0))
81		simprr 772	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ≠ 0)
82	81	neneqd 2930	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ¬ 𝑥 = 0)
83	82	iffalsed 4499	. . . . . . . . . . . . 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)))
84		ax-1cn 11126	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
85	45, 13	syldan 591	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝐴 · 𝑥) ∈ ℂ)
86		addcl 11150	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
87	84, 85, 86	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
88	87	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
89		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
90	89	dvlog2lem 26561	. . . . . . . . . . . . . . . . . 18 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
91		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
92	91	logdmss 26551	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
93	90, 92	sstri 3956	. . . . . . . . . . . . . . . . 17 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
94		eqid 2729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (abs ∘ − ) = (abs ∘ − )
95	94	cnmetdval 24658	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1 + (𝐴 · 𝑥)) ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
96	87, 84, 95	sylancl 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
97		pncan2 11428	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
98	84, 85, 97	sylancr 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
99	98	fveq2d 6862	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘((1 + (𝐴 · 𝑥)) − 1)) = (abs‘(𝐴 · 𝑥)))
100	96, 99	eqtrd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘(𝐴 · 𝑥)))
101	85	abscld 15405	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝐴 · 𝑥)) ∈ ℝ)
102	34	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((abs‘𝐴) + 1) ∈ ℝ)
103	45	abscld 15405	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘𝑥) ∈ ℝ)
104	102, 103	remulcld 11204	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) ∈ ℝ)
105		1red 11175	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 1 ∈ ℝ)
106		absmul 15260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
107	45, 106	syldan 591	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
108	32	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘𝐴) ∈ ℝ)
109	108, 33	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((abs‘𝐴) + 1) ∈ ℝ)
110	45	absge0d 15413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 0 ≤ (abs‘𝑥))
111	108	lep1d 12114	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
112	108, 109, 103, 110, 111	lemul1ad 12122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((abs‘𝐴) · (abs‘𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
113	107, 112	eqbrtrd 5129	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝐴 · 𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
114		0cn 11166	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ ℂ
115	94	cnmetdval 24658	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
116	45, 114, 115	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
117	45	subid1d 11522	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥 − 0) = 𝑥)
118	117	fveq2d 6862	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝑥 − 0)) = (abs‘𝑥))
119	116, 118	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥(abs ∘ − )0) = (abs‘𝑥))
120		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
121	120, 29	eleqtrdi 2838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
122	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs ∘ − ) ∈ (∞Met‘ℂ))
123	41	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 / ((abs‘𝐴) + 1)) ∈ ℝ^*)
124		0cnd 11167	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 0 ∈ ℂ)
125		elbl3 24280	. . . . . . . . . . . . . . . . . . . . . . . 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))))
126	122, 123, 124, 45, 125	syl22anc 838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ↔ (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1))))
127	121, 126	mpbid 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥(abs ∘ − )0) < (1 / ((abs‘𝐴) + 1)))
128	119, 127	eqbrtrrd 5131	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘𝑥) < (1 / ((abs‘𝐴) + 1)))
129	38	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 0 < ((abs‘𝐴) + 1))
130		ltmuldiv2 12057	. . . . . . . . . . . . . . . . . . . . . 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))))
131	103, 105, 109, 129, 130	syl112anc 1376	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((((abs‘𝐴) + 1) · (abs‘𝑥)) < 1 ↔ (abs‘𝑥) < (1 / ((abs‘𝐴) + 1))))
132	128, 131	mpbird 257	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) < 1)
133	101, 104, 105, 113, 132	lelttrd 11332	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝐴 · 𝑥)) < 1)
134	100, 133	eqbrtrd 5129	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1)
135		1rp 12955	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℝ⁺
136		rpxr 12961	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
137	135, 136	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 1 ∈ ℝ^*)
138		1cnd 11169	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 1 ∈ ℂ)
139		elbl3 24280	. . . . . . . . . . . . . . . . . . 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))
140	122, 137, 138, 87, 139	syl22anc 838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1))
141	134, 140	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 + (𝐴 · 𝑥)) ∈ (1(ball‘(abs ∘ − ))1))
142	93, 141	sselid 3944	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}))
143		eldifsni 4754	. . . . . . . . . . . . . . . 16 ⊢ ((1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}) → (1 + (𝐴 · 𝑥)) ≠ 0)
144	142, 143	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 + (𝐴 · 𝑥)) ≠ 0)
145	144	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 + (𝐴 · 𝑥)) ≠ 0)
146	45	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ∈ ℂ)
147	146, 81	reccld 11951	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ ℂ)
148	88, 145, 147	cxpefd 26621	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) = (exp‘((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥))))))
149	87, 144	logcld 26479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
150	149	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
151	147, 150	mulcomd 11195	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)))
152		simpll 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ∈ ℂ)
153		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ≠ 0)
154	152, 153	dividd 11956	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 / 𝐴) = 1)
155	154	oveq1d 7402	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (1 / 𝑥))
156	152, 152, 146, 153, 81	divdiv1d 11989	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
157	155, 156	eqtr3d 2766	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
158	157	oveq2d 7403	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)) = ((log‘(1 + (𝐴 · 𝑥))) · (𝐴 / (𝐴 · 𝑥))))
159	85	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 · 𝑥) ∈ ℂ)
160	78	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 · 𝑥) ≠ 0)
161	150, 152, 159, 160	div12d 11994	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((log‘(1 + (𝐴 · 𝑥))) · (𝐴 / (𝐴 · 𝑥))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
162	151, 158, 161	3eqtrd 2768	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = (𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))
163	162	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (exp‘((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥))))) = (exp‘(𝐴 · ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))))
164	83, 148, 163	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)))))
165	164	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))))))
166	80, 165	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))))))
167	166	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)))))
168	27, 28, 76, 167	ifbothda 4527	. . . . . . . 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))))))
169	168	mpteq2dva 5200	. . . . . . 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)))))))
170	44	resmptd 6011	. . . . . . 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)))))
171		1cnd 11169	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 · 𝑥) = 0) → 1 ∈ ℂ)
172	149	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (log‘(1 + (𝐴 · 𝑥))) ∈ ℂ)
173	85	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ∈ ℂ)
174		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ¬ (𝐴 · 𝑥) = 0)
175	174	neqned 2932	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ≠ 0)
176	172, 173, 175	divcld 11958	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)) ∈ ℂ)
177	171, 176	ifclda 4524	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) ∈ ℂ)
178		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)))))
179		eqidd 2730	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))))
180		oveq2 7395	. . . . . . . . . 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)))))
181	180	fveq2d 6862	. . . . . . . . 9 ⊢ (𝑦 = if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))))
182		oveq2 7395	. . . . . . . . . . 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))
183	182	fveq2d 6862	. . . . . . . . . 10 ⊢ (if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) = 1 → (exp‘(𝐴 · if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))) = (exp‘(𝐴 · 1)))
184		oveq2 7395	. . . . . . . . . . 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))))
185	184	fveq2d 6862	. . . . . . . . . 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)))))
186	183, 185	ifsb 4502	. . . . . . . . 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)))))
187	181, 186	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))))))
188	177, 178, 179, 187	fmptco 7101	. . . . . . 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)))))))
189	169, 170, 188	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))))))
190		eqidd 2730	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))))
191		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)))))
192		eqeq1 2733	. . . . . . . . . . 11 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 = 1 ↔ (1 + (𝐴 · 𝑥)) = 1))
193		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → (log‘𝑦) = (log‘(1 + (𝐴 · 𝑥))))
194		oveq1 7394	. . . . . . . . . . . 12 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 − 1) = ((1 + (𝐴 · 𝑥)) − 1))
195	193, 194	oveq12d 7405	. . . . . . . . . . 11 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → ((log‘𝑦) / (𝑦 − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)))
196	192, 195	ifbieq2d 4515	. . . . . . . . . 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))))
197	141, 190, 191, 196	fmptco 7101	. . . . . . . . 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)))))
198	59	eqeq2i 2742	. . . . . . . . . . . 12 ⊢ ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (1 + (𝐴 · 𝑥)) = 1)
199	138, 85, 124	addcand 11377	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (𝐴 · 𝑥) = 0))
200	198, 199	bitr3id 285	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥)) = 1 ↔ (𝐴 · 𝑥) = 0))
201	98	oveq2d 7403	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))
202	200, 201	ifbieq2d 4515	. . . . . . . . . 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))))
203	202	mpteq2dva 5200	. . . . . . . . 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)))))
204	197, 203	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)))))
205		eqid 2729	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
206		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
207	206	cnfldtopon 24670	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
208	207	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
209		1cnd 11169	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ)
210	208, 208, 209	cnmptc 23549	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
211		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
212	208, 208, 211	cnmptc 23549	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
213	208	cnmptid 23548	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝑥) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
214	206	mulcn 24756	. . . . . . . . . . . . . . 15 ⊢ · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
215	214	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
216	208, 212, 213, 215	cnmpt12f 23553	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
217	206	addcn 24754	. . . . . . . . . . . . . 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	208, 210, 216, 218	cnmpt12f 23553	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (1 + (𝐴 · 𝑥))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
220	205, 208, 44, 219	cnmpt1res 23563	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn (TopOpen‘ℂ_fld)))
221	141	fmpttd 7087	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))):𝑆⟶(1(ball‘(abs ∘ − ))1))
222	221	frnd 6696	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ran (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ⊆ (1(ball‘(abs ∘ − ))1))
223		difss 4099	. . . . . . . . . . . . . 14 ⊢ (ℂ ∖ {0}) ⊆ ℂ
224	93, 223	sstri 3956	. . . . . . . . . . . . 13 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ ℂ
225	224	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (1(ball‘(abs ∘ − ))1) ⊆ ℂ)
226		cnrest2 23173	. . . . . . . . . . . 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	207, 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 24301	. . . . . . . . . . . . 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 23048	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
233	207, 44, 232	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
234		toponuni 22801	. . . . . . . . . . . 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 23165	. . . . . . . . . 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 11161	. . . . . . . . . . 11 ⊢ ℂ ∈ {ℝ, ℂ}
241		logf1o 26473	. . . . . . . . . . . . . 14 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
242		f1of 6800	. . . . . . . . . . . . . 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 26471	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
245	244	ssriv 3950	. . . . . . . . . . . . 13 ⊢ ran log ⊆ ℂ
246		fss 6704	. . . . . . . . . . . . 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 6726	. . . . . . . . . . . 12 ⊢ ((log:(ℂ ∖ {0})⟶ℂ ∧ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})) → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ)
249	247, 93, 248	mp2an 692	. . . . . . . . . . 11 ⊢ (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ
250		blcntr 24301	. . . . . . . . . . . . . 14 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ⁺) → 1 ∈ (1(ball‘(abs ∘ − ))1))
251	30, 84, 135, 250	mp3an 1463	. . . . . . . . . . . . 13 ⊢ 1 ∈ (1(ball‘(abs ∘ − ))1)
252		ovex 7420	. . . . . . . . . . . . . 14 ⊢ (1 / 𝑦) ∈ V
253	89	dvlog2 26562	. . . . . . . . . . . . . 14 ⊢ (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
254	252, 253	dmmpti 6662	. . . . . . . . . . . . 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 6859	. . . . . . . . . . . . . . . . 17 ⊢ ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1) = ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1)
258		oveq2 7395	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 1 → (1 / 𝑦) = (1 / 1))
259		1div1e1 11873	. . . . . . . . . . . . . . . . . . . 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 11170	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ V
263	260, 261, 262	fvmpt 6968	. . . . . . . . . . . . . . . . . 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 6877	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) = (log‘𝑦))
268		fvres 6877	. . . . . . . . . . . . . . . . . . . 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 26494	. . . . . . . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) = ((log‘𝑦) − 0))
273	93	sseli 3942	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → 𝑦 ∈ (ℂ ∖ {0}))
274		eldifsn 4750	. . . . . . . . . . . . . . . . . . . 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 26477	. . . . . . . . . . . . . . . . . . 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 11522	. . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . 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 4510	. . . . . . . . . . . . . 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 5202	. . . . . . . . . . . . 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, 206, 282	dvcnp 25820	. . . . . . . . . . . 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 7395	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0 → (𝐴 · 𝑥) = (𝐴 · 0))
287	286	oveq2d 7403	. . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . 14 ⊢ (1 + (𝐴 · 0)) ∈ V
290	287, 288, 289	fvmpt 6968	. . . . . . . . . . . . 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 11353	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
293	292	oveq2d 7403	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = (1 + 0))
294	293, 59	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 6862	. . . . . . . . . 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 23154	. . . . . . . . 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	204, 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	208, 208, 211	cnmptc 23549	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
302	208	cnmptid 23548	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝑦) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
303	208, 301, 302, 215	cnmpt12f 23553	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
304		efcn 26353	. . . . . . . . . . 11 ⊢ exp ∈ (ℂ–cn→ℂ)
305	206	cncfcn1 24804	. . . . . . . . . . 11 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
306	304, 305	eleqtri 2826	. . . . . . . . . 10 ⊢ exp ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
307	306	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → exp ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
308	208, 303, 307	cnmpt11f 23551	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
309	177	fmpttd 7087	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))):𝑆⟶ℂ)
310	309, 231	ffvelcdmd 7057	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0) ∈ ℂ)
311		unicntop 24673	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
312	311	cncnpi 23165	. . . . . . . 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)))
313	308, 310, 312	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)))
314		cnpco 23154	. . . . . . 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))
315	300, 313, 314	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))
316	189, 315	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))
317	206	cnfldtop 24671	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ Top
318	317	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (TopOpen‘ℂ_fld) ∈ Top)
319	206	cnfldtopn 24669	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
320	319	blopn 24388	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (1 / ((abs‘𝐴) + 1)) ∈ ℝ^*) → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂ_fld))
321	30, 31, 41, 320	mp3an2i 1468	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂ_fld))
322	29, 321	eqeltrid 2832	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → 𝑆 ∈ (TopOpen‘ℂ_fld))
323		isopn3i 22969	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝑆 ∈ (TopOpen‘ℂ_fld)) → ((int‘(TopOpen‘ℂ_fld))‘𝑆) = 𝑆)
324	317, 322, 323	sylancr 587	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((int‘(TopOpen‘ℂ_fld))‘𝑆) = 𝑆)
325	231, 324	eleqtrrd 2831	. . . . . 6 ⊢ (𝐴 ∈ ℂ → 0 ∈ ((int‘(TopOpen‘ℂ_fld))‘𝑆))
326		efcl 16048	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
327	326	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ 𝑥 = 0) → (exp‘𝐴) ∈ ℂ)
328	84, 14, 86	sylancr 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
329	328, 49	cxpcld 26617	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) ∈ ℂ)
330	327, 329	ifclda 4524	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) ∈ ℂ)
331	330	fmpttd 7087	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))):ℂ⟶ℂ)
332	311, 311	cnprest 23176	. . . . . 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)))
333	318, 44, 325, 331, 332	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)))
334	316, 333	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))
335	311	cnpresti 23175	. . . 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))
336	3, 26, 334, 335	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))
337	24, 336	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))
338		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
339		rpcn 12962	. . . . . . 7 ⊢ (𝑘 ∈ ℝ⁺ → 𝑘 ∈ ℂ)
340	339	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → 𝑘 ∈ ℂ)
341		rpne0 12968	. . . . . . 7 ⊢ (𝑘 ∈ ℝ⁺ → 𝑘 ≠ 0)
342	341	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → 𝑘 ≠ 0)
343	338, 340, 342	divcld 11958	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → (𝐴 / 𝑘) ∈ ℂ)
344		addcl 11150	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (𝐴 / 𝑘) ∈ ℂ) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
345	84, 343, 344	sylancr 587	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
346	345, 340	cxpcld 26617	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → ((1 + (𝐴 / 𝑘))↑_𝑐𝑘) ∈ ℂ)
347		oveq2 7395	. . . . 5 ⊢ (𝑘 = (1 / 𝑥) → (𝐴 / 𝑘) = (𝐴 / (1 / 𝑥)))
348	347	oveq2d 7403	. . . 4 ⊢ (𝑘 = (1 / 𝑥) → (1 + (𝐴 / 𝑘)) = (1 + (𝐴 / (1 / 𝑥))))
349		id 22	. . . 4 ⊢ (𝑘 = (1 / 𝑥) → 𝑘 = (1 / 𝑥))
350	348, 349	oveq12d 7405	. . 3 ⊢ (𝑘 = (1 / 𝑥) → ((1 + (𝐴 / 𝑘))↑_𝑐𝑘) = ((1 + (𝐴 / (1 / 𝑥)))↑_𝑐(1 / 𝑥)))
351		eqid 2729	. . 3 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) = ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞))
352	326, 346, 350, 206, 351	rlimcnp3 26877	. 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)))
353	337, 352	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 3911 ⊆ wss 3914 ifcif 4488 {csn 4589 {cpr 4591 ∪ cuni 4871 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 ran crn 5639 ↾ cres 5640 ∘ ccom 5642 ⟶wf 6507 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℝ⁺crp 12951 (,]cioc 13307 [,)cico 13308 abscabs 15200 ⇝_𝑟 crli 15451 expce 16027 ↾_t crest 17383 TopOpenctopn 17384 ∞Metcxmet 21249 ballcbl 21251 ℂ_fldccnfld 21264 Topctop 22780 TopOnctopon 22797 intcnt 22904 Cn ccn 23111 CnP ccnp 23112 ×_t ctx 23447 –cn→ccncf 24769 D cdv 25764 logclog 26463 ↑_𝑐ccxp 26464

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-tan 16037 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-cxp 26466

This theorem is referenced by: (None)

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