efrlimOLD - Metamath Proof Explorer

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Theorem efrlimOLD 26947

Description: Obsolete version of efrlim 26946 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 13400	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
2		ax-resscn 11086	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
3	1, 2	sstri 3932	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
4	3	sseli 3918	. . . . . 6 ⊢ (𝑥 ∈ (0[,)+∞) → 𝑥 ∈ ℂ)
5		simpll 767	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝐴 ∈ ℂ)
6		1cnd 11130	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 1 ∈ ℂ)
7		simplr 769	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
8		ax-1ne0 11098	. . . . . . . . . . . 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 2940	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → 𝑥 ≠ 0)
12	5, 6, 7, 9, 11	divdiv2d 11954	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = ((𝐴 · 𝑥) / 1))
13		mulcl 11113	. . . . . . . . . . . 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 11914	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((𝐴 · 𝑥) / 1) = (𝐴 · 𝑥))
16	12, 15	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (𝐴 / (1 / 𝑥)) = (𝐴 · 𝑥))
17	16	oveq2d 7376	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 / (1 / 𝑥))) = (1 + (𝐴 · 𝑥)))
18	17	oveq1d 7375	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 / (1 / 𝑥)))↑_𝑐(1 / 𝑥)) = ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))
19	18	ifeq2da 4500	. . . . . 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 594	. . . . 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 5179	. . . 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 5996	. . . . 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 2790	. . 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 13402	. . . . 5 ⊢ 0 ∈ (0[,)+∞)
26	25	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℂ → 0 ∈ (0[,)+∞))
27		eqeq2 2749	. . . . . . . . 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 2749	. . . . . . . . 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 24747	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
31		0cnd 11128	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ)
32		abscl 15231	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
33		peano2re 11310	. . . . . . . . . . . . . . . . . . 19 ⊢ ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ)
35		0red 11138	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℝ)
36		absge0 15240	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
37	32	ltp1d 12077	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) < ((abs‘𝐴) + 1))
38	35, 32, 34, 36, 37	lelttrd 11295	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → 0 < ((abs‘𝐴) + 1))
39	34, 38	elrpd 12974	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 1) ∈ ℝ⁺)
40	39	rpreccld 12987	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ⁺)
41	40	rpxrd 12978	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (1 / ((abs‘𝐴) + 1)) ∈ ℝ^*)
42		blssm 24393	. . . . . . . . . . . . . . 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 1469	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ⊆ ℂ)
44	29, 43	eqsstrid 3961	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → 𝑆 ⊆ ℂ)
45	44	sselda 3922	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
46		mul0or 11781	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 · 𝑥) = 0 ↔ (𝐴 = 0 ∨ 𝑥 = 0)))
47	45, 46	syldan 592	. . . . . . . . . . 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 11915	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
50	45, 49	syldanl 603	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
51	50	adantlr 716	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 / 𝑥) ∈ ℂ)
52	51	1cxpd 26684	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1↑_𝑐(1 / 𝑥)) = 1)
53		simplr 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝐴 = 0)
54	53	oveq1d 7375	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = (0 · 𝑥))
55	45	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ ℂ)
56	55	mul02d 11335	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (0 · 𝑥) = 0)
57	54, 56	eqtrd 2772	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (𝐴 · 𝑥) = 0)
58	57	oveq2d 7376	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = (1 + 0))
59		1p0e1 12291	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 0) = 1
60	58, 59	eqtrdi 2788	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) = 1)
61	60	oveq1d 7375	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) = (1↑_𝑐(1 / 𝑥)))
62	53	fveq2d 6838	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = (exp‘0))
63		ef0 16047	. . . . . . . . . . . . . . . 16 ⊢ (exp‘0) = 1
64	62, 63	eqtrdi 2788	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → (exp‘𝐴) = 1)
65	52, 61, 64	3eqtr4d 2782	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) = (exp‘𝐴))
66	65	ifeq2da 4500	. . . . . . . . . . . . 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 4508	. . . . . . . . . . . . 13 ⊢ if(𝑥 = 0, (exp‘𝐴), (exp‘𝐴)) = (exp‘𝐴)
68	66, 67	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ 𝐴 = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘𝐴))
69		iftrue 4473	. . . . . . . . . . . . 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 960	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘𝐴))
72		mulrid 11133	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
73	72	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (𝐴 · 1) = 𝐴)
74	73	fveq2d 6838	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 = 0 ∨ 𝑥 = 0)) → (exp‘(𝐴 · 1)) = (exp‘𝐴))
75	71, 74	eqtr4d 2775	. . . . . . . . . 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 592	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 · 𝑥) = 0) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) = (exp‘(𝐴 · 1)))
77		mulne0b 11782	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
78	45, 77	syldan 592	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((𝐴 ≠ 0 ∧ 𝑥 ≠ 0) ↔ (𝐴 · 𝑥) ≠ 0))
79		df-ne 2934	. . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝑥 ≠ 0)
82	81	neneqd 2938	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ¬ 𝑥 = 0)
83	82	iffalsed 4478	. . . . . . . . . . . . 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 11087	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
85	45, 13	syldan 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝐴 · 𝑥) ∈ ℂ)
86		addcl 11111	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
87	84, 85, 86	sylancr 588	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
90	89	dvlog2lem 26629	. . . . . . . . . . . . . . . . . 18 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
91		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
92	91	logdmss 26619	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
93	90, 92	sstri 3932	. . . . . . . . . . . . . . . . 17 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
94		eqid 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (abs ∘ − ) = (abs ∘ − )
95	94	cnmetdval 24745	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1 + (𝐴 · 𝑥)) ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
96	87, 84, 95	sylancl 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘((1 + (𝐴 · 𝑥)) − 1)))
97		pncan2 11391	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℂ ∧ (𝐴 · 𝑥) ∈ ℂ) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
98	84, 85, 97	sylancr 588	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥)) − 1) = (𝐴 · 𝑥))
99	98	fveq2d 6838	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘((1 + (𝐴 · 𝑥)) − 1)) = (abs‘(𝐴 · 𝑥)))
100	96, 99	eqtrd 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) = (abs‘(𝐴 · 𝑥)))
101	85	abscld 15392	. . . . . . . . . . . . . . . . . . . 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 15392	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘𝑥) ∈ ℝ)
104	102, 103	remulcld 11166	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (((abs‘𝐴) + 1) · (abs‘𝑥)) ∈ ℝ)
105		1red 11136	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 1 ∈ ℝ)
106		absmul 15247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(𝐴 · 𝑥)) = ((abs‘𝐴) · (abs‘𝑥)))
107	45, 106	syldan 592	. . . . . . . . . . . . . . . . . . . . 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 15400	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 0 ≤ (abs‘𝑥))
111	108	lep1d 12078	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
112	108, 109, 103, 110, 111	lemul1ad 12086	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((abs‘𝐴) · (abs‘𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
113	107, 112	eqbrtrd 5108	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝐴 · 𝑥)) ≤ (((abs‘𝐴) + 1) · (abs‘𝑥)))
114		0cn 11127	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ ℂ
115	94	cnmetdval 24745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
116	45, 114, 115	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥(abs ∘ − )0) = (abs‘(𝑥 − 0)))
117	45	subid1d 11485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥 − 0) = 𝑥)
118	117	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝑥 − 0)) = (abs‘𝑥))
119	116, 118	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (𝑥(abs ∘ − )0) = (abs‘𝑥))
120		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
121	120, 29	eleqtrdi 2847	. . . . . . . . . . . . . . . . . . . . . . 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 11128	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 0 ∈ ℂ)
125		elbl3 24367	. . . . . . . . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . . . . . . . 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 12021	. . . . . . . . . . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . . . . . . . . . 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 11295	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (abs‘(𝐴 · 𝑥)) < 1)
134	100, 133	eqbrtrd 5108	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((1 + (𝐴 · 𝑥))(abs ∘ − )1) < 1)
135		1rp 12937	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℝ⁺
136		rpxr 12943	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
137	135, 136	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 1 ∈ ℝ^*)
138		1cnd 11130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → 1 ∈ ℂ)
139		elbl3 24367	. . . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → (1 + (𝐴 · 𝑥)) ∈ (ℂ ∖ {0}))
143		eldifsni 4734	. . . . . . . . . . . . . . . 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 11915	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ ℂ)
148	88, 145, 147	cxpefd 26689	. . . . . . . . . . . . 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 26547	. . . . . . . . . . . . . . . . 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 11157	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((1 / 𝑥) · (log‘(1 + (𝐴 · 𝑥)))) = ((log‘(1 + (𝐴 · 𝑥))) · (1 / 𝑥)))
152		simpll 767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ∈ ℂ)
153		simprl 771	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → 𝐴 ≠ 0)
154	152, 153	dividd 11920	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (𝐴 / 𝐴) = 1)
155	154	oveq1d 7375	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (1 / 𝑥))
156	152, 152, 146, 153, 81	divdiv1d 11953	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → ((𝐴 / 𝐴) / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
157	155, 156	eqtr3d 2774	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ (𝐴 ≠ 0 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) = (𝐴 / (𝐴 · 𝑥)))
158	157	oveq2d 7376	. . . . . . . . . . . . . . 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 11958	. . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 4506	. . . . . . . 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 5179	. . . . . . 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 5999	. . . . . . 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 11130	. . . . . . . . 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 2940	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → (𝐴 · 𝑥) ≠ 0)
176	172, 173, 175	divcld 11922	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) ∧ ¬ (𝐴 · 𝑥) = 0) → ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)) ∈ ℂ)
177	171, 176	ifclda 4503	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))) ∈ ℂ)
178		eqidd 2738	. . . . . . . 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 2738	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))))
180		oveq2 7368	. . . . . . . . . 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 6838	. . . . . . . . 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 7368	. . . . . . . . . . 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 6838	. . . . . . . . . 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 7368	. . . . . . . . . . 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 6838	. . . . . . . . . 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 4481	. . . . . . . . 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 2788	. . . . . . . 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 7076	. . . . . . 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 2782	. . . . . 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 2738	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))))
191		eqidd 2738	. . . . . . . . . 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 2741	. . . . . . . . . . 11 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 = 1 ↔ (1 + (𝐴 · 𝑥)) = 1))
193		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → (log‘𝑦) = (log‘(1 + (𝐴 · 𝑥))))
194		oveq1 7367	. . . . . . . . . . . 12 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → (𝑦 − 1) = ((1 + (𝐴 · 𝑥)) − 1))
195	193, 194	oveq12d 7378	. . . . . . . . . . 11 ⊢ (𝑦 = (1 + (𝐴 · 𝑥)) → ((log‘𝑦) / (𝑦 − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)))
196	192, 195	ifbieq2d 4494	. . . . . . . . . 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 7076	. . . . . . . . 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 2750	. . . . . . . . . . . 12 ⊢ ((1 + (𝐴 · 𝑥)) = (1 + 0) ↔ (1 + (𝐴 · 𝑥)) = 1)
199	138, 85, 124	addcand 11340	. . . . . . . . . . . 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 7376	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((log‘(1 + (𝐴 · 𝑥))) / ((1 + (𝐴 · 𝑥)) − 1)) = ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))
202	200, 201	ifbieq2d 4494	. . . . . . . . . 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 5179	. . . . . . . . 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 2772	. . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
206		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
207	206	cnfldtopon 24757	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
208	207	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
209		1cnd 11130	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ)
210	208, 208, 209	cnmptc 23637	. . . . . . . . . . . . 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 23637	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
213	208	cnmptid 23636	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ 𝑥) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
214	206	mulcn 24843	. . . . . . . . . . . . . . 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 23641	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
217	206	addcn 24841	. . . . . . . . . . . . . 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 23641	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (1 + (𝐴 · 𝑥))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
220	205, 208, 44, 219	cnmpt1res 23651	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn (TopOpen‘ℂ_fld)))
221	141	fmpttd 7061	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))):𝑆⟶(1(ball‘(abs ∘ − ))1))
222	221	frnd 6670	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ran (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) ⊆ (1(ball‘(abs ∘ − ))1))
223		difss 4077	. . . . . . . . . . . . . 14 ⊢ (ℂ ∖ {0}) ⊆ ℂ
224	93, 223	sstri 3932	. . . . . . . . . . . . 13 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ ℂ
225	224	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (1(ball‘(abs ∘ − ))1) ⊆ ℂ)
226		cnrest2 23261	. . . . . . . . . . . 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 1469	. . . . . . . . . . 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 24388	. . . . . . . . . . . . 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 1469	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → 0 ∈ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))))
231	230, 29	eleqtrrdi 2848	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → 0 ∈ 𝑆)
232		resttopon 23136	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
233	207, 44, 232	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
234		toponuni 22889	. . . . . . . . . . . 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 2839	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → 0 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
237		eqid 2737	. . . . . . . . . . 11 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)
238	237	cncnpi 23253	. . . . . . . . . 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 585	. . . . . . . . 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 11122	. . . . . . . . . . 11 ⊢ ℂ ∈ {ℝ, ℂ}
241		logf1o 26541	. . . . . . . . . . . . . 14 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
242		f1of 6774	. . . . . . . . . . . . . 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 26539	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
245	244	ssriv 3926	. . . . . . . . . . . . 13 ⊢ ran log ⊆ ℂ
246		fss 6678	. . . . . . . . . . . . 13 ⊢ ((log:(ℂ ∖ {0})⟶ran log ∧ ran log ⊆ ℂ) → log:(ℂ ∖ {0})⟶ℂ)
247	243, 245, 246	mp2an 693	. . . . . . . . . . . 12 ⊢ log:(ℂ ∖ {0})⟶ℂ
248		fssres 6700	. . . . . . . . . . . 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 693	. . . . . . . . . . 11 ⊢ (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ℂ
250		blcntr 24388	. . . . . . . . . . . . . 14 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ⁺) → 1 ∈ (1(ball‘(abs ∘ − ))1))
251	30, 84, 135, 250	mp3an 1464	. . . . . . . . . . . . 13 ⊢ 1 ∈ (1(ball‘(abs ∘ − ))1)
252		ovex 7393	. . . . . . . . . . . . . 14 ⊢ (1 / 𝑦) ∈ V
253	89	dvlog2 26630	. . . . . . . . . . . . . 14 ⊢ (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
254	252, 253	dmmpti 6636	. . . . . . . . . . . . 13 ⊢ dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (1(ball‘(abs ∘ − ))1)
255	251, 254	eleqtrri 2836	. . . . . . . . . . . 12 ⊢ 1 ∈ dom (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))
256		eqid 2737	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1)) = ((TopOpen‘ℂ_fld) ↾_t (1(ball‘(abs ∘ − ))1))
257	253	fveq1i 6835	. . . . . . . . . . . . . . . . 17 ⊢ ((ℂ D (log ↾ (1(ball‘(abs ∘ − ))1)))‘1) = ((𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))‘1)
258		oveq2 7368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 1 → (1 / 𝑦) = (1 / 1))
259		1div1e1 11836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 / 1) = 1
260	258, 259	eqtrdi 2788	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 1 → (1 / 𝑦) = 1)
261		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦)) = (𝑦 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑦))
262		1ex 11131	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ V
263	260, 261, 262	fvmpt 6941	. . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . 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 6853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) = (log‘𝑦))
268		fvres 6853	. . . . . . . . . . . . . . . . . . . 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 26562	. . . . . . . . . . . . . . . . . . 19 ⊢ (log‘1) = 0
271	269, 270	eqtrdi 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘1) = 0)
272	267, 271	oveq12d 7378	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)) = ((log‘𝑦) − 0))
273	93	sseli 3918	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → 𝑦 ∈ (ℂ ∖ {0}))
274		eldifsn 4730	. . . . . . . . . . . . . . . . . . . 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 26545	. . . . . . . . . . . . . . . . . . 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 11485	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → ((log‘𝑦) − 0) = (log‘𝑦))
279	272, 278	eqtr2d 2773	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑦) = (((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑦) − ((log ↾ (1(ball‘(abs ∘ − ))1))‘1)))
280	279	oveq1d 7375	. . . . . . . . . . . . . . 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 4489	. . . . . . . . . . . . . 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 5181	. . . . . . . . . . . . 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 25896	. . . . . . . . . . . 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 692	. . . . . . . . . . 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 1464	. . . . . . . . . 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 7368	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0 → (𝐴 · 𝑥) = (𝐴 · 0))
287	286	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → (1 + (𝐴 · 𝑥)) = (1 + (𝐴 · 0)))
288		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥))) = (𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))
289		ovex 7393	. . . . . . . . . . . . . 14 ⊢ (1 + (𝐴 · 0)) ∈ V
290	287, 288, 289	fvmpt 6941	. . . . . . . . . . . . 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 11316	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
293	292	oveq2d 7376	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = (1 + 0))
294	293, 59	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (1 + (𝐴 · 0)) = 1)
295	291, 294	eqtrd 2772	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ 𝑆 ↦ (1 + (𝐴 · 𝑥)))‘0) = 1)
296	295	fveq2d 6838	. . . . . . . . . 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 2844	. . . . . . . . 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 23242	. . . . . . . . 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 585	. . . . . . . 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 2838	. . . . . . 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 23637	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝐴) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
302	208	cnmptid 23636	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝑦) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
303	208, 301, 302, 215	cnmpt12f 23641	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
304		efcn 26421	. . . . . . . . . . 11 ⊢ exp ∈ (ℂ–cn→ℂ)
305	206	cncfcn1 24888	. . . . . . . . . . 11 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
306	304, 305	eleqtri 2835	. . . . . . . . . 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 23639	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
309	177	fmpttd 7061	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥)))):𝑆⟶ℂ)
310	309, 231	ffvelcdmd 7031	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ 𝑆 ↦ if((𝐴 · 𝑥) = 0, 1, ((log‘(1 + (𝐴 · 𝑥))) / (𝐴 · 𝑥))))‘0) ∈ ℂ)
311		unicntop 24760	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
312	311	cncnpi 23253	. . . . . . . 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 585	. . . . . . 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 23242	. . . . . . 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 585	. . . . . 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 2837	. . . . 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 24758	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ Top
318	317	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (TopOpen‘ℂ_fld) ∈ Top)
319	206	cnfldtopn 24756	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
320	319	blopn 24475	. . . . . . . . . 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 1469	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) ∈ (TopOpen‘ℂ_fld))
322	29, 321	eqeltrid 2841	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → 𝑆 ∈ (TopOpen‘ℂ_fld))
323		isopn3i 23057	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝑆 ∈ (TopOpen‘ℂ_fld)) → ((int‘(TopOpen‘ℂ_fld))‘𝑆) = 𝑆)
324	317, 322, 323	sylancr 588	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((int‘(TopOpen‘ℂ_fld))‘𝑆) = 𝑆)
325	231, 324	eleqtrrd 2840	. . . . . 6 ⊢ (𝐴 ∈ ℂ → 0 ∈ ((int‘(TopOpen‘ℂ_fld))‘𝑆))
326		efcl 16038	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
327	326	ad2antrr 727	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ 𝑥 = 0) → (exp‘𝐴) ∈ ℂ)
328	84, 14, 86	sylancr 588	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → (1 + (𝐴 · 𝑥)) ∈ ℂ)
329	328, 49	cxpcld 26685	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑥 = 0) → ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)) ∈ ℂ)
330	327, 329	ifclda 4503	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥))) ∈ ℂ)
331	330	fmpttd 7061	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ if(𝑥 = 0, (exp‘𝐴), ((1 + (𝐴 · 𝑥))↑_𝑐(1 / 𝑥)))):ℂ⟶ℂ)
332	311, 311	cnprest 23264	. . . . . 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 839	. . . . 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 23263	. . . 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 1469	. . 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 2837	. 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 12944	. . . . . . 7 ⊢ (𝑘 ∈ ℝ⁺ → 𝑘 ∈ ℂ)
340	339	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → 𝑘 ∈ ℂ)
341		rpne0 12950	. . . . . . 7 ⊢ (𝑘 ∈ ℝ⁺ → 𝑘 ≠ 0)
342	341	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → 𝑘 ≠ 0)
343	338, 340, 342	divcld 11922	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → (𝐴 / 𝑘) ∈ ℂ)
344		addcl 11111	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (𝐴 / 𝑘) ∈ ℂ) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
345	84, 343, 344	sylancr 588	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → (1 + (𝐴 / 𝑘)) ∈ ℂ)
346	345, 340	cxpcld 26685	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℝ⁺) → ((1 + (𝐴 / 𝑘))↑_𝑐𝑘) ∈ ℂ)
347		oveq2 7368	. . . . 5 ⊢ (𝑘 = (1 / 𝑥) → (𝐴 / 𝑘) = (𝐴 / (1 / 𝑥)))
348	347	oveq2d 7376	. . . 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 7378	. . 3 ⊢ (𝑘 = (1 / 𝑥) → ((1 + (𝐴 / 𝑘))↑_𝑐𝑘) = ((1 + (𝐴 / (1 / 𝑥)))↑_𝑐(1 / 𝑥)))
351		eqid 2737	. . 3 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) = ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞))
352	326, 346, 350, 206, 351	rlimcnp3 26944	. 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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ⊆ wss 3890 ifcif 4467 {csn 4568 {cpr 4570 ∪ cuni 4851 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5624 ran crn 5625 ↾ cres 5626 ∘ ccom 5628 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 +∞cpnf 11167 -∞cmnf 11168 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℝ⁺crp 12933 (,]cioc 13290 [,)cico 13291 abscabs 15187 ⇝_𝑟 crli 15438 expce 16017 ↾_t crest 17374 TopOpenctopn 17375 ∞Metcxmet 21329 ballcbl 21331 ℂ_fldccnfld 21344 Topctop 22868 TopOnctopon 22885 intcnt 22992 Cn ccn 23199 CnP ccnp 23200 ×_t ctx 23535 –cn→ccncf 24853 D cdv 25840 logclog 26531 ↑_𝑐ccxp 26532

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-tan 16027 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-cmp 23362 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-limc 25843 df-dv 25844 df-log 26533 df-cxp 26534

This theorem is referenced by: (None)

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