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Theorem efopn 25718

Description: The exponential map is an open map. (Contributed by Mario Carneiro, 23-Apr-2015.)

**Hypothesis**
Ref	Expression
efopn.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
efopn	⊢ (𝑆 ∈ 𝐽 → (exp “ 𝑆) ∈ 𝐽)

Proof of Theorem efopn Dummy variables 𝑤 𝑟 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efopn.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
2	1	cnfldtopon 23852	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
3		toponss 21984	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ ℂ)
4	2, 3	mpan 686	. . . . . 6 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ℂ)
5	4	sselda 3917	. . . . 5 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
6		cnxmet 23842	. . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
7		pirp 25523	. . . . . . 7 ⊢ π ∈ ℝ⁺
8	1	cnfldtopn 23851	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
9	8	mopni3 23556	. . . . . . 7 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) ∧ π ∈ ℝ⁺) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
10	7, 9	mpan2 687	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
11	6, 10	mp3an1 1446	. . . . 5 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
12		imass2 5999	. . . . . . . 8 ⊢ ((𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆 → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆))
13		imassrn 5969	. . . . . . . . . . . . . 14 ⊢ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ran exp
14		eff 15719	. . . . . . . . . . . . . . 15 ⊢ exp:ℂ⟶ℂ
15		frn 6591	. . . . . . . . . . . . . . 15 ⊢ (exp:ℂ⟶ℂ → ran exp ⊆ ℂ)
16	14, 15	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ran exp ⊆ ℂ
17	13, 16	sstri 3926	. . . . . . . . . . . . 13 ⊢ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ℂ
18		sseqin2 4146	. . . . . . . . . . . . 13 ⊢ ((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ℂ ↔ (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)))
19	17, 18	mpbi 229	. . . . . . . . . . . 12 ⊢ (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))
20		rpxr 12668	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
21		blssm 23479	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
22	6, 21	mp3an1 1446	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
23	20, 22	sylan2 592	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
24	23	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
25	24	sselda 3917	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑦 ∈ ℂ)
26		simp-4l 779	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ)
27	25, 26	subcld 11262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 − 𝑥) ∈ ℂ)
28	27	subid1d 11251	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥) − 0) = (𝑦 − 𝑥))
29	28	fveq2d 6760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (abs‘((𝑦 − 𝑥) − 0)) = (abs‘(𝑦 − 𝑥)))
30		0cn 10898	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℂ
31		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (abs ∘ − ) = (abs ∘ − )
32	31	cnmetdval 23840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 − 𝑥) ∈ ℂ ∧ 0 ∈ ℂ) → ((𝑦 − 𝑥)(abs ∘ − )0) = (abs‘((𝑦 − 𝑥) − 0)))
33	27, 30, 32	sylancl 585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥)(abs ∘ − )0) = (abs‘((𝑦 − 𝑥) − 0)))
34	31	cnmetdval 23840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦(abs ∘ − )𝑥) = (abs‘(𝑦 − 𝑥)))
35	25, 26, 34	syl2anc 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦(abs ∘ − )𝑥) = (abs‘(𝑦 − 𝑥)))
36	29, 33, 35	3eqtr4d 2788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥)(abs ∘ − )0) = (𝑦(abs ∘ − )𝑥))
37		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
38	6	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
39		simpllr 772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → 𝑟 ∈ ℝ⁺)
40	39	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ⁺)
41	40	rpxrd 12702	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ^*)
42		elbl3 23453	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ^*) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ (𝑦(abs ∘ − )𝑥) < 𝑟))
43	38, 41, 26, 25, 42	syl22anc 835	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ (𝑦(abs ∘ − )𝑥) < 𝑟))
44	37, 43	mpbid 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦(abs ∘ − )𝑥) < 𝑟)
45	36, 44	eqbrtrd 5092	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥)(abs ∘ − )0) < 𝑟)
46		0cnd 10899	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 0 ∈ ℂ)
47		elbl3 23453	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ (𝑦 − 𝑥) ∈ ℂ)) → ((𝑦 − 𝑥) ∈ (0(ball‘(abs ∘ − ))𝑟) ↔ ((𝑦 − 𝑥)(abs ∘ − )0) < 𝑟))
48	38, 41, 46, 27, 47	syl22anc 835	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥) ∈ (0(ball‘(abs ∘ − ))𝑟) ↔ ((𝑦 − 𝑥)(abs ∘ − )0) < 𝑟))
49	45, 48	mpbird 256	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 − 𝑥) ∈ (0(ball‘(abs ∘ − ))𝑟))
50		efsub 15737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥)))
51	25, 26, 50	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥)))
52		fveqeq2 6765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝑦 − 𝑥) → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥))))
53	52	rspcev 3552	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 − 𝑥) ∈ (0(ball‘(abs ∘ − ))𝑟) ∧ (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥))) → ∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)))
54	49, 51, 53	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)))
55		oveq1 7262	. . . . . . . . . . . . . . . . . . 19 ⊢ ((exp‘𝑦) = 𝑧 → ((exp‘𝑦) / (exp‘𝑥)) = (𝑧 / (exp‘𝑥)))
56	55	eqeq2d 2749	. . . . . . . . . . . . . . . . . 18 ⊢ ((exp‘𝑦) = 𝑧 → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘𝑤) = (𝑧 / (exp‘𝑥))))
57	56	rexbidv 3225	. . . . . . . . . . . . . . . . 17 ⊢ ((exp‘𝑦) = 𝑧 → (∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ ∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥))))
58	54, 57	syl5ibcom 244	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((exp‘𝑦) = 𝑧 → ∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥))))
59	58	rexlimdva 3212	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧 → ∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥))))
60		eqcom 2745	. . . . . . . . . . . . . . . . . 18 ⊢ ((exp‘𝑤) = (𝑧 / (exp‘𝑥)) ↔ (𝑧 / (exp‘𝑥)) = (exp‘𝑤))
61		simplr 765	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑧 ∈ ℂ)
62		simp-4l 779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ)
63		efcl 15720	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
64	62, 63	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (exp‘𝑥) ∈ ℂ)
65	39	rpxrd 12702	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → 𝑟 ∈ ℝ^*)
66		blssm 23479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
67	6, 30, 65, 66	mp3an12i 1463	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
68	67	sselda 3917	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑤 ∈ ℂ)
69		efcl 15720	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ ℂ → (exp‘𝑤) ∈ ℂ)
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (exp‘𝑤) ∈ ℂ)
71		efne0 15734	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ≠ 0)
72	62, 71	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (exp‘𝑥) ≠ 0)
73	61, 64, 70, 72	divmuld 11703	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑧 / (exp‘𝑥)) = (exp‘𝑤) ↔ ((exp‘𝑥) · (exp‘𝑤)) = 𝑧))
74	60, 73	syl5bb 282	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((exp‘𝑤) = (𝑧 / (exp‘𝑥)) ↔ ((exp‘𝑥) · (exp‘𝑤)) = 𝑧))
75	62, 68	pncan2d 11264	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = 𝑤)
76	68	subid1d 11251	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤 − 0) = 𝑤)
77	75, 76	eqtr4d 2781	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = (𝑤 − 0))
78	77	fveq2d 6760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (abs‘((𝑥 + 𝑤) − 𝑥)) = (abs‘(𝑤 − 0)))
79	62, 68	addcld 10925	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑥 + 𝑤) ∈ ℂ)
80	31	cnmetdval 23840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 + 𝑤) ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (abs‘((𝑥 + 𝑤) − 𝑥)))
81	79, 62, 80	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (abs‘((𝑥 + 𝑤) − 𝑥)))
82	31	cnmetdval 23840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑤(abs ∘ − )0) = (abs‘(𝑤 − 0)))
83	68, 30, 82	sylancl 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤(abs ∘ − )0) = (abs‘(𝑤 − 0)))
84	78, 81, 83	3eqtr4d 2788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (𝑤(abs ∘ − )0))
85		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟))
86	6	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
87	39	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ⁺)
88	87	rpxrd 12702	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ^*)
89		0cnd 10899	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 0 ∈ ℂ)
90		elbl3 23453	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟) ↔ (𝑤(abs ∘ − )0) < 𝑟))
91	86, 88, 89, 68, 90	syl22anc 835	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟) ↔ (𝑤(abs ∘ − )0) < 𝑟))
92	85, 91	mpbid 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤(abs ∘ − )0) < 𝑟)
93	84, 92	eqbrtrd 5092	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟)
94		elbl3 23453	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ^*) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝑤) ∈ ℂ)) → ((𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟))
95	86, 88, 62, 79, 94	syl22anc 835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟))
96	93, 95	mpbird 256	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
97		efadd 15731	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤)))
98	62, 68, 97	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤)))
99		fveqeq2 6765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 + 𝑤) → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤))))
100	99	rspcev 3552	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ∧ (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤))) → ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)))
101	96, 98, 100	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)))
102		eqeq2 2750	. . . . . . . . . . . . . . . . . . 19 ⊢ (((exp‘𝑥) · (exp‘𝑤)) = 𝑧 → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘𝑦) = 𝑧))
103	102	rexbidv 3225	. . . . . . . . . . . . . . . . . 18 ⊢ (((exp‘𝑥) · (exp‘𝑤)) = 𝑧 → (∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧))
104	101, 103	syl5ibcom 244	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (((exp‘𝑥) · (exp‘𝑤)) = 𝑧 → ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧))
105	74, 104	sylbid 239	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((exp‘𝑤) = (𝑧 / (exp‘𝑥)) → ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧))
106	105	rexlimdva 3212	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥)) → ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧))
107	59, 106	impbid 211	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧 ↔ ∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥))))
108		ffn 6584	. . . . . . . . . . . . . . . 16 ⊢ (exp:ℂ⟶ℂ → exp Fn ℂ)
109	14, 108	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ exp Fn ℂ
110		fvelimab 6823	. . . . . . . . . . . . . . 15 ⊢ ((exp Fn ℂ ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ) → (𝑧 ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ↔ ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧))
111	109, 24, 110	sylancr 586	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (𝑧 ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ↔ ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧))
112		fvelimab 6823	. . . . . . . . . . . . . . 15 ⊢ ((exp Fn ℂ ∧ (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ) → ((𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟)) ↔ ∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥))))
113	109, 67, 112	sylancr 586	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → ((𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟)) ↔ ∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥))))
114	107, 111, 113	3bitr4d 310	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (𝑧 ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ↔ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟))))
115	114	rabbi2dva 4148	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = {𝑧 ∈ ℂ ∣ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟))})
116	19, 115	eqtr3id 2793	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) = {𝑧 ∈ ℂ ∣ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟))})
117		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) = (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥)))
118	117	mptpreima 6130	. . . . . . . . . . 11 ⊢ (^◡(𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs ∘ − ))𝑟))) = {𝑧 ∈ ℂ ∣ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟))}
119	116, 118	eqtr4di 2797	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) = (^◡(𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs ∘ − ))𝑟))))
120	63	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp‘𝑥) ∈ ℂ)
121	71	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp‘𝑥) ≠ 0)
122	117	divccncf 23975	. . . . . . . . . . . . 13 ⊢ (((exp‘𝑥) ∈ ℂ ∧ (exp‘𝑥) ≠ 0) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (ℂ–cn→ℂ))
123	120, 121, 122	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (ℂ–cn→ℂ))
124	1	cncfcn1 23980	. . . . . . . . . . . 12 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
125	123, 124	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (𝐽 Cn 𝐽))
126	1	efopnlem2 25717	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 𝑟 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
127	126	adantll 710	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
128		cnima 22324	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (𝐽 Cn 𝐽) ∧ (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽) → (^◡(𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs ∘ − ))𝑟))) ∈ 𝐽)
129	125, 127, 128	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (^◡(𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs ∘ − ))𝑟))) ∈ 𝐽)
130	119, 129	eqeltrd 2839	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
131		blcntr 23474	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
132	6, 131	mp3an1 1446	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
133		ffun 6587	. . . . . . . . . . . . 13 ⊢ (exp:ℂ⟶ℂ → Fun exp)
134	14, 133	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun exp
135	14	fdmi 6596	. . . . . . . . . . . . 13 ⊢ dom exp = ℂ
136	23, 135	sseqtrrdi 3968	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ dom exp)
137		funfvima2 7089	. . . . . . . . . . . 12 ⊢ ((Fun exp ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ dom exp) → (𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟) → (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))))
138	134, 136, 137	sylancr 586	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → (𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟) → (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))))
139	132, 138	mpd 15	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)))
140	139	adantr 480	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)))
141		eleq2 2827	. . . . . . . . . . . 12 ⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((exp‘𝑥) ∈ 𝑦 ↔ (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))))
142		sseq1 3942	. . . . . . . . . . . 12 ⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 ⊆ (exp “ 𝑆) ↔ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆)))
143	141, 142	anbi12d 630	. . . . . . . . . . 11 ⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → (((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ((exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∧ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆))))
144	143	rspcev 3552	. . . . . . . . . 10 ⊢ (((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽 ∧ ((exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∧ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆))) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))
145	144	expr 456	. . . . . . . . 9 ⊢ (((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽 ∧ (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) → ((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
146	130, 140, 145	syl2anc 583	. . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → ((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
147	12, 146	syl5 34	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → ((𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆 → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
148	147	expimpd 453	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → ((𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
149	148	rexlimdva 3212	. . . . 5 ⊢ (𝑥 ∈ ℂ → (∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
150	5, 11, 149	sylc 65	. . . 4 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))
151	150	ralrimiva 3107	. . 3 ⊢ (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))
152		eleq1 2826	. . . . . . 7 ⊢ (𝑧 = (exp‘𝑥) → (𝑧 ∈ 𝑦 ↔ (exp‘𝑥) ∈ 𝑦))
153	152	anbi1d 629	. . . . . 6 ⊢ (𝑧 = (exp‘𝑥) → ((𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
154	153	rexbidv 3225	. . . . 5 ⊢ (𝑧 = (exp‘𝑥) → (∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
155	154	ralima 7096	. . . 4 ⊢ ((exp Fn ℂ ∧ 𝑆 ⊆ ℂ) → (∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ∀𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
156	109, 4, 155	sylancr 586	. . 3 ⊢ (𝑆 ∈ 𝐽 → (∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ∀𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
157	151, 156	mpbird 256	. 2 ⊢ (𝑆 ∈ 𝐽 → ∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))
158	1	cnfldtop 23853	. . 3 ⊢ 𝐽 ∈ Top
159		eltop2 22033	. . 3 ⊢ (𝐽 ∈ Top → ((exp “ 𝑆) ∈ 𝐽 ↔ ∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
160	158, 159	ax-mp 5	. 2 ⊢ ((exp “ 𝑆) ∈ 𝐽 ↔ ∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))
161	157, 160	sylibr 233	1 ⊢ (𝑆 ∈ 𝐽 → (exp “ 𝑆) ∈ 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 ∩ cin 3882 ⊆ wss 3883 class class class wbr 5070 ↦ cmpt 5153 ^◡ccnv 5579 dom cdm 5580 ran crn 5581 “ cima 5583 ∘ ccom 5584 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 0cc0 10802 + caddc 10805 · cmul 10807 ℝ^*cxr 10939 < clt 10940 − cmin 11135 / cdiv 11562 ℝ⁺crp 12659 abscabs 14873 expce 15699 πcpi 15704 TopOpenctopn 17049 ∞Metcxmet 20495 ballcbl 20497 ℂ_fldccnfld 20510 Topctop 21950 TopOnctopon 21967 Cn ccn 22283 –cn→ccncf 23945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-tan 15709 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617

This theorem is referenced by: (None)

W3C validator