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Theorem efopn 26574

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 24677	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
3		toponss 22821	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ ℂ)
4	2, 3	mpan 690	. . . . . 6 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ℂ)
5	4	sselda 3949	. . . . 5 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
6		cnxmet 24667	. . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
7		pirp 26377	. . . . . . 7 ⊢ π ∈ ℝ⁺
8	1	cnfldtopn 24676	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
9	8	mopni3 24389	. . . . . . 7 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) ∧ π ∈ ℝ⁺) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
10	7, 9	mpan2 691	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
11	6, 10	mp3an1 1450	. . . . 5 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
12		imass2 6076	. . . . . . . 8 ⊢ ((𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆 → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆))
13		imassrn 6045	. . . . . . . . . . . . . 14 ⊢ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ran exp
14		eff 16054	. . . . . . . . . . . . . . 15 ⊢ exp:ℂ⟶ℂ
15		frn 6698	. . . . . . . . . . . . . . 15 ⊢ (exp:ℂ⟶ℂ → ran exp ⊆ ℂ)
16	14, 15	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ran exp ⊆ ℂ
17	13, 16	sstri 3959	. . . . . . . . . . . . 13 ⊢ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ℂ
18		sseqin2 4189	. . . . . . . . . . . . 13 ⊢ ((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ℂ ↔ (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)))
19	17, 18	mpbi 230	. . . . . . . . . . . 12 ⊢ (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))
20		rpxr 12968	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
21		blssm 24313	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
22	6, 21	mp3an1 1450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
23	20, 22	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
24	23	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
25	24	sselda 3949	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑦 ∈ ℂ)
26		simp-4l 782	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ)
27	25, 26	subcld 11540	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 − 𝑥) ∈ ℂ)
28	27	subid1d 11529	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥) − 0) = (𝑦 − 𝑥))
29	28	fveq2d 6865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (abs‘((𝑦 − 𝑥) − 0)) = (abs‘(𝑦 − 𝑥)))
30		0cn 11173	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℂ
31		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (abs ∘ − ) = (abs ∘ − )
32	31	cnmetdval 24665	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 − 𝑥) ∈ ℂ ∧ 0 ∈ ℂ) → ((𝑦 − 𝑥)(abs ∘ − )0) = (abs‘((𝑦 − 𝑥) − 0)))
33	27, 30, 32	sylancl 586	. . . . . . . . . . . . . . . . . . . . 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 24665	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦(abs ∘ − )𝑥) = (abs‘(𝑦 − 𝑥)))
35	25, 26, 34	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . . . 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 13003	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ^*)
42		elbl3 24287	. . . . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦(abs ∘ − )𝑥) < 𝑟)
45	36, 44	eqbrtrd 5132	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥)(abs ∘ − )0) < 𝑟)
46		0cnd 11174	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 0 ∈ ℂ)
47		elbl3 24287	. . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 − 𝑥) ∈ (0(ball‘(abs ∘ − ))𝑟))
50		efsub 16075	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥)))
51	25, 26, 50	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥)))
52		fveqeq2 6870	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝑦 − 𝑥) → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥))))
53	52	rspcev 3591	. . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 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 7397	. . . . . . . . . . . . . . . . . . 19 ⊢ ((exp‘𝑦) = 𝑧 → ((exp‘𝑦) / (exp‘𝑥)) = (𝑧 / (exp‘𝑥)))
56	55	eqeq2d 2741	. . . . . . . . . . . . . . . . . 18 ⊢ ((exp‘𝑦) = 𝑧 → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘𝑤) = (𝑧 / (exp‘𝑥))))
57	56	rexbidv 3158	. . . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . . 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 3135	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . 18 ⊢ ((exp‘𝑤) = (𝑧 / (exp‘𝑥)) ↔ (𝑧 / (exp‘𝑥)) = (exp‘𝑤))
61		simplr 768	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑧 ∈ ℂ)
62		simp-4l 782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ)
63		efcl 16055	. . . . . . . . . . . . . . . . . . . 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 13003	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → 𝑟 ∈ ℝ^*)
66		blssm 24313	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
67	6, 30, 65, 66	mp3an12i 1467	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
68	67	sselda 3949	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑤 ∈ ℂ)
69		efcl 16055	. . . . . . . . . . . . . . . . . . . 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 16071	. . . . . . . . . . . . . . . . . . . 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 11987	. . . . . . . . . . . . . . . . . 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	bitrid 283	. . . . . . . . . . . . . . . . 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 11542	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = 𝑤)
76	68	subid1d 11529	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤 − 0) = 𝑤)
77	75, 76	eqtr4d 2768	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = (𝑤 − 0))
78	77	fveq2d 6865	. . . . . . . . . . . . . . . . . . . . . 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 11200	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑥 + 𝑤) ∈ ℂ)
80	31	cnmetdval 24665	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 + 𝑤) ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (abs‘((𝑥 + 𝑤) − 𝑥)))
81	79, 62, 80	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 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 24665	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑤(abs ∘ − )0) = (abs‘(𝑤 − 0)))
83	68, 30, 82	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . . . . . 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 13003	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ^*)
89		0cnd 11174	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 0 ∈ ℂ)
90		elbl3 24287	. . . . . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤(abs ∘ − )0) < 𝑟)
93	84, 92	eqbrtrd 5132	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟)
94		elbl3 24287	. . . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
97		efadd 16067	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤)))
98	62, 68, 97	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤)))
99		fveqeq2 6870	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 + 𝑤) → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤))))
100	99	rspcev 3591	. . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . 19 ⊢ (((exp‘𝑥) · (exp‘𝑤)) = 𝑧 → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘𝑦) = 𝑧))
103	102	rexbidv 3158	. . . . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . 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 3135	. . . . . . . . . . . . . . 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 212	. . . . . . . . . . . . . 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 6691	. . . . . . . . . . . . . . . 16 ⊢ (exp:ℂ⟶ℂ → exp Fn ℂ)
109	14, 108	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ exp Fn ℂ
110		fvelimab 6936	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 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 6936	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 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 311	. . . . . . . . . . . . 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 4192	. . . . . . . . . . . 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 2779	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) = {𝑧 ∈ ℂ ∣ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟))})
117		eqid 2730	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) = (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥)))
118	117	mptpreima 6214	. . . . . . . . . . 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 2783	. . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp‘𝑥) ∈ ℂ)
121	71	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp‘𝑥) ≠ 0)
122	117	divccncf 24806	. . . . . . . . . . . . 13 ⊢ (((exp‘𝑥) ∈ ℂ ∧ (exp‘𝑥) ≠ 0) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (ℂ–cn→ℂ))
123	120, 121, 122	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (ℂ–cn→ℂ))
124	1	cncfcn1 24811	. . . . . . . . . . . 12 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
125	123, 124	eleqtrdi 2839	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (𝐽 Cn 𝐽))
126	1	efopnlem2 26573	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 𝑟 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
127	126	adantll 714	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
128		cnima 23159	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (^◡(𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs ∘ − ))𝑟))) ∈ 𝐽)
130	119, 129	eqeltrd 2829	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
131		blcntr 24308	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
132	6, 131	mp3an1 1450	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
133		ffun 6694	. . . . . . . . . . . . 13 ⊢ (exp:ℂ⟶ℂ → Fun exp)
134	14, 133	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun exp
135	14	fdmi 6702	. . . . . . . . . . . . 13 ⊢ dom exp = ℂ
136	23, 135	sseqtrrdi 3991	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ dom exp)
137		funfvima2 7208	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 2818	. . . . . . . . . . . 12 ⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((exp‘𝑥) ∈ 𝑦 ↔ (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))))
142		sseq1 3975	. . . . . . . . . . . 12 ⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 ⊆ (exp “ 𝑆) ↔ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆)))
143	141, 142	anbi12d 632	. . . . . . . . . . 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 3591	. . . . . . . . . 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 584	. . . . . . . 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 3135	. . . . 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 3126	. . 3 ⊢ (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))
152		eleq1 2817	. . . . . . 7 ⊢ (𝑧 = (exp‘𝑥) → (𝑧 ∈ 𝑦 ↔ (exp‘𝑥) ∈ 𝑦))
153	152	anbi1d 631	. . . . . 6 ⊢ (𝑧 = (exp‘𝑥) → ((𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
154	153	rexbidv 3158	. . . . 5 ⊢ (𝑧 = (exp‘𝑥) → (∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
155	154	ralima 7214	. . . 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 587	. . 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 257	. 2 ⊢ (𝑆 ∈ 𝐽 → ∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))
158	1	cnfldtop 24678	. . 3 ⊢ 𝐽 ∈ Top
159		eltop2 22869	. . 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 234	1 ⊢ (𝑆 ∈ 𝐽 → (exp “ 𝑆) ∈ 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 {crab 3408 ∩ cin 3916 ⊆ wss 3917 class class class wbr 5110 ↦ cmpt 5191 ^◡ccnv 5640 dom cdm 5641 ran crn 5642 “ cima 5644 ∘ ccom 5645 Fun wfun 6508 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 0cc0 11075 + caddc 11078 · cmul 11080 ℝ^*cxr 11214 < clt 11215 − cmin 11412 / cdiv 11842 ℝ⁺crp 12958 abscabs 15207 expce 16034 πcpi 16039 TopOpenctopn 17391 ∞Metcxmet 21256 ballcbl 21258 ℂ_fldccnfld 21271 Topctop 22787 TopOnctopon 22804 Cn ccn 23118 –cn→ccncf 24776

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040 df-sin 16042 df-cos 16043 df-tan 16044 df-pi 16045 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-cmp 23281 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775 df-log 26472

This theorem is referenced by: (None)

W3C validator