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Theorem efopn 25801

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 23934	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
3		toponss 22064	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ ℂ)
4	2, 3	mpan 687	. . . . . 6 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ℂ)
5	4	sselda 3921	. . . . 5 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
6		cnxmet 23924	. . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
7		pirp 25606	. . . . . . 7 ⊢ π ∈ ℝ⁺
8	1	cnfldtopn 23933	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
9	8	mopni3 23638	. . . . . . 7 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) ∧ π ∈ ℝ⁺) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
10	7, 9	mpan2 688	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
11	6, 10	mp3an1 1447	. . . . 5 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
12		imass2 6004	. . . . . . . 8 ⊢ ((𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆 → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆))
13		imassrn 5974	. . . . . . . . . . . . . 14 ⊢ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ran exp
14		eff 15779	. . . . . . . . . . . . . . 15 ⊢ exp:ℂ⟶ℂ
15		frn 6600	. . . . . . . . . . . . . . 15 ⊢ (exp:ℂ⟶ℂ → ran exp ⊆ ℂ)
16	14, 15	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ran exp ⊆ ℂ
17	13, 16	sstri 3930	. . . . . . . . . . . . 13 ⊢ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ℂ
18		sseqin2 4150	. . . . . . . . . . . . 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 12727	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
21		blssm 23559	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
22	6, 21	mp3an1 1447	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
25	24	sselda 3921	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑦 ∈ ℂ)
26		simp-4l 780	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ)
27	25, 26	subcld 11320	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 − 𝑥) ∈ ℂ)
28	27	subid1d 11309	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥) − 0) = (𝑦 − 𝑥))
29	28	fveq2d 6771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (abs‘((𝑦 − 𝑥) − 0)) = (abs‘(𝑦 − 𝑥)))
30		0cn 10955	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℂ
31		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (abs ∘ − ) = (abs ∘ − )
32	31	cnmetdval 23922	. . . . . . . . . . . . . . . . . . . . . 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 23922	. . . . . . . . . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥)(abs ∘ − )0) = (𝑦(abs ∘ − )𝑥))
37		simpr 485	. . . . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → 𝑟 ∈ ℝ⁺)
40	39	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ⁺)
41	40	rpxrd 12761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ^*)
42		elbl3 23533	. . . . . . . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . . . . . . 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 5096	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥)(abs ∘ − )0) < 𝑟)
46		0cnd 10956	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 0 ∈ ℂ)
47		elbl3 23533	. . . . . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . . . . 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 15797	. . . . . . . . . . . . . . . . . . 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 6776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝑦 − 𝑥) → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥))))
53	52	rspcev 3560	. . . . . . . . . . . . . . . . . 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 7275	. . . . . . . . . . . . . . . . . . 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 3224	. . . . . . . . . . . . . . . . 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 3211	. . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑧 ∈ ℂ)
62		simp-4l 780	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ)
63		efcl 15780	. . . . . . . . . . . . . . . . . . . 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 12761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → 𝑟 ∈ ℝ^*)
66		blssm 23559	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
67	6, 30, 65, 66	mp3an12i 1464	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
68	67	sselda 3921	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑤 ∈ ℂ)
69		efcl 15780	. . . . . . . . . . . . . . . . . . . 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 15794	. . . . . . . . . . . . . . . . . . . 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 11761	. . . . . . . . . . . . . . . . . 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 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 11322	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = 𝑤)
76	68	subid1d 11309	. . . . . . . . . . . . . . . . . . . . . . . 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 6771	. . . . . . . . . . . . . . . . . . . . . 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 10982	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑥 + 𝑤) ∈ ℂ)
80	31	cnmetdval 23922	. . . . . . . . . . . . . . . . . . . . . . 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 23922	. . . . . . . . . . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (𝑤(abs ∘ − )0))
85		simpr 485	. . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ⁺)
88	87	rpxrd 12761	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ^*)
89		0cnd 10956	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 0 ∈ ℂ)
90		elbl3 23533	. . . . . . . . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . . . . . . . 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 5096	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟)
94		elbl3 23533	. . . . . . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . . . . . 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 15791	. . . . . . . . . . . . . . . . . . . 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 6776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 + 𝑤) → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤))))
100	99	rspcev 3560	. . . . . . . . . . . . . . . . . . 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 2750	. . . . . . . . . . . . . . . . . . 19 ⊢ (((exp‘𝑥) · (exp‘𝑤)) = 𝑧 → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘𝑦) = 𝑧))
103	102	rexbidv 3224	. . . . . . . . . . . . . . . . . 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 3211	. . . . . . . . . . . . . . 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 6593	. . . . . . . . . . . . . . . 16 ⊢ (exp:ℂ⟶ℂ → exp Fn ℂ)
109	14, 108	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ exp Fn ℂ
110		fvelimab 6834	. . . . . . . . . . . . . . 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 6834	. . . . . . . . . . . . . . 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 4152	. . . . . . . . . . . 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 2792	. . . . . . . . . . 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 6135	. . . . . . . . . . 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 2796	. . . . . . . . . 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 723	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp‘𝑥) ∈ ℂ)
121	71	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp‘𝑥) ≠ 0)
122	117	divccncf 24057	. . . . . . . . . . . . 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 24062	. . . . . . . . . . . 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 25800	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 𝑟 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
127	126	adantll 711	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
128		cnima 22404	. . . . . . . . . . 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 2839	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
131		blcntr 23554	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
132	6, 131	mp3an1 1447	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
133		ffun 6596	. . . . . . . . . . . . 13 ⊢ (exp:ℂ⟶ℂ → Fun exp)
134	14, 133	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun exp
135	14	fdmi 6605	. . . . . . . . . . . . 13 ⊢ dom exp = ℂ
136	23, 135	sseqtrrdi 3972	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ dom exp)
137		funfvima2 7100	. . . . . . . . . . . 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 481	. . . . . . . . 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 3946	. . . . . . . . . . . 12 ⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 ⊆ (exp “ 𝑆) ↔ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆)))
143	141, 142	anbi12d 631	. . . . . . . . . . 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 3560	. . . . . . . . . 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 457	. . . . . . . . 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 454	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → ((𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
149	148	rexlimdva 3211	. . . . 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 3113	. . 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 630	. . . . . 6 ⊢ (𝑧 = (exp‘𝑥) → ((𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
154	153	rexbidv 3224	. . . . 5 ⊢ (𝑧 = (exp‘𝑥) → (∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
155	154	ralima 7107	. . . 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 256	. 2 ⊢ (𝑆 ∈ 𝐽 → ∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))
158	1	cnfldtop 23935	. . 3 ⊢ 𝐽 ∈ Top
159		eltop2 22113	. . 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 ∩ cin 3886 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 ^◡ccnv 5584 dom cdm 5585 ran crn 5586 “ cima 5588 ∘ ccom 5589 Fun wfun 6421 Fn wfn 6422 ⟶wf 6423 ‘cfv 6427 (class class class)co 7268 ℂcc 10857 0cc0 10859 + caddc 10862 · cmul 10864 ℝ^*cxr 10996 < clt 10997 − cmin 11193 / cdiv 11620 ℝ⁺crp 12718 abscabs 14933 expce 15759 πcpi 15764 TopOpenctopn 17120 ∞Metcxmet 20570 ballcbl 20572 ℂ_fldccnfld 20585 Topctop 22030 TopOnctopon 22047 Cn ccn 22363 –cn→ccncf 24027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-inf2 9387 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 ax-addf 10938 ax-mulf 10939

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7704 df-1st 7821 df-2nd 7822 df-supp 7966 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-2o 8286 df-er 8486 df-map 8605 df-pm 8606 df-ixp 8674 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-z 12308 df-dec 12426 df-uz 12571 df-q 12677 df-rp 12719 df-xneg 12836 df-xadd 12837 df-xmul 12838 df-ioo 13071 df-ioc 13072 df-ico 13073 df-icc 13074 df-fz 13228 df-fzo 13371 df-fl 13500 df-mod 13578 df-seq 13710 df-exp 13771 df-fac 13976 df-bc 14005 df-hash 14033 df-shft 14766 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-limsup 15168 df-clim 15185 df-rlim 15186 df-sum 15386 df-ef 15765 df-sin 15767 df-cos 15768 df-tan 15769 df-pi 15770 df-struct 16836 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-starv 16965 df-sca 16966 df-vsca 16967 df-ip 16968 df-tset 16969 df-ple 16970 df-ds 16972 df-unif 16973 df-hom 16974 df-cco 16975 df-rest 17121 df-topn 17122 df-0g 17140 df-gsum 17141 df-topgen 17142 df-pt 17143 df-prds 17146 df-xrs 17201 df-qtop 17206 df-imas 17207 df-xps 17209 df-mre 17283 df-mrc 17284 df-acs 17286 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-submnd 18419 df-mulg 18689 df-cntz 18911 df-cmn 19376 df-psmet 20577 df-xmet 20578 df-met 20579 df-bl 20580 df-mopn 20581 df-fbas 20582 df-fg 20583 df-cnfld 20586 df-top 22031 df-topon 22048 df-topsp 22070 df-bases 22084 df-cld 22158 df-ntr 22159 df-cls 22160 df-nei 22237 df-lp 22275 df-perf 22276 df-cn 22366 df-cnp 22367 df-haus 22454 df-cmp 22526 df-tx 22701 df-hmeo 22894 df-fil 22985 df-fm 23077 df-flim 23078 df-flf 23079 df-xms 23461 df-ms 23462 df-tms 23463 df-cncf 24029 df-limc 25018 df-dv 25019 df-log 25700

This theorem is referenced by: (None)

W3C validator