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Theorem efopn 26647

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 24772	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
3		toponss 22917	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ ℂ)
4	2, 3	mpan 696	. . . . . 6 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ℂ)
5	4	sselda 3922	. . . . 5 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
6		cnxmet 24762	. . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
7		pirp 26450	. . . . . . 7 ⊢ π ∈ ℝ⁺
8	1	cnfldtopn 24771	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
9	8	mopni3 24484	. . . . . . 7 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) ∧ π ∈ ℝ⁺) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
10	7, 9	mpan2 697	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
11	6, 10	mp3an1 1456	. . . . 5 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ⁺ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
12		imass2 6061	. . . . . . . 8 ⊢ ((𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆 → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆))
13		imassrn 6030	. . . . . . . . . . . . . 14 ⊢ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ran exp
14		eff 16044	. . . . . . . . . . . . . . 15 ⊢ exp:ℂ⟶ℂ
15		frn 6669	. . . . . . . . . . . . . . 15 ⊢ (exp:ℂ⟶ℂ → ran exp ⊆ ℂ)
16	14, 15	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ran exp ⊆ ℂ
17	13, 16	sstri 3931	. . . . . . . . . . . . 13 ⊢ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ℂ
18		sseqin2 4159	. . . . . . . . . . . . 13 ⊢ ((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ℂ ↔ (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)))
19	17, 18	mpbi 231	. . . . . . . . . . . 12 ⊢ (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))
20		rpxr 12950	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
21		blssm 24408	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
22	6, 21	mp3an1 1456	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
23	20, 22	sylan2 599	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
24	23	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
25	24	sselda 3922	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑦 ∈ ℂ)
26		simp-4l 788	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ)
27	25, 26	subcld 11503	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 − 𝑥) ∈ ℂ)
28	27	subid1d 11492	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥) − 0) = (𝑦 − 𝑥))
29	28	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (abs‘((𝑦 − 𝑥) − 0)) = (abs‘(𝑦 − 𝑥)))
30		0cn 11134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℂ
31		eqid 2740	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (abs ∘ − ) = (abs ∘ − )
32	31	cnmetdval 24760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 − 𝑥) ∈ ℂ ∧ 0 ∈ ℂ) → ((𝑦 − 𝑥)(abs ∘ − )0) = (abs‘((𝑦 − 𝑥) − 0)))
33	27, 30, 32	sylancl 592	. . . . . . . . . . . . . . . . . . . . 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 24760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦(abs ∘ − )𝑥) = (abs‘(𝑦 − 𝑥)))
35	25, 26, 34	syl2anc 590	. . . . . . . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . . . . . . . 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 12985	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ^*)
42		elbl3 24382	. . . . . . . . . . . . . . . . . . . . . 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 844	. . . . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦(abs ∘ − )𝑥) < 𝑟)
45	36, 44	eqbrtrd 5101	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥)(abs ∘ − )0) < 𝑟)
46		0cnd 11135	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 0 ∈ ℂ)
47		elbl3 24382	. . . . . . . . . . . . . . . . . . . 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 844	. . . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 − 𝑥) ∈ (0(ball‘(abs ∘ − ))𝑟))
50		efsub 16065	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥)))
51	25, 26, 50	syl2anc 590	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥)))
52		fveqeq2 6843	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝑦 − 𝑥) → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥))))
53	52	rspcev 3567	. . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . 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 7370	. . . . . . . . . . . . . . . . . . 19 ⊢ ((exp‘𝑦) = 𝑧 → ((exp‘𝑦) / (exp‘𝑥)) = (𝑧 / (exp‘𝑥)))
56	55	eqeq2d 2751	. . . . . . . . . . . . . . . . . 18 ⊢ ((exp‘𝑦) = 𝑧 → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘𝑤) = (𝑧 / (exp‘𝑥))))
57	56	rexbidv 3164	. . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . 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 3141	. . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . . . . 18 ⊢ ((exp‘𝑤) = (𝑧 / (exp‘𝑥)) ↔ (𝑧 / (exp‘𝑥)) = (exp‘𝑤))
61		simplr 774	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑧 ∈ ℂ)
62		simp-4l 788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ)
63		efcl 16045	. . . . . . . . . . . . . . . . . . . 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 12985	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → 𝑟 ∈ ℝ^*)
66		blssm 24408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
67	6, 30, 65, 66	mp3an12i 1473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
68	67	sselda 3922	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑤 ∈ ℂ)
69		efcl 16045	. . . . . . . . . . . . . . . . . . . 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 16061	. . . . . . . . . . . . . . . . . . . 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 11951	. . . . . . . . . . . . . . . . . 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 284	. . . . . . . . . . . . . . . . 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 11505	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = 𝑤)
76	68	subid1d 11492	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤 − 0) = 𝑤)
77	75, 76	eqtr4d 2778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = (𝑤 − 0))
78	77	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . . 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 11162	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑥 + 𝑤) ∈ ℂ)
80	31	cnmetdval 24760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 + 𝑤) ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (abs‘((𝑥 + 𝑤) − 𝑥)))
81	79, 62, 80	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . 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 24760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑤(abs ∘ − )0) = (abs‘(𝑤 − 0)))
83	68, 30, 82	sylancl 592	. . . . . . . . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . . . . . . . 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 12985	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ^*)
89		0cnd 11135	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 0 ∈ ℂ)
90		elbl3 24382	. . . . . . . . . . . . . . . . . . . . . . 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 844	. . . . . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤(abs ∘ − )0) < 𝑟)
93	84, 92	eqbrtrd 5101	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟)
94		elbl3 24382	. . . . . . . . . . . . . . . . . . . . 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 844	. . . . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
97		efadd 16057	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤)))
98	62, 68, 97	syl2anc 590	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤)))
99		fveqeq2 6843	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 + 𝑤) → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤))))
100	99	rspcev 3567	. . . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . . . . . 19 ⊢ (((exp‘𝑥) · (exp‘𝑤)) = 𝑧 → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘𝑦) = 𝑧))
103	102	rexbidv 3164	. . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . . . 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 3141	. . . . . . . . . . . . . . 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 213	. . . . . . . . . . . . . 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 6662	. . . . . . . . . . . . . . . 16 ⊢ (exp:ℂ⟶ℂ → exp Fn ℂ)
109	14, 108	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ exp Fn ℂ
110		fvelimab 6906	. . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . 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 312	. . . . . . . . . . . . 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 4161	. . . . . . . . . . . 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 2789	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) = {𝑧 ∈ ℂ ∣ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟))})
117		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) = (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥)))
118	117	mptpreima 6196	. . . . . . . . . . 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 2793	. . . . . . . . . 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 732	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp‘𝑥) ∈ ℂ)
121	71	ad2antrr 732	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp‘𝑥) ≠ 0)
122	117	divccncf 24898	. . . . . . . . . . . . 13 ⊢ (((exp‘𝑥) ∈ ℂ ∧ (exp‘𝑥) ≠ 0) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (ℂ–cn→ℂ))
123	120, 121, 122	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (ℂ–cn→ℂ))
124	1	cncfcn1 24903	. . . . . . . . . . . 12 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
125	123, 124	eleqtrdi 2850	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (𝐽 Cn 𝐽))
126	1	efopnlem2 26646	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 𝑟 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
127	126	adantll 720	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
128		cnima 23255	. . . . . . . . . . 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 590	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (^◡(𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs ∘ − ))𝑟))) ∈ 𝐽)
130	119, 129	eqeltrd 2840	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
131		blcntr 24403	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
132	6, 131	mp3an1 1456	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
133		ffun 6665	. . . . . . . . . . . . 13 ⊢ (exp:ℂ⟶ℂ → Fun exp)
134	14, 133	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun exp
135	14	fdmi 6673	. . . . . . . . . . . . 13 ⊢ dom exp = ℂ
136	23, 135	sseqtrrdi 3963	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ dom exp)
137		funfvima2 7182	. . . . . . . . . . . 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 593	. . . . . . . . . . 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 2829	. . . . . . . . . . . 12 ⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((exp‘𝑥) ∈ 𝑦 ↔ (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))))
142		sseq1 3947	. . . . . . . . . . . 12 ⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 ⊆ (exp “ 𝑆) ↔ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆)))
143	141, 142	anbi12d 638	. . . . . . . . . . 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 3567	. . . . . . . . . 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 590	. . . . . . . 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 3141	. . . . 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 3132	. . 3 ⊢ (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))
152		eleq1 2828	. . . . . . 7 ⊢ (𝑧 = (exp‘𝑥) → (𝑧 ∈ 𝑦 ↔ (exp‘𝑥) ∈ 𝑦))
153	152	anbi1d 637	. . . . . 6 ⊢ (𝑧 = (exp‘𝑥) → ((𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
154	153	rexbidv 3164	. . . . 5 ⊢ (𝑧 = (exp‘𝑥) → (∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))))
155	154	ralima 7188	. . . 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 593	. . 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 258	. 2 ⊢ (𝑆 ∈ 𝐽 → ∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))
158	1	cnfldtop 24773	. . 3 ⊢ 𝐽 ∈ Top
159		eltop2 22965	. . 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 235	1 ⊢ (𝑆 ∈ 𝐽 → (exp “ 𝑆) ∈ 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∀wral 3054 ∃wrex 3064 {crab 3392 ∩ cin 3889 ⊆ wss 3890 class class class wbr 5079 ↦ cmpt 5160 ^◡ccnv 5624 dom cdm 5625 ran crn 5626 “ cima 5628 ∘ ccom 5629 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 0cc0 11036 + caddc 11039 · cmul 11041 ℝ^*cxr 11176 < clt 11177 − cmin 11375 / cdiv 11805 ℝ⁺crp 12940 abscabs 15194 expce 16024 πcpi 16029 TopOpenctopn 17382 ∞Metcxmet 21339 ballcbl 21341 ℂ_fldccnfld 21354 Topctop 22883 TopOnctopon 22900 Cn ccn 23214 –cn→ccncf 24868

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ioc 13301 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-mod 13827 df-seq 13962 df-exp 14022 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-ef 16030 df-sin 16032 df-cos 16033 df-tan 16034 df-pi 16035 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-cmp 23377 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-limc 25858 df-dv 25859 df-log 26545

This theorem is referenced by: (None)

W3C validator