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Theorem efopn 24311
Description: The exponential map is an open map. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
efopn.j 𝐽 = (TopOpen‘ℂfld)
Assertion
Ref Expression
efopn (𝑆𝐽 → (exp “ 𝑆) ∈ 𝐽)

Proof of Theorem efopn
Dummy variables 𝑤 𝑟 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efopn.j . . . . . . . 8 𝐽 = (TopOpen‘ℂfld)
21cnfldtopon 22499 . . . . . . 7 𝐽 ∈ (TopOn‘ℂ)
3 toponss 20643 . . . . . . 7 ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆𝐽) → 𝑆 ⊆ ℂ)
42, 3mpan 705 . . . . . 6 (𝑆𝐽𝑆 ⊆ ℂ)
54sselda 3584 . . . . 5 ((𝑆𝐽𝑥𝑆) → 𝑥 ∈ ℂ)
6 cnxmet 22489 . . . . . 6 (abs ∘ − ) ∈ (∞Met‘ℂ)
7 pirp 24124 . . . . . . 7 π ∈ ℝ+
81cnfldtopn 22498 . . . . . . . 8 𝐽 = (MetOpen‘(abs ∘ − ))
98mopni3 22212 . . . . . . 7 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆𝐽𝑥𝑆) ∧ π ∈ ℝ+) → ∃𝑟 ∈ ℝ+ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
107, 9mpan2 706 . . . . . 6 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆𝐽𝑥𝑆) → ∃𝑟 ∈ ℝ+ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
116, 10mp3an1 1408 . . . . 5 ((𝑆𝐽𝑥𝑆) → ∃𝑟 ∈ ℝ+ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆))
12 imass2 5462 . . . . . . . 8 ((𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆 → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆))
13 imassrn 5438 . . . . . . . . . . . . . 14 (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ran exp
14 eff 14740 . . . . . . . . . . . . . . 15 exp:ℂ⟶ℂ
15 frn 6012 . . . . . . . . . . . . . . 15 (exp:ℂ⟶ℂ → ran exp ⊆ ℂ)
1614, 15ax-mp 5 . . . . . . . . . . . . . 14 ran exp ⊆ ℂ
1713, 16sstri 3593 . . . . . . . . . . . . 13 (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ℂ
18 sseqin2 3797 . . . . . . . . . . . . 13 ((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ ℂ ↔ (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)))
1917, 18mpbi 220 . . . . . . . . . . . 12 (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))
20 rpxr 11787 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
21 blssm 22136 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ*) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
226, 21mp3an1 1408 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ*) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
2320, 22sylan2 491 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
2423ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
2524sselda 3584 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑦 ∈ ℂ)
26 simp-4l 805 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ)
2725, 26subcld 10339 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦𝑥) ∈ ℂ)
2827subid1d 10328 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦𝑥) − 0) = (𝑦𝑥))
2928fveq2d 6154 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (abs‘((𝑦𝑥) − 0)) = (abs‘(𝑦𝑥)))
30 0cn 9979 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℂ
31 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . 23 (abs ∘ − ) = (abs ∘ − )
3231cnmetdval 22487 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦𝑥) ∈ ℂ ∧ 0 ∈ ℂ) → ((𝑦𝑥)(abs ∘ − )0) = (abs‘((𝑦𝑥) − 0)))
3327, 30, 32sylancl 693 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦𝑥)(abs ∘ − )0) = (abs‘((𝑦𝑥) − 0)))
3431cnmetdval 22487 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦(abs ∘ − )𝑥) = (abs‘(𝑦𝑥)))
3525, 26, 34syl2anc 692 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦(abs ∘ − )𝑥) = (abs‘(𝑦𝑥)))
3629, 33, 353eqtr4d 2665 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦𝑥)(abs ∘ − )0) = (𝑦(abs ∘ − )𝑥))
37 simpr 477 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
386a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
39 simpllr 798 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → 𝑟 ∈ ℝ+)
4039adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ+)
4140rpxrd 11820 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ*)
42 elbl3 22110 . . . . . . . . . . . . . . . . . . . . . 22 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ*) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ (𝑦(abs ∘ − )𝑥) < 𝑟))
4338, 41, 26, 25, 42syl22anc 1324 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ (𝑦(abs ∘ − )𝑥) < 𝑟))
4437, 43mpbid 222 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦(abs ∘ − )𝑥) < 𝑟)
4536, 44eqbrtrd 4637 . . . . . . . . . . . . . . . . . . 19 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦𝑥)(abs ∘ − )0) < 𝑟)
46 0cnd 9980 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 0 ∈ ℂ)
47 elbl3 22110 . . . . . . . . . . . . . . . . . . . 20 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ*) ∧ (0 ∈ ℂ ∧ (𝑦𝑥) ∈ ℂ)) → ((𝑦𝑥) ∈ (0(ball‘(abs ∘ − ))𝑟) ↔ ((𝑦𝑥)(abs ∘ − )0) < 𝑟))
4838, 41, 46, 27, 47syl22anc 1324 . . . . . . . . . . . . . . . . . . 19 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦𝑥) ∈ (0(ball‘(abs ∘ − ))𝑟) ↔ ((𝑦𝑥)(abs ∘ − )0) < 𝑟))
4945, 48mpbird 247 . . . . . . . . . . . . . . . . . 18 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦𝑥) ∈ (0(ball‘(abs ∘ − ))𝑟))
50 efsub 14758 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (exp‘(𝑦𝑥)) = ((exp‘𝑦) / (exp‘𝑥)))
5125, 26, 50syl2anc 692 . . . . . . . . . . . . . . . . . 18 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (exp‘(𝑦𝑥)) = ((exp‘𝑦) / (exp‘𝑥)))
52 fveq2 6150 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝑦𝑥) → (exp‘𝑤) = (exp‘(𝑦𝑥)))
5352eqeq1d 2623 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝑦𝑥) → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘(𝑦𝑥)) = ((exp‘𝑦) / (exp‘𝑥))))
5453rspcev 3295 . . . . . . . . . . . . . . . . . 18 (((𝑦𝑥) ∈ (0(ball‘(abs ∘ − ))𝑟) ∧ (exp‘(𝑦𝑥)) = ((exp‘𝑦) / (exp‘𝑥))) → ∃𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)))
5549, 51, 54syl2anc 692 . . . . . . . . . . . . . . . . 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)))
56 oveq1 6614 . . . . . . . . . . . . . . . . . . 19 ((exp‘𝑦) = 𝑧 → ((exp‘𝑦) / (exp‘𝑥)) = (𝑧 / (exp‘𝑥)))
5756eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 ((exp‘𝑦) = 𝑧 → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘𝑤) = (𝑧 / (exp‘𝑥))))
5857rexbidv 3045 . . . . . . . . . . . . . . . . 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))))
5955, 58syl5ibcom 235 . . . . . . . . . . . . . . . 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))))
6059rexlimdva 3024 . . . . . . . . . . . . . . 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))))
61 eqcom 2628 . . . . . . . . . . . . . . . . . 18 ((exp‘𝑤) = (𝑧 / (exp‘𝑥)) ↔ (𝑧 / (exp‘𝑥)) = (exp‘𝑤))
62 simplr 791 . . . . . . . . . . . . . . . . . . 19 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑧 ∈ ℂ)
63 simp-4l 805 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ)
64 efcl 14741 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
6563, 64syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (exp‘𝑥) ∈ ℂ)
6639rpxrd 11820 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → 𝑟 ∈ ℝ*)
67 blssm 22136 . . . . . . . . . . . . . . . . . . . . . . 23 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑟 ∈ ℝ*) → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
686, 30, 67mp3an12 1411 . . . . . . . . . . . . . . . . . . . . . 22 (𝑟 ∈ ℝ* → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
6966, 68syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ)
7069sselda 3584 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑤 ∈ ℂ)
71 efcl 14741 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ ℂ → (exp‘𝑤) ∈ ℂ)
7270, 71syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (exp‘𝑤) ∈ ℂ)
73 efne0 14755 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (exp‘𝑥) ≠ 0)
7463, 73syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (exp‘𝑥) ≠ 0)
7562, 65, 72, 74divmuld 10770 . . . . . . . . . . . . . . . . . 18 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑧 / (exp‘𝑥)) = (exp‘𝑤) ↔ ((exp‘𝑥) · (exp‘𝑤)) = 𝑧))
7661, 75syl5bb 272 . . . . . . . . . . . . . . . . 17 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((exp‘𝑤) = (𝑧 / (exp‘𝑥)) ↔ ((exp‘𝑥) · (exp‘𝑤)) = 𝑧))
7763, 70pncan2d 10341 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = 𝑤)
7870subid1d 10328 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤 − 0) = 𝑤)
7977, 78eqtr4d 2658 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = (𝑤 − 0))
8079fveq2d 6154 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (abs‘((𝑥 + 𝑤) − 𝑥)) = (abs‘(𝑤 − 0)))
8163, 70addcld 10006 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑥 + 𝑤) ∈ ℂ)
8231cnmetdval 22487 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 + 𝑤) ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (abs‘((𝑥 + 𝑤) − 𝑥)))
8381, 63, 82syl2anc 692 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (abs‘((𝑥 + 𝑤) − 𝑥)))
8431cnmetdval 22487 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑤 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑤(abs ∘ − )0) = (abs‘(𝑤 − 0)))
8570, 30, 84sylancl 693 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤(abs ∘ − )0) = (abs‘(𝑤 − 0)))
8680, 83, 853eqtr4d 2665 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (𝑤(abs ∘ − )0))
87 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟))
886a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
8939adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ+)
9089rpxrd 11820 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ*)
91 0cnd 9980 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → 0 ∈ ℂ)
92 elbl3 22110 . . . . . . . . . . . . . . . . . . . . . . 23 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ*) ∧ (0 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟) ↔ (𝑤(abs ∘ − )0) < 𝑟))
9388, 90, 91, 70, 92syl22anc 1324 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟) ↔ (𝑤(abs ∘ − )0) < 𝑟))
9487, 93mpbid 222 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑤(abs ∘ − )0) < 𝑟)
9586, 94eqbrtrd 4637 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟)
96 elbl3 22110 . . . . . . . . . . . . . . . . . . . . 21 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ*) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝑤) ∈ ℂ)) → ((𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟))
9788, 90, 63, 81, 96syl22anc 1324 . . . . . . . . . . . . . . . . . . . 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))
9895, 97mpbird 247 . . . . . . . . . . . . . . . . . . 19 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
99 efadd 14752 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤)))
10063, 70, 99syl2anc 692 . . . . . . . . . . . . . . . . . . 19 (((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ − ))𝑟)) → (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤)))
101 fveq2 6150 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑥 + 𝑤) → (exp‘𝑦) = (exp‘(𝑥 + 𝑤)))
102101eqeq1d 2623 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑥 + 𝑤) → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤))))
103102rspcev 3295 . . . . . . . . . . . . . . . . . . 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)))
10498, 100, 103syl2anc 692 . . . . . . . . . . . . . . . . . 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)))
105 eqeq2 2632 . . . . . . . . . . . . . . . . . . 19 (((exp‘𝑥) · (exp‘𝑤)) = 𝑧 → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘𝑦) = 𝑧))
106105rexbidv 3045 . . . . . . . . . . . . . . . . . 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))
107104, 106syl5ibcom 235 . . . . . . . . . . . . . . . . 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))
10876, 107sylbid 230 . . . . . . . . . . . . . . . 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))
109108rexlimdva 3024 . . . . . . . . . . . . . . 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))
11060, 109impbid 202 . . . . . . . . . . . . . 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))))
111 ffn 6004 . . . . . . . . . . . . . . . 16 (exp:ℂ⟶ℂ → exp Fn ℂ)
11214, 111ax-mp 5 . . . . . . . . . . . . . . 15 exp Fn ℂ
113 fvelimab 6212 . . . . . . . . . . . . . . 15 ((exp Fn ℂ ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ) → (𝑧 ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ↔ ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧))
114112, 24, 113sylancr 694 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (𝑧 ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ↔ ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧))
115 fvelimab 6212 . . . . . . . . . . . . . . 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))))
116112, 69, 115sylancr 694 . . . . . . . . . . . . . 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))))
117110, 114, 1163bitr4d 300 . . . . . . . . . . . . 13 ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) → (𝑧 ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ↔ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟))))
118117rabbi2dva 3801 . . . . . . . . . . . 12 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = {𝑧 ∈ ℂ ∣ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟))})
11919, 118syl5eqr 2669 . . . . . . . . . . 11 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) = {𝑧 ∈ ℂ ∣ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟))})
120 eqid 2621 . . . . . . . . . . . 12 (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) = (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥)))
121120mptpreima 5589 . . . . . . . . . . 11 ((𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs ∘ − ))𝑟))) = {𝑧 ∈ ℂ ∣ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs ∘ − ))𝑟))}
122119, 121syl6eqr 2673 . . . . . . . . . 10 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) = ((𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs ∘ − ))𝑟))))
12364ad2antrr 761 . . . . . . . . . . . . 13 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → (exp‘𝑥) ∈ ℂ)
12473ad2antrr 761 . . . . . . . . . . . . 13 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → (exp‘𝑥) ≠ 0)
125120divccncf 22622 . . . . . . . . . . . . 13 (((exp‘𝑥) ∈ ℂ ∧ (exp‘𝑥) ≠ 0) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (ℂ–cn→ℂ))
126123, 124, 125syl2anc 692 . . . . . . . . . . . 12 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (ℂ–cn→ℂ))
1271cncfcn1 22626 . . . . . . . . . . . 12 (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
128126, 127syl6eleq 2708 . . . . . . . . . . 11 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (𝐽 Cn 𝐽))
1291efopnlem2 24310 . . . . . . . . . . . 12 ((𝑟 ∈ ℝ+𝑟 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
130129adantll 749 . . . . . . . . . . 11 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
131 cnima 20982 . . . . . . . . . . 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)
132128, 130, 131syl2anc 692 . . . . . . . . . 10 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → ((𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs ∘ − ))𝑟))) ∈ 𝐽)
133122, 132eqeltrd 2698 . . . . . . . . 9 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽)
134 blcntr 22131 . . . . . . . . . . . 12 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
1356, 134mp3an1 1408 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟))
136 ffun 6007 . . . . . . . . . . . . 13 (exp:ℂ⟶ℂ → Fun exp)
13714, 136ax-mp 5 . . . . . . . . . . . 12 Fun exp
13814fdmi 6011 . . . . . . . . . . . . 13 dom exp = ℂ
13923, 138syl6sseqr 3633 . . . . . . . . . . . 12 ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ dom exp)
140 funfvima2 6450 . . . . . . . . . . . 12 ((Fun exp ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ dom exp) → (𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟) → (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))))
141137, 139, 140sylancr 694 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → (𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟) → (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))))
142135, 141mpd 15 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)))
143142adantr 481 . . . . . . . . 9 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)))
144 eleq2 2687 . . . . . . . . . . . 12 (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((exp‘𝑥) ∈ 𝑦 ↔ (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))))
145 sseq1 3607 . . . . . . . . . . . 12 (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 ⊆ (exp “ 𝑆) ↔ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆)))
146144, 145anbi12d 746 . . . . . . . . . . 11 (𝑦 = (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) → (((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆)) ↔ ((exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∧ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆))))
147146rspcev 3295 . . . . . . . . . 10 (((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽 ∧ ((exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∧ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆))) → ∃𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆)))
148147expr 642 . . . . . . . . 9 (((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽 ∧ (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) → ((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆) → ∃𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆))))
149133, 143, 148syl2anc 692 . . . . . . . 8 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → ((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆) → ∃𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆))))
15012, 149syl5 34 . . . . . . 7 (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < π) → ((𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆 → ∃𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆))))
151150expimpd 628 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → ((𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆) → ∃𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆))))
152151rexlimdva 3024 . . . . 5 (𝑥 ∈ ℂ → (∃𝑟 ∈ ℝ+ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆) → ∃𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆))))
1535, 11, 152sylc 65 . . . 4 ((𝑆𝐽𝑥𝑆) → ∃𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆)))
154153ralrimiva 2960 . . 3 (𝑆𝐽 → ∀𝑥𝑆𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆)))
155 eleq1 2686 . . . . . . 7 (𝑧 = (exp‘𝑥) → (𝑧𝑦 ↔ (exp‘𝑥) ∈ 𝑦))
156155anbi1d 740 . . . . . 6 (𝑧 = (exp‘𝑥) → ((𝑧𝑦𝑦 ⊆ (exp “ 𝑆)) ↔ ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆))))
157156rexbidv 3045 . . . . 5 (𝑧 = (exp‘𝑥) → (∃𝑦𝐽 (𝑧𝑦𝑦 ⊆ (exp “ 𝑆)) ↔ ∃𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆))))
158157ralima 6455 . . . 4 ((exp Fn ℂ ∧ 𝑆 ⊆ ℂ) → (∀𝑧 ∈ (exp “ 𝑆)∃𝑦𝐽 (𝑧𝑦𝑦 ⊆ (exp “ 𝑆)) ↔ ∀𝑥𝑆𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆))))
159112, 4, 158sylancr 694 . . 3 (𝑆𝐽 → (∀𝑧 ∈ (exp “ 𝑆)∃𝑦𝐽 (𝑧𝑦𝑦 ⊆ (exp “ 𝑆)) ↔ ∀𝑥𝑆𝑦𝐽 ((exp‘𝑥) ∈ 𝑦𝑦 ⊆ (exp “ 𝑆))))
160154, 159mpbird 247 . 2 (𝑆𝐽 → ∀𝑧 ∈ (exp “ 𝑆)∃𝑦𝐽 (𝑧𝑦𝑦 ⊆ (exp “ 𝑆)))
1611cnfldtop 22500 . . 3 𝐽 ∈ Top
162 eltop2 20693 . . 3 (𝐽 ∈ Top → ((exp “ 𝑆) ∈ 𝐽 ↔ ∀𝑧 ∈ (exp “ 𝑆)∃𝑦𝐽 (𝑧𝑦𝑦 ⊆ (exp “ 𝑆))))
163161, 162ax-mp 5 . 2 ((exp “ 𝑆) ∈ 𝐽 ↔ ∀𝑧 ∈ (exp “ 𝑆)∃𝑦𝐽 (𝑧𝑦𝑦 ⊆ (exp “ 𝑆)))
164160, 163sylibr 224 1 (𝑆𝐽 → (exp “ 𝑆) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cin 3555  wss 3556   class class class wbr 4615  cmpt 4675  ccnv 5075  dom cdm 5076  ran crn 5077  cima 5079  ccom 5080  Fun wfun 5843   Fn wfn 5844  wf 5845  cfv 5849  (class class class)co 6607  cc 9881  0cc0 9883   + caddc 9886   · cmul 9888  *cxr 10020   < clt 10021  cmin 10213   / cdiv 10631  +crp 11779  abscabs 13911  expce 14720  πcpi 14725  TopOpenctopn 16006  ∞Metcxmt 19653  ballcbl 19655  fldccnfld 19668  Topctop 20620  TopOnctopon 20637   Cn ccn 20941  cnccncf 22592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961  ax-addf 9962  ax-mulf 9963
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-of 6853  df-om 7016  df-1st 7116  df-2nd 7117  df-supp 7244  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-ixp 7856  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-fsupp 8223  df-fi 8264  df-sup 8295  df-inf 8296  df-oi 8362  df-card 8712  df-cda 8937  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-7 11031  df-8 11032  df-9 11033  df-n0 11240  df-z 11325  df-dec 11441  df-uz 11635  df-q 11736  df-rp 11780  df-xneg 11893  df-xadd 11894  df-xmul 11895  df-ioo 12124  df-ioc 12125  df-ico 12126  df-icc 12127  df-fz 12272  df-fzo 12410  df-fl 12536  df-mod 12612  df-seq 12745  df-exp 12804  df-fac 13004  df-bc 13033  df-hash 13061  df-shft 13744  df-cj 13776  df-re 13777  df-im 13778  df-sqrt 13912  df-abs 13913  df-limsup 14139  df-clim 14156  df-rlim 14157  df-sum 14354  df-ef 14726  df-sin 14728  df-cos 14729  df-tan 14730  df-pi 14731  df-struct 15786  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-ress 15791  df-plusg 15878  df-mulr 15879  df-starv 15880  df-sca 15881  df-vsca 15882  df-ip 15883  df-tset 15884  df-ple 15885  df-ds 15888  df-unif 15889  df-hom 15890  df-cco 15891  df-rest 16007  df-topn 16008  df-0g 16026  df-gsum 16027  df-topgen 16028  df-pt 16029  df-prds 16032  df-xrs 16086  df-qtop 16091  df-imas 16092  df-xps 16094  df-mre 16170  df-mrc 16171  df-acs 16173  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-submnd 17260  df-mulg 17465  df-cntz 17674  df-cmn 18119  df-psmet 19660  df-xmet 19661  df-met 19662  df-bl 19663  df-mopn 19664  df-fbas 19665  df-fg 19666  df-cnfld 19669  df-top 20621  df-topon 20638  df-topsp 20651  df-bases 20664  df-cld 20736  df-ntr 20737  df-cls 20738  df-nei 20815  df-lp 20853  df-perf 20854  df-cn 20944  df-cnp 20945  df-haus 21032  df-cmp 21103  df-tx 21278  df-hmeo 21471  df-fil 21563  df-fm 21655  df-flim 21656  df-flf 21657  df-xms 22038  df-ms 22039  df-tms 22040  df-cncf 22594  df-limc 23543  df-dv 23544  df-log 24214
This theorem is referenced by: (None)
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