xrlimcnp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xrlimcnp		Structured version Visualization version GIF version

Theorem xrlimcnp 24695

Description: Relate a limit of a real-valued sequence at infinity to the continuity of the corresponding extended real function at +∞. Since any ⇝_𝑟 limit can be written in the form on the left side of the implication, this shows that real limits are a special case of topological continuity at a point. (Contributed by Mario Carneiro, 8-Sep-2015.)

**Hypotheses**
Ref	Expression
xrlimcnp.a	⊢ (𝜑 → 𝐴 = (𝐵 ∪ {+∞}))
xrlimcnp.b	⊢ (𝜑 → 𝐵 ⊆ ℝ)
xrlimcnp.r	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ ℂ)
xrlimcnp.c	⊢ (𝑥 = +∞ → 𝑅 = 𝐶)
xrlimcnp.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
xrlimcnp.k	⊢ 𝐾 = ((ordTop‘ ≤ ) ↾_t 𝐴)

**Assertion**
Ref	Expression
xrlimcnp	⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)))

Distinct variable groups: 𝑥,𝐵 𝜑,𝑥 𝑥,𝐴 𝑥,𝐶 Allowed substitution hints: 𝑅(𝑥) 𝐽(𝑥) 𝐾(𝑥)

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

Step	Hyp	Ref	Expression
1		xrlimcnp.r	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ ℂ)
2		eqid 2622	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)
3	1, 2	fmptd 6385	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶ℂ)
4	3	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶ℂ)
5		ssun2 3777	. . . . . . . . . 10 ⊢ {+∞} ⊆ (𝐵 ∪ {+∞})
6		pnfex 10093	. . . . . . . . . . 11 ⊢ +∞ ∈ V
7	6	snid 4208	. . . . . . . . . 10 ⊢ +∞ ∈ {+∞}
8	5, 7	sselii 3600	. . . . . . . . 9 ⊢ +∞ ∈ (𝐵 ∪ {+∞})
9		xrlimcnp.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = (𝐵 ∪ {+∞}))
10	8, 9	syl5eleqr 2708	. . . . . . . 8 ⊢ (𝜑 → +∞ ∈ 𝐴)
11	1	ralrimiva 2966	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ ℂ)
12		xrlimcnp.c	. . . . . . . . . . 11 ⊢ (𝑥 = +∞ → 𝑅 = 𝐶)
13	12	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (𝑅 ∈ ℂ ↔ 𝐶 ∈ ℂ))
14	13	rspcv 3305	. . . . . . . . 9 ⊢ (+∞ ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑅 ∈ ℂ → 𝐶 ∈ ℂ))
15	10, 11, 14	sylc 65	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
16	12, 2	fvmptg 6280	. . . . . . . 8 ⊢ ((+∞ ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) = 𝐶)
17	10, 15, 16	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) = 𝐶)
18	17	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) ∧ 𝑦 ∈ 𝐽) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) = 𝐶)
19	18	eleq1d 2686	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) ∧ 𝑦 ∈ 𝐽) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) ∈ 𝑦 ↔ 𝐶 ∈ 𝑦))
20		cnxmet 22576	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
21		xrlimcnp.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
22	21	cnfldtopn 22585	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
23	22	mopni2 22298	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ 𝐽 ∧ 𝐶 ∈ 𝑦) → ∃𝑟 ∈ ℝ⁺ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)
24	20, 23	mp3an1 1411	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐽 ∧ 𝐶 ∈ 𝑦) → ∃𝑟 ∈ ℝ⁺ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)
25		ssun1 3776	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ (𝐵 ∪ {+∞})
26	25, 9	syl5sseqr 3654	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
27		ssralv 3666	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 𝑅 ∈ ℂ → ∀𝑥 ∈ 𝐵 𝑅 ∈ ℂ))
28	26, 11, 27	sylc 65	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝑅 ∈ ℂ)
29	28	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) → ∀𝑥 ∈ 𝐵 𝑅 ∈ ℂ)
30		simprl 794	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) → 𝑟 ∈ ℝ⁺)
31		simplr 792	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) → (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶)
32	29, 30, 31	rlimi 14244	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))
33		letop 21010	. . . . . . . . . . . . . . 15 ⊢ (ordTop‘ ≤ ) ∈ Top
34	33	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → (ordTop‘ ≤ ) ∈ Top)
35		xrlimcnp.b	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
36		ressxr 10083	. . . . . . . . . . . . . . . . . . 19 ⊢ ℝ ⊆ ℝ^*
37	35, 36	syl6ss 3615	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
38		pnfxr 10092	. . . . . . . . . . . . . . . . . . . 20 ⊢ +∞ ∈ ℝ^*
39	38	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → +∞ ∈ ℝ^*)
40	39	snssd 4340	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {+∞} ⊆ ℝ^*)
41	37, 40	unssd 3789	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ∪ {+∞}) ⊆ ℝ^*)
42	9, 41	eqsstrd 3639	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
43		xrex 11829	. . . . . . . . . . . . . . . . 17 ⊢ ℝ^* ∈ V
44	43	ssex 4802	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ ℝ^* → 𝐴 ∈ V)
45	42, 44	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ V)
46	45	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → 𝐴 ∈ V)
47		iocpnfordt 21019	. . . . . . . . . . . . . . 15 ⊢ (𝑘(,]+∞) ∈ (ordTop‘ ≤ )
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → (𝑘(,]+∞) ∈ (ordTop‘ ≤ ))
49		elrestr 16089	. . . . . . . . . . . . . 14 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ (𝑘(,]+∞) ∈ (ordTop‘ ≤ )) → ((𝑘(,]+∞) ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾_t 𝐴))
50	34, 46, 48, 49	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ((𝑘(,]+∞) ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾_t 𝐴))
51		xrlimcnp.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾_t 𝐴)
52	50, 51	syl6eleqr 2712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ((𝑘(,]+∞) ∩ 𝐴) ∈ 𝐾)
53		simprl 794	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → 𝑘 ∈ ℝ)
54	53	rexrd 10089	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → 𝑘 ∈ ℝ^*)
55	38	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → +∞ ∈ ℝ^*)
56		ltpnf 11954	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℝ → 𝑘 < +∞)
57	53, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → 𝑘 < +∞)
58		ubioc1 12227	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑘 < +∞) → +∞ ∈ (𝑘(,]+∞))
59	54, 55, 57, 58	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → +∞ ∈ (𝑘(,]+∞))
60	10	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → +∞ ∈ 𝐴)
61	59, 60	elind 3798	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → +∞ ∈ ((𝑘(,]+∞) ∩ 𝐴))
62		simplr 792	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℝ)
63	62	rexrd 10089	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℝ^*)
64		elioc1 12217	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑥 ∈ (𝑘(,]+∞) ↔ (𝑥 ∈ ℝ^ ∧ 𝑘 < 𝑥 ∧ 𝑥 ≤ +∞)))
65	63, 38, 64	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝑘(,]+∞) ↔ (𝑥 ∈ ℝ^* ∧ 𝑘 < 𝑥 ∧ 𝑥 ≤ +∞)))
66		simp2 1062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑘 < 𝑥 ∧ 𝑥 ≤ +∞) → 𝑘 < 𝑥)
67	65, 66	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝑘(,]+∞) → 𝑘 < 𝑥))
68	35	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) → 𝐵 ⊆ ℝ)
69	68	sselda 3603	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ ℝ)
70		ltle 10126	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 < 𝑥 → 𝑘 ≤ 𝑥))
71	62, 69, 70	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (𝑘 < 𝑥 → 𝑘 ≤ 𝑥))
72	67, 71	syld 47	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝑘(,]+∞) → 𝑘 ≤ 𝑥))
73	20	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (abs ∘ − ) ∈ (∞Met‘ℂ))
74		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) → 𝑟 ∈ ℝ⁺)
75	74	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝑟 ∈ ℝ⁺)
76		rpxr 11840	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
77	75, 76	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝑟 ∈ ℝ^*)
78	15	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
79	28	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) → ∀𝑥 ∈ 𝐵 𝑅 ∈ ℂ)
80	79	r19.21bi 2932	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ ℂ)
81		elbl3 22197	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ^*) ∧ (𝐶 ∈ ℂ ∧ 𝑅 ∈ ℂ)) → (𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ (𝑅(abs ∘ − )𝐶) < 𝑟))
82	73, 77, 78, 80, 81	syl22anc 1327	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ (𝑅(abs ∘ − )𝐶) < 𝑟))
83		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (abs ∘ − ) = (abs ∘ − )
84	83	cnmetdval 22574	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑅 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝑅(abs ∘ − )𝐶) = (abs‘(𝑅 − 𝐶)))
85	80, 78, 84	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (𝑅(abs ∘ − )𝐶) = (abs‘(𝑅 − 𝐶)))
86	85	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → ((𝑅(abs ∘ − )𝐶) < 𝑟 ↔ (abs‘(𝑅 − 𝐶)) < 𝑟))
87	82, 86	bitrd 268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ (abs‘(𝑅 − 𝐶)) < 𝑟))
88	87	biimprd 238	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → ((abs‘(𝑅 − 𝐶)) < 𝑟 → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
89	72, 88	imim12d 81	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → ((𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟) → (𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))))
90	89	ralimdva 2962	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ 𝑘 ∈ ℝ) → (∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟) → ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))))
91	90	impr 649	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
92	20	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → (abs ∘ − ) ∈ (∞Met‘ℂ))
93	15	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → 𝐶 ∈ ℂ)
94		simplrl 800	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → 𝑟 ∈ ℝ⁺)
95		blcntr 22218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑟 ∈ ℝ⁺) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))
96	92, 93, 94, 95	syl3anc 1326	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))
97	96	a1d 25	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → (+∞ ∈ (𝑘(,]+∞) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
98		eleq1 2689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = +∞ → (𝑥 ∈ (𝑘(,]+∞) ↔ +∞ ∈ (𝑘(,]+∞)))
99	12	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = +∞ → (𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
100	98, 99	imbi12d 334	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = +∞ → ((𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) ↔ (+∞ ∈ (𝑘(,]+∞) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))))
101	6, 100	ralsn 4222	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ {+∞} (𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) ↔ (+∞ ∈ (𝑘(,]+∞) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
102	97, 101	sylibr 224	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ∀𝑥 ∈ {+∞} (𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
103		ralunb 3794	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ (𝐵 ∪ {+∞})(𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) ↔ (∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) ∧ ∀𝑥 ∈ {+∞} (𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))))
104	91, 102, 103	sylanbrc 698	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ∀𝑥 ∈ (𝐵 ∪ {+∞})(𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
105	9	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → 𝐴 = (𝐵 ∪ {+∞}))
106	105	raleqdv 3144	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → (∀𝑥 ∈ 𝐴 (𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) ↔ ∀𝑥 ∈ (𝐵 ∪ {+∞})(𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))))
107	104, 106	mpbird 247	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
108		ss2rab 3678	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑥 ∈ (𝑘(,]+∞)} ⊆ {𝑥 ∈ 𝐴 ∣ 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)} ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ (𝑘(,]+∞) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
109	107, 108	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ (𝑘(,]+∞)} ⊆ {𝑥 ∈ 𝐴 ∣ 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)})
110		incom 3805	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘(,]+∞) ∩ 𝐴) = (𝐴 ∩ (𝑘(,]+∞))
111		dfin5 3582	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∩ (𝑘(,]+∞)) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ (𝑘(,]+∞)}
112	110, 111	eqtri 2644	. . . . . . . . . . . . . . 15 ⊢ ((𝑘(,]+∞) ∩ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ (𝑘(,]+∞)}
113	2	mptpreima 5628	. . . . . . . . . . . . . . 15 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝐶(ball‘(abs ∘ − ))𝑟)) = {𝑥 ∈ 𝐴 ∣ 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)}
114	109, 112, 113	3sstr4g 3646	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ((𝑘(,]+∞) ∩ 𝐴) ⊆ (^◡(𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝐶(ball‘(abs ∘ − ))𝑟)))
115		funmpt 5926	. . . . . . . . . . . . . . 15 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝑅)
116		inss2 3834	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘(,]+∞) ∩ 𝐴) ⊆ 𝐴
117	3	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶ℂ)
118		fdm 6051	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶ℂ → dom (𝑥 ∈ 𝐴 ↦ 𝑅) = 𝐴)
119	117, 118	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → dom (𝑥 ∈ 𝐴 ↦ 𝑅) = 𝐴)
120	116, 119	syl5sseqr 3654	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ((𝑘(,]+∞) ∩ 𝐴) ⊆ dom (𝑥 ∈ 𝐴 ↦ 𝑅))
121		funimass3 6333	. . . . . . . . . . . . . . 15 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝑅) ∧ ((𝑘(,]+∞) ∩ 𝐴) ⊆ dom (𝑥 ∈ 𝐴 ↦ 𝑅)) → (((𝑥 ∈ 𝐴 ↦ 𝑅) “ ((𝑘(,]+∞) ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ ((𝑘(,]+∞) ∩ 𝐴) ⊆ (^◡(𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝐶(ball‘(abs ∘ − ))𝑟))))
122	115, 120, 121	sylancr 695	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → (((𝑥 ∈ 𝐴 ↦ 𝑅) “ ((𝑘(,]+∞) ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ ((𝑘(,]+∞) ∩ 𝐴) ⊆ (^◡(𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝐶(ball‘(abs ∘ − ))𝑟))))
123	114, 122	mpbird 247	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ((𝑥 ∈ 𝐴 ↦ 𝑅) “ ((𝑘(,]+∞) ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟))
124		simplrr 801	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)
125	123, 124	sstrd 3613	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ((𝑥 ∈ 𝐴 ↦ 𝑅) “ ((𝑘(,]+∞) ∩ 𝐴)) ⊆ 𝑦)
126		eleq2 2690	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑘(,]+∞) ∩ 𝐴) → (+∞ ∈ 𝑧 ↔ +∞ ∈ ((𝑘(,]+∞) ∩ 𝐴)))
127		imaeq2 5462	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ((𝑘(,]+∞) ∩ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝑅) “ ((𝑘(,]+∞) ∩ 𝐴)))
128	127	sseq1d 3632	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((𝑘(,]+∞) ∩ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ ((𝑘(,]+∞) ∩ 𝐴)) ⊆ 𝑦))
129	126, 128	anbi12d 747	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑘(,]+∞) ∩ 𝐴) → ((+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦) ↔ (+∞ ∈ ((𝑘(,]+∞) ∩ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ ((𝑘(,]+∞) ∩ 𝐴)) ⊆ 𝑦)))
130	129	rspcev 3309	. . . . . . . . . . . 12 ⊢ ((((𝑘(,]+∞) ∩ 𝐴) ∈ 𝐾 ∧ (+∞ ∈ ((𝑘(,]+∞) ∩ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ ((𝑘(,]+∞) ∩ 𝐴)) ⊆ 𝑦)) → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦))
131	52, 61, 125, 130	syl12anc 1324	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) ∧ (𝑘 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))) → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦))
132	131	rexlimdvaa 3032	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) → (∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟) → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦)))
133	132	adantlr 751	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) → (∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 ≤ 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟) → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦)))
134	32, 133	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦)) → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦))
135	134	rexlimdvaa 3032	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) → (∃𝑟 ∈ ℝ⁺ (𝐶(ball‘(abs ∘ − ))𝑟) ⊆ 𝑦 → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦)))
136	24, 135	syl5 34	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) → ((𝑦 ∈ 𝐽 ∧ 𝐶 ∈ 𝑦) → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦)))
137	136	expdimp 453	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) ∧ 𝑦 ∈ 𝐽) → (𝐶 ∈ 𝑦 → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦)))
138	19, 137	sylbid 230	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) ∧ 𝑦 ∈ 𝐽) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) ∈ 𝑦 → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦)))
139	138	ralrimiva 2966	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) → ∀𝑦 ∈ 𝐽 (((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) ∈ 𝑦 → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦)))
140		letopon 21009	. . . . . . 7 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
141		resttopon 20965	. . . . . . 7 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ 𝐴 ⊆ ℝ^) → ((ordTop‘ ≤ ) ↾_t 𝐴) ∈ (TopOn‘𝐴))
142	140, 42, 141	sylancr 695	. . . . . 6 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾_t 𝐴) ∈ (TopOn‘𝐴))
143	51, 142	syl5eqel 2705	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐴))
144	21	cnfldtopon 22586	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
145	144	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘ℂ))
146		iscnp 21041	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝐴) ∧ 𝐽 ∈ (TopOn‘ℂ) ∧ +∞ ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞) ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶ℂ ∧ ∀𝑦 ∈ 𝐽 (((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) ∈ 𝑦 → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦)))))
147	143, 145, 10, 146	syl3anc 1326	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞) ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶ℂ ∧ ∀𝑦 ∈ 𝐽 (((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) ∈ 𝑦 → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦)))))
148	147	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞) ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶ℂ ∧ ∀𝑦 ∈ 𝐽 (((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) ∈ 𝑦 → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ 𝑦)))))
149	4, 139, 148	mpbir2and 957	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞))
150		simplr 792	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞))
151	20	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → (abs ∘ − ) ∈ (∞Met‘ℂ))
152	15	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
153	76	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ^*)
154	22	blopn 22305	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝑟 ∈ ℝ^*) → (𝐶(ball‘(abs ∘ − ))𝑟) ∈ 𝐽)
155	151, 152, 153, 154	syl3anc 1326	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → (𝐶(ball‘(abs ∘ − ))𝑟) ∈ 𝐽)
156	17	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) = 𝐶)
157		simpr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
158	151, 152, 157, 95	syl3anc 1326	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → 𝐶 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))
159	156, 158	eqeltrd 2701	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) ∈ (𝐶(ball‘(abs ∘ − ))𝑟))
160		cnpimaex 21060	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞) ∧ (𝐶(ball‘(abs ∘ − ))𝑟) ∈ 𝐽 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘+∞) ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟)))
161	150, 155, 159, 160	syl3anc 1326	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟)))
162		vex 3203	. . . . . . . . 9 ⊢ 𝑤 ∈ V
163	162	inex1 4799	. . . . . . . 8 ⊢ (𝑤 ∩ 𝐴) ∈ V
164	163	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑤 ∈ (ordTop‘ ≤ )) → (𝑤 ∩ 𝐴) ∈ V)
165	51	eleq2i 2693	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐾 ↔ 𝑧 ∈ ((ordTop‘ ≤ ) ↾_t 𝐴))
166	45	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → 𝐴 ∈ V)
167		elrest 16088	. . . . . . . . 9 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ ((ordTop‘ ≤ ) ↾_t 𝐴) ↔ ∃𝑤 ∈ (ordTop‘ ≤ )𝑧 = (𝑤 ∩ 𝐴)))
168	33, 166, 167	sylancr 695	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → (𝑧 ∈ ((ordTop‘ ≤ ) ↾_t 𝐴) ↔ ∃𝑤 ∈ (ordTop‘ ≤ )𝑧 = (𝑤 ∩ 𝐴)))
169	165, 168	syl5bb 272	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → (𝑧 ∈ 𝐾 ↔ ∃𝑤 ∈ (ordTop‘ ≤ )𝑧 = (𝑤 ∩ 𝐴)))
170		eleq2 2690	. . . . . . . . 9 ⊢ (𝑧 = (𝑤 ∩ 𝐴) → (+∞ ∈ 𝑧 ↔ +∞ ∈ (𝑤 ∩ 𝐴)))
171		imaeq2 5462	. . . . . . . . . 10 ⊢ (𝑧 = (𝑤 ∩ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)))
172	171	sseq1d 3632	. . . . . . . . 9 ⊢ (𝑧 = (𝑤 ∩ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟)))
173	170, 172	anbi12d 747	. . . . . . . 8 ⊢ (𝑧 = (𝑤 ∩ 𝐴) → ((+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟)) ↔ (+∞ ∈ (𝑤 ∩ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟))))
174	173	adantl 482	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑧 = (𝑤 ∩ 𝐴)) → ((+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟)) ↔ (+∞ ∈ (𝑤 ∩ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟))))
175	164, 169, 174	rexxfr2d 4883	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → (∃𝑧 ∈ 𝐾 (+∞ ∈ 𝑧 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ 𝑧) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟)) ↔ ∃𝑤 ∈ (ordTop‘ ≤ )(+∞ ∈ (𝑤 ∩ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟))))
176	161, 175	mpbid 222	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑤 ∈ (ordTop‘ ≤ )(+∞ ∈ (𝑤 ∩ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟)))
177		inss1 3833	. . . . . . . . . . . 12 ⊢ (𝑤 ∩ 𝐴) ⊆ 𝑤
178	177	sseli 3599	. . . . . . . . . . 11 ⊢ (+∞ ∈ (𝑤 ∩ 𝐴) → +∞ ∈ 𝑤)
179		pnfnei 21024	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝑤) → ∃𝑘 ∈ ℝ (𝑘(,]+∞) ⊆ 𝑤)
180	178, 179	sylan2 491	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ (𝑤 ∩ 𝐴)) → ∃𝑘 ∈ ℝ (𝑘(,]+∞) ⊆ 𝑤)
181		df-ima 5127	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) = ran ((𝑥 ∈ 𝐴 ↦ 𝑅) ↾ (𝑤 ∩ 𝐴))
182		inss2 3834	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∩ 𝐴) ⊆ 𝐴
183		resmpt 5449	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∩ 𝐴) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝑅) ↾ (𝑤 ∩ 𝐴)) = (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅))
184	182, 183	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅) ↾ (𝑤 ∩ 𝐴)) = (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅)
185	184	rneqi 5352	. . . . . . . . . . . . . . . 16 ⊢ ran ((𝑥 ∈ 𝐴 ↦ 𝑅) ↾ (𝑤 ∩ 𝐴)) = ran (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅)
186	181, 185	eqtri 2644	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) = ran (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅)
187	186	sseq1i 3629	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ ran (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟))
188		dfss3 3592	. . . . . . . . . . . . . 14 ⊢ (ran (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ ∀𝑧 ∈ ran (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅)𝑧 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))
189	187, 188	bitri 264	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ ∀𝑧 ∈ ran (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅)𝑧 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))
190	11	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∀𝑥 ∈ 𝐴 𝑅 ∈ ℂ)
191		ssralv 3666	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∩ 𝐴) ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 𝑅 ∈ ℂ → ∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ ℂ))
192	182, 190, 191	mpsyl 68	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ ℂ)
193		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅) = (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅)
194		eleq1 2689	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑅 → (𝑧 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
195	193, 194	ralrnmpt 6368	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ ℂ → (∀𝑧 ∈ ran (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅)𝑧 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ ∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
196	192, 195	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (∀𝑧 ∈ ran (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅)𝑧 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ ∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
197	196	biimpd 219	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (∀𝑧 ∈ ran (𝑥 ∈ (𝑤 ∩ 𝐴) ↦ 𝑅)𝑧 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) → ∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
198	189, 197	syl5bi 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) → ∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)))
199		simplrr 801	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → (𝑘(,]+∞) ⊆ 𝑤)
200	37	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝐵 ⊆ ℝ^*)
201		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑥 ∈ 𝐵)
202	200, 201	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑥 ∈ ℝ^*)
203		simprr 796	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑘 < 𝑥)
204		pnfge 11964	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ +∞)
205	202, 204	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑥 ≤ +∞)
206		simplrl 800	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑘 ∈ ℝ)
207	206	rexrd 10089	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑘 ∈ ℝ^*)
208	207, 38, 64	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → (𝑥 ∈ (𝑘(,]+∞) ↔ (𝑥 ∈ ℝ^* ∧ 𝑘 < 𝑥 ∧ 𝑥 ≤ +∞)))
209	202, 203, 205, 208	mpbir3and 1245	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑥 ∈ (𝑘(,]+∞))
210	199, 209	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑥 ∈ 𝑤)
211	26	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) → 𝐵 ⊆ 𝐴)
212	211	sselda 3603	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
213	212	adantrr 753	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑥 ∈ 𝐴)
214	210, 213	elind 3798	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑥 ∈ (𝑤 ∩ 𝐴))
215	214	ex 450	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) → ((𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥) → 𝑥 ∈ (𝑤 ∩ 𝐴)))
216	215	imim1d 82	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) → ((𝑥 ∈ (𝑤 ∩ 𝐴) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) → ((𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟))))
217	20	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
218	76	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ^*)
219	218	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑟 ∈ ℝ^*)
220	15	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝐶 ∈ ℂ)
221	28	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) → ∀𝑥 ∈ 𝐵 𝑅 ∈ ℂ)
222	221	r19.21bi 2932	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ ℂ)
223	222	adantrr 753	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → 𝑅 ∈ ℂ)
224	217, 219, 220, 223, 81	syl22anc 1327	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → (𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ (𝑅(abs ∘ − )𝐶) < 𝑟))
225	223, 220, 84	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → (𝑅(abs ∘ − )𝐶) = (abs‘(𝑅 − 𝐶)))
226	225	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → ((𝑅(abs ∘ − )𝐶) < 𝑟 ↔ (abs‘(𝑅 − 𝐶)) < 𝑟))
227	224, 226	bitrd 268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥)) → (𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) ↔ (abs‘(𝑅 − 𝐶)) < 𝑟))
228	227	pm5.74da 723	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) → (((𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥) → (abs‘(𝑅 − 𝐶)) < 𝑟)))
229	216, 228	sylibd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) → ((𝑥 ∈ (𝑤 ∩ 𝐴) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) → ((𝑥 ∈ 𝐵 ∧ 𝑘 < 𝑥) → (abs‘(𝑅 − 𝐶)) < 𝑟)))
230	229	exp4a 633	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) → ((𝑥 ∈ (𝑤 ∩ 𝐴) → 𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) → (𝑥 ∈ 𝐵 → (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))))
231	230	ralimdv2 2961	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) → (∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) → ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟)))
232	231	imp 445	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) ∧ ∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) → ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))
233	232	an32s 846	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ ∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) ∧ (𝑘 ∈ ℝ ∧ (𝑘(,]+∞) ⊆ 𝑤)) → ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))
234	233	expr 643	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ ∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) ∧ 𝑘 ∈ ℝ) → ((𝑘(,]+∞) ⊆ 𝑤 → ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟)))
235	234	reximdva 3017	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ ∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟)) → (∃𝑘 ∈ ℝ (𝑘(,]+∞) ⊆ 𝑤 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟)))
236	235	ex 450	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ (𝑤 ∩ 𝐴)𝑅 ∈ (𝐶(ball‘(abs ∘ − ))𝑟) → (∃𝑘 ∈ ℝ (𝑘(,]+∞) ⊆ 𝑤 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))))
237	198, 236	syld 47	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) → (∃𝑘 ∈ ℝ (𝑘(,]+∞) ⊆ 𝑤 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))))
238	237	com23 86	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (∃𝑘 ∈ ℝ (𝑘(,]+∞) ⊆ 𝑤 → (((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))))
239	180, 238	syl5 34	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑤 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ (𝑤 ∩ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))))
240	239	impl 650	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑤 ∈ (ordTop‘ ≤ )) ∧ +∞ ∈ (𝑤 ∩ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟)))
241	240	expimpd 629	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑤 ∈ (ordTop‘ ≤ )) → ((+∞ ∈ (𝑤 ∩ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟)) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟)))
242	241	rexlimdva 3031	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (∃𝑤 ∈ (ordTop‘ ≤ )(+∞ ∈ (𝑤 ∩ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟)) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟)))
243	242	adantlr 751	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → (∃𝑤 ∈ (ordTop‘ ≤ )(+∞ ∈ (𝑤 ∩ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅) “ (𝑤 ∩ 𝐴)) ⊆ (𝐶(ball‘(abs ∘ − ))𝑟)) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟)))
244	176, 243	mpd 15	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))
245	244	ralrimiva 2966	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) → ∀𝑟 ∈ ℝ⁺ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟))
246	28, 35, 15	rlim2lt 14228	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶 ↔ ∀𝑟 ∈ ℝ⁺ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟)))
247	246	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) → ((𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶 ↔ ∀𝑟 ∈ ℝ⁺ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑘 < 𝑥 → (abs‘(𝑅 − 𝐶)) < 𝑟)))
248	245, 247	mpbird 247	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)) → (𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶)
249	149, 248	impbida 877	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝑅) ⇝_𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ ((𝐾 CnP 𝐽)‘+∞)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 ∘ ccom 5118 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 ℝ⁺crp 11832 (,]cioc 12176 abscabs 13974 ⇝_𝑟 crli 14216 ↾_t crest 16081 TopOpenctopn 16082 ordTopcordt 16159 ∞Metcxmt 19731 ballcbl 19733 ℂ_fldccnfld 19746 Topctop 20698 TopOnctopon 20715 CnP ccnp 21029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-rlim 14220 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-ordt 16161 df-ps 17200 df-tsr 17201 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cnp 21032 df-xms 22125 df-ms 22126

This theorem is referenced by: (None)

W3C validator