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Theorem limcimolemlt 13273

Description: Lemma for limcimo 13274. (Contributed by Jim Kingdon, 3-Jul-2023.)

**Hypotheses**
Ref	Expression
limcflf.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
limcflf.a	⊢ (𝜑 → 𝐴 ⊆ ℂ)
limcimo.b	⊢ (𝜑 → 𝐵 ∈ ℂ)
limcimo.bc	⊢ (𝜑 → 𝐵 ∈ 𝐶)
limcimo.bs	⊢ (𝜑 → 𝐵 ∈ 𝑆)
limcimo.c	⊢ (𝜑 → 𝐶 ∈ (𝐾 ↾_t 𝑆))
limcimo.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
limcimo.ca	⊢ (𝜑 → {𝑞 ∈ 𝐶 ∣ 𝑞 # 𝐵} ⊆ 𝐴)
limcflfcntop.k	⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
limcimo.d	⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
limcimo.x	⊢ (𝜑 → 𝑋 ∈ (𝐹 lim_ℂ 𝐵))
limcimo.y	⊢ (𝜑 → 𝑌 ∈ (𝐹 lim_ℂ 𝐵))
limcimo.z	⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝐷) → (abs‘((𝐹‘𝑧) − 𝑋)) < ((abs‘(𝑋 − 𝑌)) / 2)))
limcimo.g	⊢ (𝜑 → 𝐺 ∈ ℝ⁺)
limcimo.w	⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ((𝑤 # 𝐵 ∧ (abs‘(𝑤 − 𝐵)) < 𝐺) → (abs‘((𝐹‘𝑤) − 𝑌)) < ((abs‘(𝑋 − 𝑌)) / 2)))

**Assertion**
Ref	Expression
limcimolemlt	⊢ (𝜑 → (abs‘(𝑋 − 𝑌)) < (abs‘(𝑋 − 𝑌)))

Distinct variable groups: 𝑤,𝐴 𝑧,𝐴 𝐵,𝑞 𝑤,𝐵 𝑧,𝐵 𝐶,𝑞 𝑧,𝐷 𝑤,𝐹 𝑧,𝐹 𝑤,𝐺 𝑤,𝑋 𝑧,𝑋 𝑤,𝑌 𝑧,𝑌 Allowed substitution hints: 𝜑(𝑧,𝑤,𝑞) 𝐴(𝑞) 𝐶(𝑧,𝑤) 𝐷(𝑤,𝑞) 𝑆(𝑧,𝑤,𝑞) 𝐹(𝑞) 𝐺(𝑧,𝑞) 𝐾(𝑧,𝑤,𝑞) 𝑋(𝑞) 𝑌(𝑞)

Proof of Theorem limcimolemlt Dummy variables 𝑎 𝑏 𝑐 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnxmet 13171	. . . 4 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
2		ax-resscn 7845	. . . . . . 7 ⊢ ℝ ⊆ ℂ
3		sseq1 3165	. . . . . . 7 ⊢ (𝑆 = ℝ → (𝑆 ⊆ ℂ ↔ ℝ ⊆ ℂ))
4	2, 3	mpbiri 167	. . . . . 6 ⊢ (𝑆 = ℝ → 𝑆 ⊆ ℂ)
5	4	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 = ℝ) → 𝑆 ⊆ ℂ)
6		eqimss 3196	. . . . . 6 ⊢ (𝑆 = ℂ → 𝑆 ⊆ ℂ)
7	6	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 = ℂ) → 𝑆 ⊆ ℂ)
8		limcimo.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
9		elpri 3599	. . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ))
10	8, 9	syl 14	. . . . 5 ⊢ (𝜑 → (𝑆 = ℝ ∨ 𝑆 = ℂ))
11	5, 7, 10	mpjaodan 788	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
12		xmetres2 13019	. . . 4 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
13	1, 11, 12	sylancr 411	. . 3 ⊢ (𝜑 → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
14		limcimo.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐾 ↾_t 𝑆))
15		eqid 2165	. . . . . 6 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
16		limcflfcntop.k	. . . . . 6 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
17		eqid 2165	. . . . . 6 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
18	15, 16, 17	metrest 13146	. . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾_t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
19	1, 11, 18	sylancr 411	. . . 4 ⊢ (𝜑 → (𝐾 ↾_t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
20	14, 19	eleqtrd 2245	. . 3 ⊢ (𝜑 → 𝐶 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
21		limcimo.bc	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
22		limcimo.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
23		limcimo.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ℝ⁺)
24		rpmincl 11179	. . . 4 ⊢ ((𝐷 ∈ ℝ⁺ ∧ 𝐺 ∈ ℝ⁺) → inf({𝐷, 𝐺}, ℝ, < ) ∈ ℝ⁺)
25	22, 23, 24	syl2anc 409	. . 3 ⊢ (𝜑 → inf({𝐷, 𝐺}, ℝ, < ) ∈ ℝ⁺)
26	17	mopni3 13124	. . 3 ⊢ (((((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆) ∧ 𝐶 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) ∧ 𝐵 ∈ 𝐶) ∧ inf({𝐷, 𝐺}, ℝ, < ) ∈ ℝ⁺) → ∃𝑟 ∈ ℝ⁺ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))
27	13, 20, 21, 25, 26	syl31anc 1231	. 2 ⊢ (𝜑 → ∃𝑟 ∈ ℝ⁺ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))
28		limcimo.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐹 lim_ℂ 𝐵))
29		limcrcl 13267	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝐹 lim_ℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
30	28, 29	syl 14	. . . . . . . 8 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
31	30	simp1d 999	. . . . . . 7 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
32	30	simp2d 1000	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ)
33		limcimo.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
34	31, 32, 33	ellimc3ap 13270	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝑋 ∈ ℂ ∧ ∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ ∀𝑐 ∈ dom 𝐹((𝑐 # 𝐵 ∧ (abs‘(𝑐 − 𝐵)) < 𝑏) → (abs‘((𝐹‘𝑐) − 𝑋)) < 𝑎))))
35	28, 34	mpbid 146	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ ℂ ∧ ∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ ∀𝑐 ∈ dom 𝐹((𝑐 # 𝐵 ∧ (abs‘(𝑐 − 𝐵)) < 𝑏) → (abs‘((𝐹‘𝑐) − 𝑋)) < 𝑎)))
36	35	simpld 111	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ)
37	36	adantr 274	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝑋 ∈ ℂ)
38		limcimo.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐹 lim_ℂ 𝐵))
39	31, 32, 33	ellimc3ap 13270	. . . . . 6 ⊢ (𝜑 → (𝑌 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝑌 ∈ ℂ ∧ ∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ ∀𝑐 ∈ dom 𝐹((𝑐 # 𝐵 ∧ (abs‘(𝑐 − 𝐵)) < 𝑏) → (abs‘((𝐹‘𝑐) − 𝑌)) < 𝑎))))
40	38, 39	mpbid 146	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ ℂ ∧ ∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ ∀𝑐 ∈ dom 𝐹((𝑐 # 𝐵 ∧ (abs‘(𝑐 − 𝐵)) < 𝑏) → (abs‘((𝐹‘𝑐) − 𝑌)) < 𝑎)))
41	40	simpld 111	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ℂ)
42	41	adantr 274	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝑌 ∈ ℂ)
43		limcflf.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
44	43	adantr 274	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝐹:𝐴⟶ℂ)
45		breq1 3985	. . . . . 6 ⊢ (𝑞 = (𝐵 + (𝑟 / 2)) → (𝑞 # 𝐵 ↔ (𝐵 + (𝑟 / 2)) # 𝐵))
46		simprrr 530	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶)
47		limcimo.bs	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
48	47	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝐵 ∈ 𝑆)
49	47	ad2antrr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℝ) → 𝐵 ∈ 𝑆)
50		simpr 109	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℝ) → 𝑆 = ℝ)
51	49, 50	eleqtrd 2245	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℝ) → 𝐵 ∈ ℝ)
52		simprl 521	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝑟 ∈ ℝ⁺)
53	52	rphalfcld 9645	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑟 / 2) ∈ ℝ⁺)
54	53	adantr 274	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℝ) → (𝑟 / 2) ∈ ℝ⁺)
55	54	rpred 9632	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℝ) → (𝑟 / 2) ∈ ℝ)
56	51, 55	readdcld 7928	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℝ) → (𝐵 + (𝑟 / 2)) ∈ ℝ)
57	56, 50	eleqtrrd 2246	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℝ) → (𝐵 + (𝑟 / 2)) ∈ 𝑆)
58	33	ad2antrr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℂ) → 𝐵 ∈ ℂ)
59	53	adantr 274	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℂ) → (𝑟 / 2) ∈ ℝ⁺)
60	59	rpcnd 9634	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℂ) → (𝑟 / 2) ∈ ℂ)
61	58, 60	addcld 7918	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℂ) → (𝐵 + (𝑟 / 2)) ∈ ℂ)
62		simpr 109	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℂ) → 𝑆 = ℂ)
63	61, 62	eleqtrrd 2246	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) ∧ 𝑆 = ℂ) → (𝐵 + (𝑟 / 2)) ∈ 𝑆)
64	10	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑆 = ℝ ∨ 𝑆 = ℂ))
65	57, 63, 64	mpjaodan 788	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 + (𝑟 / 2)) ∈ 𝑆)
66	48, 65	ovresd 5982	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵((abs ∘ − ) ↾ (𝑆 × 𝑆))(𝐵 + (𝑟 / 2))) = (𝐵(abs ∘ − )(𝐵 + (𝑟 / 2))))
67	33	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝐵 ∈ ℂ)
68	53	rpcnd 9634	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑟 / 2) ∈ ℂ)
69	67, 68	addcld 7918	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 + (𝑟 / 2)) ∈ ℂ)
70		eqid 2165	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (abs ∘ − )
71	70	cnmetdval 13169	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℂ ∧ (𝐵 + (𝑟 / 2)) ∈ ℂ) → (𝐵(abs ∘ − )(𝐵 + (𝑟 / 2))) = (abs‘(𝐵 − (𝐵 + (𝑟 / 2)))))
72	67, 69, 71	syl2anc 409	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵(abs ∘ − )(𝐵 + (𝑟 / 2))) = (abs‘(𝐵 − (𝐵 + (𝑟 / 2)))))
73	67, 67, 68	subsub4d 8240	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((𝐵 − 𝐵) − (𝑟 / 2)) = (𝐵 − (𝐵 + (𝑟 / 2))))
74	67	subidd 8197	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 − 𝐵) = 0)
75	74	oveq1d 5857	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((𝐵 − 𝐵) − (𝑟 / 2)) = (0 − (𝑟 / 2)))
76	73, 75	eqtr3d 2200	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 − (𝐵 + (𝑟 / 2))) = (0 − (𝑟 / 2)))
77	76	fveq2d 5490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘(𝐵 − (𝐵 + (𝑟 / 2)))) = (abs‘(0 − (𝑟 / 2))))
78		0cnd 7892	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 0 ∈ ℂ)
79	78, 68	abssubd 11135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘(0 − (𝑟 / 2))) = (abs‘((𝑟 / 2) − 0)))
80	77, 79	eqtrd 2198	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘(𝐵 − (𝐵 + (𝑟 / 2)))) = (abs‘((𝑟 / 2) − 0)))
81	68	subid1d 8198	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((𝑟 / 2) − 0) = (𝑟 / 2))
82	81	fveq2d 5490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝑟 / 2) − 0)) = (abs‘(𝑟 / 2)))
83	53	rpred 9632	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑟 / 2) ∈ ℝ)
84	53	rpge0d 9636	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 0 ≤ (𝑟 / 2))
85	83, 84	absidd 11109	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘(𝑟 / 2)) = (𝑟 / 2))
86	80, 82, 85	3eqtrd 2202	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘(𝐵 − (𝐵 + (𝑟 / 2)))) = (𝑟 / 2))
87	66, 72, 86	3eqtrd 2202	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵((abs ∘ − ) ↾ (𝑆 × 𝑆))(𝐵 + (𝑟 / 2))) = (𝑟 / 2))
88		rphalflt 9619	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) < 𝑟)
89	88	ad2antrl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑟 / 2) < 𝑟)
90	87, 89	eqbrtrd 4004	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵((abs ∘ − ) ↾ (𝑆 × 𝑆))(𝐵 + (𝑟 / 2))) < 𝑟)
91	13	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
92		rpxr 9597	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
93	92	ad2antrl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝑟 ∈ ℝ^*)
94		elbl2 13033	. . . . . . . . 9 ⊢ (((((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆) ∧ 𝑟 ∈ ℝ^*) ∧ (𝐵 ∈ 𝑆 ∧ (𝐵 + (𝑟 / 2)) ∈ 𝑆)) → ((𝐵 + (𝑟 / 2)) ∈ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ↔ (𝐵((abs ∘ − ) ↾ (𝑆 × 𝑆))(𝐵 + (𝑟 / 2))) < 𝑟))
95	91, 93, 48, 65, 94	syl22anc 1229	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((𝐵 + (𝑟 / 2)) ∈ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ↔ (𝐵((abs ∘ − ) ↾ (𝑆 × 𝑆))(𝐵 + (𝑟 / 2))) < 𝑟))
96	90, 95	mpbird 166	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 + (𝑟 / 2)) ∈ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟))
97	46, 96	sseldd 3143	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 + (𝑟 / 2)) ∈ 𝐶)
98	53	rpap0d 9638	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑟 / 2) # 0)
99	67, 67	negsubdid 8224	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → -(𝐵 − 𝐵) = (-𝐵 + 𝐵))
100	74	negeqd 8093	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → -(𝐵 − 𝐵) = -0)
101		neg0 8144	. . . . . . . . . . . 12 ⊢ -0 = 0
102	100, 101	eqtrdi 2215	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → -(𝐵 − 𝐵) = 0)
103	99, 102	eqtr3d 2200	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (-𝐵 + 𝐵) = 0)
104	103	oveq1d 5857	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((-𝐵 + 𝐵) + (𝑟 / 2)) = (0 + (𝑟 / 2)))
105	67	negcld 8196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → -𝐵 ∈ ℂ)
106	105, 67, 68	addassd 7921	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((-𝐵 + 𝐵) + (𝑟 / 2)) = (-𝐵 + (𝐵 + (𝑟 / 2))))
107	68	addid2d 8048	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (0 + (𝑟 / 2)) = (𝑟 / 2))
108	104, 106, 107	3eqtr3d 2206	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (-𝐵 + (𝐵 + (𝑟 / 2))) = (𝑟 / 2))
109	98, 108, 103	3brtr4d 4014	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (-𝐵 + (𝐵 + (𝑟 / 2))) # (-𝐵 + 𝐵))
110		apadd2 8507	. . . . . . . 8 ⊢ (((𝐵 + (𝑟 / 2)) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ -𝐵 ∈ ℂ) → ((𝐵 + (𝑟 / 2)) # 𝐵 ↔ (-𝐵 + (𝐵 + (𝑟 / 2))) # (-𝐵 + 𝐵)))
111	69, 67, 105, 110	syl3anc 1228	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((𝐵 + (𝑟 / 2)) # 𝐵 ↔ (-𝐵 + (𝐵 + (𝑟 / 2))) # (-𝐵 + 𝐵)))
112	109, 111	mpbird 166	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 + (𝑟 / 2)) # 𝐵)
113	45, 97, 112	elrabd 2884	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 + (𝑟 / 2)) ∈ {𝑞 ∈ 𝐶 ∣ 𝑞 # 𝐵})
114		limcimo.ca	. . . . . . 7 ⊢ (𝜑 → {𝑞 ∈ 𝐶 ∣ 𝑞 # 𝐵} ⊆ 𝐴)
115	114	sseld 3141	. . . . . 6 ⊢ (𝜑 → ((𝐵 + (𝑟 / 2)) ∈ {𝑞 ∈ 𝐶 ∣ 𝑞 # 𝐵} → (𝐵 + (𝑟 / 2)) ∈ 𝐴))
116	115	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((𝐵 + (𝑟 / 2)) ∈ {𝑞 ∈ 𝐶 ∣ 𝑞 # 𝐵} → (𝐵 + (𝑟 / 2)) ∈ 𝐴))
117	113, 116	mpd 13	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 + (𝑟 / 2)) ∈ 𝐴)
118	44, 117	ffvelrnd 5621	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐹‘(𝐵 + (𝑟 / 2))) ∈ ℂ)
119	37, 42	subcld 8209	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑋 − 𝑌) ∈ ℂ)
120	119	abscld 11123	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘(𝑋 − 𝑌)) ∈ ℝ)
121	37, 118	abssubd 11135	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘(𝑋 − (𝐹‘(𝐵 + (𝑟 / 2))))) = (abs‘((𝐹‘(𝐵 + (𝑟 / 2))) − 𝑋)))
122	69, 67	subcld 8209	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((𝐵 + (𝑟 / 2)) − 𝐵) ∈ ℂ)
123	122	abscld 11123	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) ∈ ℝ)
124	52	rpred 9632	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝑟 ∈ ℝ)
125	22	rpred 9632	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℝ)
126	125	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝐷 ∈ ℝ)
127	67, 68	pncan2d 8211	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((𝐵 + (𝑟 / 2)) − 𝐵) = (𝑟 / 2))
128	127	fveq2d 5490	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) = (abs‘(𝑟 / 2)))
129	128, 85	eqtrd 2198	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) = (𝑟 / 2))
130	129, 89	eqbrtrd 4004	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝑟)
131	23	rpred 9632	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ ℝ)
132	131	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝐺 ∈ ℝ)
133		mincl 11172	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 𝐺 ∈ ℝ) → inf({𝐷, 𝐺}, ℝ, < ) ∈ ℝ)
134	126, 132, 133	syl2anc 409	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → inf({𝐷, 𝐺}, ℝ, < ) ∈ ℝ)
135		simprrl 529	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝑟 < inf({𝐷, 𝐺}, ℝ, < ))
136		min1inf 11173	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 𝐺 ∈ ℝ) → inf({𝐷, 𝐺}, ℝ, < ) ≤ 𝐷)
137	126, 132, 136	syl2anc 409	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → inf({𝐷, 𝐺}, ℝ, < ) ≤ 𝐷)
138	124, 134, 126, 135, 137	ltletrd 8321	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝑟 < 𝐷)
139	123, 124, 126, 130, 138	lttrd 8024	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐷)
140		breq1 3985	. . . . . . . 8 ⊢ (𝑧 = (𝐵 + (𝑟 / 2)) → (𝑧 # 𝐵 ↔ (𝐵 + (𝑟 / 2)) # 𝐵))
141		fvoveq1 5865	. . . . . . . . 9 ⊢ (𝑧 = (𝐵 + (𝑟 / 2)) → (abs‘(𝑧 − 𝐵)) = (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)))
142	141	breq1d 3992	. . . . . . . 8 ⊢ (𝑧 = (𝐵 + (𝑟 / 2)) → ((abs‘(𝑧 − 𝐵)) < 𝐷 ↔ (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐷))
143	140, 142	anbi12d 465	. . . . . . 7 ⊢ (𝑧 = (𝐵 + (𝑟 / 2)) → ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝐷) ↔ ((𝐵 + (𝑟 / 2)) # 𝐵 ∧ (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐷)))
144	143	imbrov2fvoveq 5867	. . . . . 6 ⊢ (𝑧 = (𝐵 + (𝑟 / 2)) → (((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝐷) → (abs‘((𝐹‘𝑧) − 𝑋)) < ((abs‘(𝑋 − 𝑌)) / 2)) ↔ (((𝐵 + (𝑟 / 2)) # 𝐵 ∧ (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐷) → (abs‘((𝐹‘(𝐵 + (𝑟 / 2))) − 𝑋)) < ((abs‘(𝑋 − 𝑌)) / 2))))
145		limcimo.z	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝐷) → (abs‘((𝐹‘𝑧) − 𝑋)) < ((abs‘(𝑋 − 𝑌)) / 2)))
146	145	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝐷) → (abs‘((𝐹‘𝑧) − 𝑋)) < ((abs‘(𝑋 − 𝑌)) / 2)))
147	144, 146, 117	rspcdva 2835	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (((𝐵 + (𝑟 / 2)) # 𝐵 ∧ (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐷) → (abs‘((𝐹‘(𝐵 + (𝑟 / 2))) − 𝑋)) < ((abs‘(𝑋 − 𝑌)) / 2)))
148	112, 139, 147	mp2and 430	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝐹‘(𝐵 + (𝑟 / 2))) − 𝑋)) < ((abs‘(𝑋 − 𝑌)) / 2))
149	121, 148	eqbrtrd 4004	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘(𝑋 − (𝐹‘(𝐵 + (𝑟 / 2))))) < ((abs‘(𝑋 − 𝑌)) / 2))
150		min2inf 11174	. . . . . . 7 ⊢ ((𝐷 ∈ ℝ ∧ 𝐺 ∈ ℝ) → inf({𝐷, 𝐺}, ℝ, < ) ≤ 𝐺)
151	126, 132, 150	syl2anc 409	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → inf({𝐷, 𝐺}, ℝ, < ) ≤ 𝐺)
152	124, 134, 132, 135, 151	ltletrd 8321	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝑟 < 𝐺)
153	123, 124, 132, 130, 152	lttrd 8024	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐺)
154		breq1 3985	. . . . . . 7 ⊢ (𝑤 = (𝐵 + (𝑟 / 2)) → (𝑤 # 𝐵 ↔ (𝐵 + (𝑟 / 2)) # 𝐵))
155		fvoveq1 5865	. . . . . . . 8 ⊢ (𝑤 = (𝐵 + (𝑟 / 2)) → (abs‘(𝑤 − 𝐵)) = (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)))
156	155	breq1d 3992	. . . . . . 7 ⊢ (𝑤 = (𝐵 + (𝑟 / 2)) → ((abs‘(𝑤 − 𝐵)) < 𝐺 ↔ (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐺))
157	154, 156	anbi12d 465	. . . . . 6 ⊢ (𝑤 = (𝐵 + (𝑟 / 2)) → ((𝑤 # 𝐵 ∧ (abs‘(𝑤 − 𝐵)) < 𝐺) ↔ ((𝐵 + (𝑟 / 2)) # 𝐵 ∧ (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐺)))
158	157	imbrov2fvoveq 5867	. . . . 5 ⊢ (𝑤 = (𝐵 + (𝑟 / 2)) → (((𝑤 # 𝐵 ∧ (abs‘(𝑤 − 𝐵)) < 𝐺) → (abs‘((𝐹‘𝑤) − 𝑌)) < ((abs‘(𝑋 − 𝑌)) / 2)) ↔ (((𝐵 + (𝑟 / 2)) # 𝐵 ∧ (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐺) → (abs‘((𝐹‘(𝐵 + (𝑟 / 2))) − 𝑌)) < ((abs‘(𝑋 − 𝑌)) / 2))))
159		limcimo.w	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ((𝑤 # 𝐵 ∧ (abs‘(𝑤 − 𝐵)) < 𝐺) → (abs‘((𝐹‘𝑤) − 𝑌)) < ((abs‘(𝑋 − 𝑌)) / 2)))
160	159	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ∀𝑤 ∈ 𝐴 ((𝑤 # 𝐵 ∧ (abs‘(𝑤 − 𝐵)) < 𝐺) → (abs‘((𝐹‘𝑤) − 𝑌)) < ((abs‘(𝑋 − 𝑌)) / 2)))
161	158, 160, 117	rspcdva 2835	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (((𝐵 + (𝑟 / 2)) # 𝐵 ∧ (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐺) → (abs‘((𝐹‘(𝐵 + (𝑟 / 2))) − 𝑌)) < ((abs‘(𝑋 − 𝑌)) / 2)))
162	112, 153, 161	mp2and 430	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝐹‘(𝐵 + (𝑟 / 2))) − 𝑌)) < ((abs‘(𝑋 − 𝑌)) / 2))
163	37, 42, 118, 120, 149, 162	abs3lemd 11143	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘(𝑋 − 𝑌)) < (abs‘(𝑋 − 𝑌)))
164	27, 163	rexlimddv 2588	1 ⊢ (𝜑 → (abs‘(𝑋 − 𝑌)) < (abs‘(𝑋 − 𝑌)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 ∀wral 2444 ∃wrex 2445 {crab 2448 ⊆ wss 3116 {cpr 3577 class class class wbr 3982 × cxp 4602 dom cdm 4604 ↾ cres 4606 ∘ ccom 4608 ⟶wf 5184 ‘cfv 5188 (class class class)co 5842 infcinf 6948 ℂcc 7751 ℝcr 7752 0cc0 7753 + caddc 7756 ℝ^*cxr 7932 < clt 7933 ≤ cle 7934 − cmin 8069 -cneg 8070 # cap 8479 / cdiv 8568 2c2 8908 ℝ⁺crp 9589 abscabs 10939 ↾_t crest 12556 ∞Metcxmet 12620 ballcbl 12622 MetOpencmopn 12625 lim_ℂ climc 13263

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-map 6616 df-pm 6617 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-rest 12558 df-topgen 12577 df-psmet 12627 df-xmet 12628 df-met 12629 df-bl 12630 df-mopn 12631 df-top 12636 df-topon 12649 df-bases 12681 df-limced 13265

This theorem is referenced by: limcimo 13274

W3C validator