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Theorem limcimolemlt 13174

Description: Lemma for limcimo 13175. (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 13072	. . . 4 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
2		ax-resscn 7836	. . . . . . 7 ⊢ ℝ ⊆ ℂ
3		sseq1 3160	. . . . . . 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 3191	. . . . . 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 3593	. . . . . 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 12920	. . . 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 2164	. . . . . 6 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
16		limcflfcntop.k	. . . . . 6 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
17		eqid 2164	. . . . . 6 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
18	15, 16, 17	metrest 13047	. . . . 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 2243	. . 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 11165	. . . 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 13025	. . 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 1230	. 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 13168	. . . . . . . . 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 998	. . . . . . 7 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
32	30	simp2d 999	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ)
33		limcimo.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
34	31, 32, 33	ellimc3ap 13171	. . . . . 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 13171	. . . . . 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 3979	. . . . . 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 2243	. . . . . . . . . . . . . 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 9636	. . . . . . . . . . . . . . . 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 9623	. . . . . . . . . . . . . 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 7919	. . . . . . . . . . . . 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 2244	. . . . . . . . . . . 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 9625	. . . . . . . . . . . . . 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 7909	. . . . . . . . . . . . 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 2244	. . . . . . . . . . . 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 5973	. . . . . . . . . 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 9625	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑟 / 2) ∈ ℂ)
69	67, 68	addcld 7909	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 + (𝑟 / 2)) ∈ ℂ)
70		eqid 2164	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (abs ∘ − )
71	70	cnmetdval 13070	. . . . . . . . . . 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 8231	. . . . . . . . . . . . . 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 8188	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 − 𝐵) = 0)
75	74	oveq1d 5851	. . . . . . . . . . . . . 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 2199	. . . . . . . . . . . . 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 5484	. . . . . . . . . . . 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 7883	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 0 ∈ ℂ)
79	78, 68	abssubd 11121	. . . . . . . . . . . 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 2197	. . . . . . . . . . 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 8189	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((𝑟 / 2) − 0) = (𝑟 / 2))
82	81	fveq2d 5484	. . . . . . . . . . 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 9623	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑟 / 2) ∈ ℝ)
84	53	rpge0d 9627	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 0 ≤ (𝑟 / 2))
85	83, 84	absidd 11095	. . . . . . . . . . 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 2201	. . . . . . . . . 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 2201	. . . . . . . . 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 9610	. . . . . . . . . 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 3998	. . . . . . . 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 9588	. . . . . . . . . 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 12934	. . . . . . . . 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 1228	. . . . . . . 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 3138	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝐵 + (𝑟 / 2)) ∈ 𝐶)
98	53	rpap0d 9629	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑟 / 2) # 0)
99	67, 67	negsubdid 8215	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → -(𝐵 − 𝐵) = (-𝐵 + 𝐵))
100	74	negeqd 8084	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → -(𝐵 − 𝐵) = -0)
101		neg0 8135	. . . . . . . . . . . 12 ⊢ -0 = 0
102	100, 101	eqtrdi 2213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → -(𝐵 − 𝐵) = 0)
103	99, 102	eqtr3d 2199	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (-𝐵 + 𝐵) = 0)
104	103	oveq1d 5851	. . . . . . . . 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 8187	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → -𝐵 ∈ ℂ)
106	105, 67, 68	addassd 7912	. . . . . . . . 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 8039	. . . . . . . . 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 2205	. . . . . . . 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 4008	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (-𝐵 + (𝐵 + (𝑟 / 2))) # (-𝐵 + 𝐵))
110		apadd2 8498	. . . . . . . 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 1227	. . . . . . 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 2879	. . . . 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 3136	. . . . . 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 5615	. . 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 8200	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (𝑋 − 𝑌) ∈ ℂ)
120	119	abscld 11109	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘(𝑋 − 𝑌)) ∈ ℝ)
121	37, 118	abssubd 11121	. . . 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 8200	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → ((𝐵 + (𝑟 / 2)) − 𝐵) ∈ ℂ)
123	122	abscld 11109	. . . . . 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 9623	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → 𝑟 ∈ ℝ)
125	22	rpred 9623	. . . . . . 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 8202	. . . . . . . . 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 5484	. . . . . . . 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 2197	. . . . . . 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 3998	. . . . . 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 9623	. . . . . . . . 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 11158	. . . . . . . 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 11159	. . . . . . . 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 8312	. . . . . 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 8015	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐷)
140		breq1 3979	. . . . . . . 8 ⊢ (𝑧 = (𝐵 + (𝑟 / 2)) → (𝑧 # 𝐵 ↔ (𝐵 + (𝑟 / 2)) # 𝐵))
141		fvoveq1 5859	. . . . . . . . 9 ⊢ (𝑧 = (𝐵 + (𝑟 / 2)) → (abs‘(𝑧 − 𝐵)) = (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)))
142	141	breq1d 3986	. . . . . . . 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 5861	. . . . . 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 2830	. . . . 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 3998	. . 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 11160	. . . . . . 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 8312	. . . . 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 8015	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑟 < inf({𝐷, 𝐺}, ℝ, < ) ∧ (𝐵(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑟) ⊆ 𝐶))) → (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)) < 𝐺)
154		breq1 3979	. . . . . . 7 ⊢ (𝑤 = (𝐵 + (𝑟 / 2)) → (𝑤 # 𝐵 ↔ (𝐵 + (𝑟 / 2)) # 𝐵))
155		fvoveq1 5859	. . . . . . . 8 ⊢ (𝑤 = (𝐵 + (𝑟 / 2)) → (abs‘(𝑤 − 𝐵)) = (abs‘((𝐵 + (𝑟 / 2)) − 𝐵)))
156	155	breq1d 3986	. . . . . . 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 5861	. . . . 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 2830	. . . 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 11129	. 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 2586	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 967 = wceq 1342 ∈ wcel 2135 ∀wral 2442 ∃wrex 2443 {crab 2446 ⊆ wss 3111 {cpr 3571 class class class wbr 3976 × cxp 4596 dom cdm 4598 ↾ cres 4600 ∘ ccom 4602 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 infcinf 6939 ℂcc 7742 ℝcr 7743 0cc0 7744 + caddc 7747 ℝ^*cxr 7923 < clt 7924 ≤ cle 7925 − cmin 8060 -cneg 8061 # cap 8470 / cdiv 8559 2c2 8899 ℝ⁺crp 9580 abscabs 10925 ↾_t crest 12492 ∞Metcxmet 12521 ballcbl 12523 MetOpencmopn 12526 lim_ℂ climc 13164

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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-map 6607 df-pm 6608 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-xneg 9699 df-xadd 9700 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-rest 12494 df-topgen 12513 df-psmet 12528 df-xmet 12529 df-met 12530 df-bl 12531 df-mopn 12532 df-top 12537 df-topon 12550 df-bases 12582 df-limced 13166

This theorem is referenced by: limcimo 13175

W3C validator