unblimceq0lem - Mathbox for Asger C. Ipsen

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Theorem unblimceq0lem 35377

Description: Lemma for unblimceq0 35378. (Contributed by Asger C. Ipsen, 12-May-2021.)

**Hypotheses**
Ref	Expression
unblimceq0lem.0	⊢ (𝜑 → 𝑆 ⊆ ℂ)
unblimceq0lem.1	⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
unblimceq0lem.2	⊢ (𝜑 → 𝐴 ∈ ℂ)
unblimceq0lem.3	⊢ (𝜑 → ∀𝑏 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))))

**Assertion**
Ref	Expression
unblimceq0lem	⊢ (𝜑 → ∀𝑐 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑦 ∈ 𝑆 (𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑦))))

Distinct variable groups: 𝐴,𝑏,𝑑,𝑥 𝑦,𝐴,𝑑,𝑥 𝐹,𝑏,𝑑,𝑥 𝑦,𝐹 𝑆,𝑏,𝑑,𝑥 𝑦,𝑆 𝜑,𝑏,𝑐,𝑑,𝑥 𝜑,𝑦,𝑐 Allowed substitution hints: 𝐴(𝑐) 𝑆(𝑐) 𝐹(𝑐)

**Proof of Theorem unblimceq0lem**
Step	Hyp	Ref	Expression
1		breq1 5151	. . . . . . . 8 ⊢ (𝑏 = if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) → (𝑏 ≤ (abs‘(𝐹‘𝑥)) ↔ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))
2	1	anbi2d 629	. . . . . . 7 ⊢ (𝑏 = if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) → (((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))) ↔ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥)))))
3	2	rexbidv 3178	. . . . . 6 ⊢ (𝑏 = if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) → (∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))) ↔ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥)))))
4	3	ralbidv 3177	. . . . 5 ⊢ (𝑏 = if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) → (∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥)))))
5		unblimceq0lem.3	. . . . . 6 ⊢ (𝜑 → ∀𝑏 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))))
6	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → ∀𝑏 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥))))
7		unblimceq0lem.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
8	7	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → 𝐹:𝑆⟶ℂ)
9		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆)
10	8, 9	ffvelcdmd 7087	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) ∈ ℂ)
11	10	abscld 15382	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → (abs‘(𝐹‘𝐴)) ∈ ℝ)
12		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 𝑐 ∈ ℝ⁺)
13	12	rpred 13015	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 𝑐 ∈ ℝ)
14	13	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → 𝑐 ∈ ℝ)
15	11, 14	readdcld 11242	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → ((abs‘(𝐹‘𝐴)) + 𝑐) ∈ ℝ)
16	10	absge0d 15390	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → 0 ≤ (abs‘(𝐹‘𝐴)))
17	12	rpgt0d 13018	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 0 < 𝑐)
18	17	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → 0 < 𝑐)
19	11, 14, 16, 18	addgegt0d 11786	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → 0 < ((abs‘(𝐹‘𝐴)) + 𝑐))
20	15, 19	elrpd 13012	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → ((abs‘(𝐹‘𝐴)) + 𝑐) ∈ ℝ⁺)
21		simplrl 775	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ ¬ 𝐴 ∈ 𝑆) → 𝑐 ∈ ℝ⁺)
22	20, 21	ifclda 4563	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ∈ ℝ⁺)
23	4, 6, 22	rspcdva 3613	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → ∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))
24		simprr 771	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 𝑑 ∈ ℝ⁺)
25		rsp 3244	. . . 4 ⊢ (∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))) → (𝑑 ∈ ℝ⁺ → ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥)))))
26	23, 24, 25	sylc 65	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))
27		simprl 769	. . . 4 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) → 𝑥 ∈ 𝑆)
28		neeq1 3003	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≠ 𝐴 ↔ 𝑥 ≠ 𝐴))
29		fvoveq1 7431	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (abs‘(𝑦 − 𝐴)) = (abs‘(𝑥 − 𝐴)))
30	29	breq1d 5158	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((abs‘(𝑦 − 𝐴)) < 𝑑 ↔ (abs‘(𝑥 − 𝐴)) < 𝑑))
31		2fveq3 6896	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (abs‘(𝐹‘𝑦)) = (abs‘(𝐹‘𝑥)))
32	31	breq2d 5160	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑐 ≤ (abs‘(𝐹‘𝑦)) ↔ 𝑐 ≤ (abs‘(𝐹‘𝑥))))
33	28, 30, 32	3anbi123d 1436	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑦))) ↔ (𝑥 ≠ 𝐴 ∧ (abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑥)))))
34	33	adantl 482	. . . 4 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝑦 = 𝑥) → ((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑦))) ↔ (𝑥 ≠ 𝐴 ∧ (abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑥)))))
35	15	adantlr 713	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → ((abs‘(𝐹‘𝐴)) + 𝑐) ∈ ℝ)
36	7	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) → 𝐹:𝑆⟶ℂ)
37	36, 27	ffvelcdmd 7087	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) → (𝐹‘𝑥) ∈ ℂ)
38	37	abscld 15382	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
39	38	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
40		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆)
41	40	iftrued 4536	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) = ((abs‘(𝐹‘𝐴)) + 𝑐))
42	41	eqcomd 2738	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → ((abs‘(𝐹‘𝐴)) + 𝑐) = if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐))
43		simprrr 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) → if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥)))
44	43	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥)))
45	42, 44	eqbrtrd 5170	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → ((abs‘(𝐹‘𝐴)) + 𝑐) ≤ (abs‘(𝐹‘𝑥)))
46	35, 39, 45	lensymd 11364	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → ¬ (abs‘(𝐹‘𝑥)) < ((abs‘(𝐹‘𝐴)) + 𝑐))
47		2fveq3 6896	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘𝐴)))
48	47	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 = 𝐴) → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘𝐴)))
49	14, 11	ltaddposd 11797	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → (0 < 𝑐 ↔ (abs‘(𝐹‘𝐴)) < ((abs‘(𝐹‘𝐴)) + 𝑐)))
50	18, 49	mpbid 231	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → (abs‘(𝐹‘𝐴)) < ((abs‘(𝐹‘𝐴)) + 𝑐))
51	50	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 = 𝐴) → (abs‘(𝐹‘𝐴)) < ((abs‘(𝐹‘𝐴)) + 𝑐))
52	48, 51	eqbrtrd 5170	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) ∧ 𝑥 = 𝐴) → (abs‘(𝐹‘𝑥)) < ((abs‘(𝐹‘𝐴)) + 𝑐))
53	52	ex 413	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ 𝐴 ∈ 𝑆) → (𝑥 = 𝐴 → (abs‘(𝐹‘𝑥)) < ((abs‘(𝐹‘𝐴)) + 𝑐)))
54	53	adantlr 713	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → (𝑥 = 𝐴 → (abs‘(𝐹‘𝑥)) < ((abs‘(𝐹‘𝐴)) + 𝑐)))
55	54	necon3bd 2954	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → (¬ (abs‘(𝐹‘𝑥)) < ((abs‘(𝐹‘𝐴)) + 𝑐) → 𝑥 ≠ 𝐴))
56	46, 55	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → 𝑥 ≠ 𝐴)
57		simprrl 779	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) → (abs‘(𝑥 − 𝐴)) < 𝑑)
58	57	adantr 481	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → (abs‘(𝑥 − 𝐴)) < 𝑑)
59	14	adantlr 713	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → 𝑐 ∈ ℝ)
60	10	adantlr 713	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) ∈ ℂ)
61	60	absge0d 15390	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → 0 ≤ (abs‘(𝐹‘𝐴)))
62	11	adantlr 713	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → (abs‘(𝐹‘𝐴)) ∈ ℝ)
63	59, 62	addge02d 11802	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → (0 ≤ (abs‘(𝐹‘𝐴)) ↔ 𝑐 ≤ ((abs‘(𝐹‘𝐴)) + 𝑐)))
64	61, 63	mpbid 231	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → 𝑐 ≤ ((abs‘(𝐹‘𝐴)) + 𝑐))
65	59, 35, 39, 64, 45	letrd 11370	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → 𝑐 ≤ (abs‘(𝐹‘𝑥)))
66	56, 58, 65	3jca 1128	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ 𝐴 ∈ 𝑆) → (𝑥 ≠ 𝐴 ∧ (abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑥))))
67		simpr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → ¬ 𝐴 ∈ 𝑆)
68		simpr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
69	27	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → 𝑥 ∈ 𝑆)
70	69	adantr 481	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) ∧ 𝑥 = 𝐴) → 𝑥 ∈ 𝑆)
71	68, 70	eqeltrrd 2834	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝑆)
72	71	ex 413	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → (𝑥 = 𝐴 → 𝐴 ∈ 𝑆))
73	72	necon3bd 2954	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → (¬ 𝐴 ∈ 𝑆 → 𝑥 ≠ 𝐴))
74	67, 73	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → 𝑥 ≠ 𝐴)
75	57	adantr 481	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → (abs‘(𝑥 − 𝐴)) < 𝑑)
76	67	iffalsed 4539	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) = 𝑐)
77	76	eqcomd 2738	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → 𝑐 = if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐))
78	43	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥)))
79	77, 78	eqbrtrd 5170	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → 𝑐 ≤ (abs‘(𝐹‘𝑥)))
80	74, 75, 79	3jca 1128	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) ∧ ¬ 𝐴 ∈ 𝑆) → (𝑥 ≠ 𝐴 ∧ (abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑥))))
81	66, 80	pm2.61dan 811	. . . 4 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) → (𝑥 ≠ 𝐴 ∧ (abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑥))))
82	27, 34, 81	rspcedvd 3614	. . 3 ⊢ (((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑆 ∧ ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ if(𝐴 ∈ 𝑆, ((abs‘(𝐹‘𝐴)) + 𝑐), 𝑐) ≤ (abs‘(𝐹‘𝑥))))) → ∃𝑦 ∈ 𝑆 (𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑦))))
83	26, 82	rexlimddv 3161	. 2 ⊢ ((𝜑 ∧ (𝑐 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → ∃𝑦 ∈ 𝑆 (𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑦))))
84	83	ralrimivva 3200	1 ⊢ (𝜑 → ∀𝑐 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑦 ∈ 𝑆 (𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑦))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3948 ifcif 4528 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 ℝcr 11108 0cc0 11109 + caddc 11112 < clt 11247 ≤ cle 11248 − cmin 11443 ℝ⁺crp 12973 abscabs 15180

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182

This theorem is referenced by: unblimceq0 35378

W3C validator