Theorem cnrefiisplem 43260

Description: Lemma for cnrefiisp 43261 (some local definitions are used). (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
cnrefiisplem.a	⊢ (𝜑 → 𝐴 ∈ ℂ)
cnrefiisplem.n	⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)
cnrefiisplem.b	⊢ (𝜑 → 𝐵 ∈ Fin)
cnrefiisplem.c	⊢ 𝐶 = (ℝ ∪ 𝐵)
cnrefiisplem.d	⊢ 𝐷 = ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
cnrefiisplem.x	⊢ 𝑋 = inf(𝐷, ℝ*, < )

Assertion

Ref	Expression
cnrefiisplem	⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))

Distinct variable groups:   𝑦,𝐴,𝑥   𝑦,𝐵   𝑥,𝐶   𝑥,𝑋,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem cnrefiisplem

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 = (abs‘(ℑ‘𝐴)))
2		cnrefiisplem.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		cnrefiisplem.n	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)
4	2, 3	absimnre 42907	. . . . . . . 8 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
6	1, 5	eqeltrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
7	6	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐷) ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
8		simpll 763	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐷) ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → 𝜑)
9		cnrefiisplem.d	. . . . . . . . . . . 12 ⊢ 𝐷 = ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
10	9	eleq2i 2830	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐷 ↔ 𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}))
11	10	biimpi 215	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝐷 → 𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}))
12		nelsn 4598	. . . . . . . . . 10 ⊢ (𝑤 ≠ (abs‘(ℑ‘𝐴)) → ¬ 𝑤 ∈ {(abs‘(ℑ‘𝐴))})
13		elunnel1 4080	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}) ∧ ¬ 𝑤 ∈ {(abs‘(ℑ‘𝐴))}) → 𝑤 ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
14	11, 12, 13	syl2an 595	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐷 ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
15		eliun 4925	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ↔ ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦 − 𝐴))})
16	14, 15	sylib 217	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐷 ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦 − 𝐴))})
17		velsn 4574	. . . . . . . . 9 ⊢ (𝑤 ∈ {(abs‘(𝑦 − 𝐴))} ↔ 𝑤 = (abs‘(𝑦 − 𝐴)))
18	17	rexbii 3177	. . . . . . . 8 ⊢ (∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦 − 𝐴))} ↔ ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦 − 𝐴)))
19	16, 18	sylib 217	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐷 ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦 − 𝐴)))
20	19	adantll 710	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐷) ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦 − 𝐴)))
21		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑤 = (abs‘(𝑦 − 𝐴)))
22		eldifi 4057	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ (𝐵 ∩ ℂ))
23	22	elin2d 4129	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ ℂ)
24	23	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑦 ∈ ℂ)
25	2	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝐴 ∈ ℂ)
26	24, 25	subcld 11262	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (𝑦 − 𝐴) ∈ ℂ)
27		eldifsni 4720	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ≠ 𝐴)
28	27	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑦 ≠ 𝐴)
29	24, 25, 28	subne0d 11271	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (𝑦 − 𝐴) ≠ 0)
30	26, 29	absrpcld 15088	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (abs‘(𝑦 − 𝐴)) ∈ ℝ+)
31	21, 30	eqeltrd 2839	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑤 ∈ ℝ+)
32	31	rexlimdva2 3215	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦 − 𝐴)) → 𝑤 ∈ ℝ+))
33	8, 20, 32	sylc 65	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐷) ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
34	7, 33	pm2.61dane 3031	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ ℝ+)
35	34	ssd 42519	. . 3 ⊢ (𝜑 → 𝐷 ⊆ ℝ+)
36		cnrefiisplem.x	. . . 4 ⊢ 𝑋 = inf(𝐷, ℝ*, < )
37		xrltso 12804	. . . . . 6 ⊢ < Or ℝ*
38	37	a1i 11	. . . . 5 ⊢ (𝜑 → < Or ℝ*)
39		snfi 8788	. . . . . . . 8 ⊢ {(abs‘(ℑ‘𝐴))} ∈ Fin
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → {(abs‘(ℑ‘𝐴))} ∈ Fin)
41		cnrefiisplem.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin)
42		inss1 4159	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ℂ) ⊆ 𝐵
43	42	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ ℂ) ⊆ 𝐵)
44	43	ssdifssd 4073	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ⊆ 𝐵)
45	41, 44	ssfid 8971	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin)
46		snfi 8788	. . . . . . . . 9 ⊢ {(abs‘(𝑦 − 𝐴))} ∈ Fin
47	46	rgenw 3075	. . . . . . . 8 ⊢ ∀𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ∈ Fin
48		iunfi 9037	. . . . . . . 8 ⊢ ((((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin ∧ ∀𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ∈ Fin) → ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ∈ Fin)
49	45, 47, 48	sylancl 585	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ∈ Fin)
50	40, 49	unfid 42591	. . . . . 6 ⊢ (𝜑 → ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}) ∈ Fin)
51	9, 50	eqeltrid 2843	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Fin)
52		fvex 6769	. . . . . . . . . 10 ⊢ (abs‘(ℑ‘𝐴)) ∈ V
53	52	snid 4594	. . . . . . . . 9 ⊢ (abs‘(ℑ‘𝐴)) ∈ {(abs‘(ℑ‘𝐴))}
54		elun1 4106	. . . . . . . . 9 ⊢ ((abs‘(ℑ‘𝐴)) ∈ {(abs‘(ℑ‘𝐴))} → (abs‘(ℑ‘𝐴)) ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}))
55	53, 54	ax-mp 5	. . . . . . . 8 ⊢ (abs‘(ℑ‘𝐴)) ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
56	55, 9	eleqtrri 2838	. . . . . . 7 ⊢ (abs‘(ℑ‘𝐴)) ∈ 𝐷
57	56	a1i 11	. . . . . 6 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ 𝐷)
58	57	ne0d 4266	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ ∅)
59		rpssxr 42911	. . . . . 6 ⊢ ℝ+ ⊆ ℝ*
60	35, 59	sstrdi 3929	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ℝ*)
61		fiinfcl 9190	. . . . 5 ⊢ (( < Or ℝ* ∧ (𝐷 ∈ Fin ∧ 𝐷 ≠ ∅ ∧ 𝐷 ⊆ ℝ*)) → inf(𝐷, ℝ*, < ) ∈ 𝐷)
62	38, 51, 58, 60, 61	syl13anc 1370	. . . 4 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ 𝐷)
63	36, 62	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
64	35, 63	sseldd 3918	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ+)
65	35, 62	sseldd 3918	. . . . . . . . . 10 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ+)
66	65	rpred 12701	. . . . . . . . 9 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ)
67	66	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ∈ ℝ)
68	2	imcld 14834	. . . . . . . . . . 11 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ)
69	68	recnd 10934	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℂ)
70	69	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ)
71	70	abscld 15076	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
72		recn 10892	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
73	72	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
74	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐴 ∈ ℂ)
75	73, 74	subcld 11262	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 − 𝐴) ∈ ℂ)
76	75	abscld 15076	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(𝑦 − 𝐴)) ∈ ℝ)
77	60	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐷 ⊆ ℝ*)
78		infxrlb 12997	. . . . . . . . 9 ⊢ ((𝐷 ⊆ ℝ* ∧ (abs‘(ℑ‘𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
79	77, 56, 78	sylancl 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
80		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
81	74, 80	absimlere 42910	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝑦 − 𝐴)))
82	67, 71, 76, 79, 81	letrd 11062	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
83	36, 82	eqbrtrid 5105	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
84	83	ad4ant14 748	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
85		cnrefiisplem.c	. . . . . . . . 9 ⊢ 𝐶 = (ℝ ∪ 𝐵)
86	85	eleq2i 2830	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ (ℝ ∪ 𝐵))
87		elunnel1 4080	. . . . . . . 8 ⊢ ((𝑦 ∈ (ℝ ∪ 𝐵) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
88	86, 87	sylanb 580	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐶 ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
89	88	ad4ant24 750	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
90	60	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝐷 ⊆ ℝ*)
91		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
92		simpll 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ℂ)
93	91, 92	elind 4124	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐵 ∩ ℂ))
94		nelsn 4598	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ 𝐴 → ¬ 𝑦 ∈ {𝐴})
95	94	ad2antlr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦 ∈ {𝐴})
96	93, 95	eldifd 3894	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}))
97		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (abs‘(𝑦 − 𝐴)) ∈ V
98	97	snid 4594	. . . . . . . . . . . . . 14 ⊢ (abs‘(𝑦 − 𝐴)) ∈ {(abs‘(𝑦 − 𝐴))}
99		fvoveq1 7278	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → (abs‘(𝑤 − 𝐴)) = (abs‘(𝑦 − 𝐴)))
100	99	sneqd 4570	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → {(abs‘(𝑤 − 𝐴))} = {(abs‘(𝑦 − 𝐴))})
101	100	eliuni 4927	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) ∧ (abs‘(𝑦 − 𝐴)) ∈ {(abs‘(𝑦 − 𝐴))}) → (abs‘(𝑦 − 𝐴)) ∈ ∪ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤 − 𝐴))})
102	96, 98, 101	sylancl 585	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ ∪ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤 − 𝐴))})
103	100	cbviunv 4966	. . . . . . . . . . . . 13 ⊢ ∪ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤 − 𝐴))} = ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}
104	102, 103	eleqtrdi 2849	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
105		elun2 4107	. . . . . . . . . . . 12 ⊢ ((abs‘(𝑦 − 𝐴)) ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} → (abs‘(𝑦 − 𝐴)) ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}))
106	104, 105	syl 17	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}))
107	106, 9	eleqtrrdi 2850	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ 𝐷)
108	107	adantll 710	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ 𝐷)
109		infxrlb 12997	. . . . . . . . 9 ⊢ ((𝐷 ⊆ ℝ* ∧ (abs‘(𝑦 − 𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
110	90, 108, 109	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
111	36, 110	eqbrtrid 5105	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
112	111	adantllr 715	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
113	89, 112	syldan 590	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
114	84, 113	pm2.61dan 809	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
115	114	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴))))
116	115	ralrimiva 3107	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴))))
117		breq1 5073	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ (abs‘(𝑦 − 𝐴)) ↔ 𝑋 ≤ (abs‘(𝑦 − 𝐴))))
118	117	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))) ↔ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))))
119	118	ralbidv 3120	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))) ↔ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))))
120	119	rspcev 3552	. 2 ⊢ ((𝑋 ∈ ℝ+ ∧ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))
121	64, 116, 120	syl2anc 583	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ ciun 4921   class class class wbr 5070   Or wor 5493  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  infcinf 9130  ℂcc 10800  ℝcr 10801  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   − cmin 11135  ℝ⁺crp 12659  ℑcim 14737  abscabs 14873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875

This theorem is referenced by:  cnrefiisp  43261