Theorem cnrefiisplem 45814

Description: Lemma for cnrefiisp 45815 (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 45459	. . . . . . . 8 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
6	1, 5	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
7	6	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐷) ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
8		simpll 766	. . . . . 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 2820	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐷 ↔ 𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}))
11	10	biimpi 216	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝐷 → 𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}))
12		nelsn 4620	. . . . . . . . . 10 ⊢ (𝑤 ≠ (abs‘(ℑ‘𝐴)) → ¬ 𝑤 ∈ {(abs‘(ℑ‘𝐴))})
13		elunnel1 4107	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐷 ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
15		eliun 4948	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ↔ ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦 − 𝐴))})
16	14, 15	sylib 218	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐷 ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦 − 𝐴))})
17		velsn 4595	. . . . . . . . 9 ⊢ (𝑤 ∈ {(abs‘(𝑦 − 𝐴))} ↔ 𝑤 = (abs‘(𝑦 − 𝐴)))
18	17	rexbii 3076	. . . . . . . 8 ⊢ (∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦 − 𝐴))} ↔ ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦 − 𝐴)))
19	16, 18	sylib 218	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐷 ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦 − 𝐴)))
20	19	adantll 714	. . . . . 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 4084	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ (𝐵 ∩ ℂ))
23	22	elin2d 4158	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ ℂ)
24	23	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑦 ∈ ℂ)
25	2	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝐴 ∈ ℂ)
26	24, 25	subcld 11493	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (𝑦 − 𝐴) ∈ ℂ)
27		eldifsni 4744	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ≠ 𝐴)
28	27	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑦 ≠ 𝐴)
29	24, 25, 28	subne0d 11502	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (𝑦 − 𝐴) ≠ 0)
30	26, 29	absrpcld 15376	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (abs‘(𝑦 − 𝐴)) ∈ ℝ+)
31	21, 30	eqeltrd 2828	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑤 ∈ ℝ+)
32	31	rexlimdva2 3132	. . . . . 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 3012	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ ℝ+)
35	34	ssd 45061	. . 3 ⊢ (𝜑 → 𝐷 ⊆ ℝ+)
36		cnrefiisplem.x	. . . 4 ⊢ 𝑋 = inf(𝐷, ℝ*, < )
37		xrltso 13061	. . . . . 6 ⊢ < Or ℝ*
38	37	a1i 11	. . . . 5 ⊢ (𝜑 → < Or ℝ*)
39		snfi 8975	. . . . . . . 8 ⊢ {(abs‘(ℑ‘𝐴))} ∈ Fin
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → {(abs‘(ℑ‘𝐴))} ∈ Fin)
41		cnrefiisplem.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin)
42		inss1 4190	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ℂ) ⊆ 𝐵
43	42	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ ℂ) ⊆ 𝐵)
44	43	ssdifssd 4100	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ⊆ 𝐵)
45	41, 44	ssfid 9170	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin)
46		snfi 8975	. . . . . . . . 9 ⊢ {(abs‘(𝑦 − 𝐴))} ∈ Fin
47	46	rgenw 3048	. . . . . . . 8 ⊢ ∀𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ∈ Fin
48		iunfi 9252	. . . . . . . 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 586	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ∈ Fin)
50	40, 49	unfid 9096	. . . . . 6 ⊢ (𝜑 → ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}) ∈ Fin)
51	9, 50	eqeltrid 2832	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Fin)
52		fvex 6839	. . . . . . . . . 10 ⊢ (abs‘(ℑ‘𝐴)) ∈ V
53	52	snid 4616	. . . . . . . . 9 ⊢ (abs‘(ℑ‘𝐴)) ∈ {(abs‘(ℑ‘𝐴))}
54		elun1 4135	. . . . . . . . 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 2827	. . . . . . 7 ⊢ (abs‘(ℑ‘𝐴)) ∈ 𝐷
57	56	a1i 11	. . . . . 6 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ 𝐷)
58	57	ne0d 4295	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ ∅)
59		rpssxr 45463	. . . . . 6 ⊢ ℝ+ ⊆ ℝ*
60	35, 59	sstrdi 3950	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ℝ*)
61		fiinfcl 9412	. . . . 5 ⊢ (( < Or ℝ* ∧ (𝐷 ∈ Fin ∧ 𝐷 ≠ ∅ ∧ 𝐷 ⊆ ℝ*)) → inf(𝐷, ℝ*, < ) ∈ 𝐷)
62	38, 51, 58, 60, 61	syl13anc 1374	. . . 4 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ 𝐷)
63	36, 62	eqeltrid 2832	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
64	35, 63	sseldd 3938	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ+)
65	35, 62	sseldd 3938	. . . . . . . . . 10 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ+)
66	65	rpred 12955	. . . . . . . . 9 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ)
67	66	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ∈ ℝ)
68	2	imcld 15120	. . . . . . . . . . 11 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ)
69	68	recnd 11162	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℂ)
70	69	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ)
71	70	abscld 15364	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
72		recn 11118	. . . . . . . . . . 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 11493	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 − 𝐴) ∈ ℂ)
76	75	abscld 15364	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(𝑦 − 𝐴)) ∈ ℝ)
77	60	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐷 ⊆ ℝ*)
78		infxrlb 13255	. . . . . . . . 9 ⊢ ((𝐷 ⊆ ℝ* ∧ (abs‘(ℑ‘𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
79	77, 56, 78	sylancl 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
80		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
81	74, 80	absimlere 45462	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝑦 − 𝐴)))
82	67, 71, 76, 79, 81	letrd 11291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
83	36, 82	eqbrtrid 5130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
84	83	ad4ant14 752	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
85		cnrefiisplem.c	. . . . . . . . 9 ⊢ 𝐶 = (ℝ ∪ 𝐵)
86	85	eleq2i 2820	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ (ℝ ∪ 𝐵))
87		elunnel1 4107	. . . . . . . 8 ⊢ ((𝑦 ∈ (ℝ ∪ 𝐵) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
88	86, 87	sylanb 581	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐶 ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
89	88	ad4ant24 754	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
90	60	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝐷 ⊆ ℝ*)
91		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
92		simpll 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ℂ)
93	91, 92	elind 4153	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐵 ∩ ℂ))
94		nelsn 4620	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ 𝐴 → ¬ 𝑦 ∈ {𝐴})
95	94	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦 ∈ {𝐴})
96	93, 95	eldifd 3916	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}))
97		fvex 6839	. . . . . . . . . . . . . . 15 ⊢ (abs‘(𝑦 − 𝐴)) ∈ V
98	97	snid 4616	. . . . . . . . . . . . . 14 ⊢ (abs‘(𝑦 − 𝐴)) ∈ {(abs‘(𝑦 − 𝐴))}
99		fvoveq1 7376	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → (abs‘(𝑤 − 𝐴)) = (abs‘(𝑦 − 𝐴)))
100	99	sneqd 4591	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → {(abs‘(𝑤 − 𝐴))} = {(abs‘(𝑦 − 𝐴))})
101	100	eliuni 4950	. . . . . . . . . . . . . 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 586	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ ∪ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤 − 𝐴))})
103	100	cbviunv 4992	. . . . . . . . . . . . 13 ⊢ ∪ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤 − 𝐴))} = ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}
104	102, 103	eleqtrdi 2838	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
105		elun2 4136	. . . . . . . . . . . 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 2839	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ 𝐷)
108	107	adantll 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ 𝐷)
109		infxrlb 13255	. . . . . . . . 9 ⊢ ((𝐷 ⊆ ℝ* ∧ (abs‘(𝑦 − 𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
110	90, 108, 109	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
111	36, 110	eqbrtrid 5130	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
112	111	adantllr 719	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
113	89, 112	syldan 591	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
114	84, 113	pm2.61dan 812	. . . 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 3121	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴))))
117		breq1 5098	. . . . 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 3152	. . 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 3579	. 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 584	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  {csn 4579  ∪ ciun 4944   class class class wbr 5095   Or wor 5530  ‘cfv 6486  (class class class)co 7353  Fincfn 8879  infcinf 9350  ℂcc 11026  ℝcr 11027  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169   − cmin 11365  ℝ⁺crp 12911  ℑcim 15023  abscabs 15159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-z 12490  df-uz 12754  df-rp 12912  df-seq 13927  df-exp 13987  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161

This theorem is referenced by:  cnrefiisp  45815