Theorem cnrefiisplem 45130

Description: Lemma for cnrefiisp 45131 (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 44772	. . . . . . . 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 714	. . . . 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 215	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝐷 → 𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}))
12		nelsn 4664	. . . . . . . . . 10 ⊢ (𝑤 ≠ (abs‘(ℑ‘𝐴)) → ¬ 𝑤 ∈ {(abs‘(ℑ‘𝐴))})
13		elunnel1 4145	. . . . . . . . . 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 4995	. . . . . . . . 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 4640	. . . . . . . . 9 ⊢ (𝑤 ∈ {(abs‘(𝑦 − 𝐴))} ↔ 𝑤 = (abs‘(𝑦 − 𝐴)))
18	17	rexbii 3089	. . . . . . . 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 713	. . . . . 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 4122	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ (𝐵 ∩ ℂ))
23	22	elin2d 4195	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ ℂ)
24	23	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑦 ∈ ℂ)
25	2	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝐴 ∈ ℂ)
26	24, 25	subcld 11587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (𝑦 − 𝐴) ∈ ℂ)
27		eldifsni 4789	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ≠ 𝐴)
28	27	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑦 ≠ 𝐴)
29	24, 25, 28	subne0d 11596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (𝑦 − 𝐴) ≠ 0)
30	26, 29	absrpcld 15413	. . . . . . . 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 3152	. . . . . 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 3024	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ ℝ+)
35	34	ssd 44359	. . 3 ⊢ (𝜑 → 𝐷 ⊆ ℝ+)
36		cnrefiisplem.x	. . . 4 ⊢ 𝑋 = inf(𝐷, ℝ*, < )
37		xrltso 13138	. . . . . 6 ⊢ < Or ℝ*
38	37	a1i 11	. . . . 5 ⊢ (𝜑 → < Or ℝ*)
39		snfi 9058	. . . . . . . 8 ⊢ {(abs‘(ℑ‘𝐴))} ∈ Fin
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → {(abs‘(ℑ‘𝐴))} ∈ Fin)
41		cnrefiisplem.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin)
42		inss1 4224	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ℂ) ⊆ 𝐵
43	42	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ ℂ) ⊆ 𝐵)
44	43	ssdifssd 4138	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ⊆ 𝐵)
45	41, 44	ssfid 9281	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin)
46		snfi 9058	. . . . . . . . 9 ⊢ {(abs‘(𝑦 − 𝐴))} ∈ Fin
47	46	rgenw 3060	. . . . . . . 8 ⊢ ∀𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ∈ Fin
48		iunfi 9354	. . . . . . . 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 44432	. . . . . 6 ⊢ (𝜑 → ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}) ∈ Fin)
51	9, 50	eqeltrid 2832	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Fin)
52		fvex 6904	. . . . . . . . . 10 ⊢ (abs‘(ℑ‘𝐴)) ∈ V
53	52	snid 4660	. . . . . . . . 9 ⊢ (abs‘(ℑ‘𝐴)) ∈ {(abs‘(ℑ‘𝐴))}
54		elun1 4172	. . . . . . . . 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 4331	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ ∅)
59		rpssxr 44776	. . . . . 6 ⊢ ℝ+ ⊆ ℝ*
60	35, 59	sstrdi 3990	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ℝ*)
61		fiinfcl 9510	. . . . 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 2832	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
64	35, 63	sseldd 3979	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ+)
65	35, 62	sseldd 3979	. . . . . . . . . 10 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ+)
66	65	rpred 13034	. . . . . . . . 9 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ)
67	66	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ∈ ℝ)
68	2	imcld 15160	. . . . . . . . . . 11 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ)
69	68	recnd 11258	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℂ)
70	69	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ)
71	70	abscld 15401	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
72		recn 11214	. . . . . . . . . . 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 11587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 − 𝐴) ∈ ℂ)
76	75	abscld 15401	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(𝑦 − 𝐴)) ∈ ℝ)
77	60	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐷 ⊆ ℝ*)
78		infxrlb 13331	. . . . . . . . 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 44775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝑦 − 𝐴)))
82	67, 71, 76, 79, 81	letrd 11387	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
83	36, 82	eqbrtrid 5177	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
84	83	ad4ant14 751	. . . . 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 4145	. . . . . . . 8 ⊢ ((𝑦 ∈ (ℝ ∪ 𝐵) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
88	86, 87	sylanb 580	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐶 ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
89	88	ad4ant24 753	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
90	60	ad2antrr 725	. . . . . . . . 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 4190	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐵 ∩ ℂ))
94		nelsn 4664	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ 𝐴 → ¬ 𝑦 ∈ {𝐴})
95	94	ad2antlr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦 ∈ {𝐴})
96	93, 95	eldifd 3955	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}))
97		fvex 6904	. . . . . . . . . . . . . . 15 ⊢ (abs‘(𝑦 − 𝐴)) ∈ V
98	97	snid 4660	. . . . . . . . . . . . . 14 ⊢ (abs‘(𝑦 − 𝐴)) ∈ {(abs‘(𝑦 − 𝐴))}
99		fvoveq1 7437	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → (abs‘(𝑤 − 𝐴)) = (abs‘(𝑦 − 𝐴)))
100	99	sneqd 4636	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → {(abs‘(𝑤 − 𝐴))} = {(abs‘(𝑦 − 𝐴))})
101	100	eliuni 4997	. . . . . . . . . . . . . 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 5037	. . . . . . . . . . . . 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 4173	. . . . . . . . . . . 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 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ 𝐷)
109		infxrlb 13331	. . . . . . . . 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 5177	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
112	111	adantllr 718	. . . . . 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 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 3141	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴))))
117		breq1 5145	. . . . 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 3172	. . 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 3607	. 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 1534   ∈ wcel 2099   ≠ wne 2935  ∀wral 3056  ∃wrex 3065   ∖ cdif 3941   ∪ cun 3942   ∩ cin 3943   ⊆ wss 3944  ∅c0 4318  {csn 4624  ∪ ciun 4991   class class class wbr 5142   Or wor 5583  ‘cfv 6542  (class class class)co 7414  Fincfn 8953  infcinf 9450  ℂcc 11122  ℝcr 11123  ℝ^*cxr 11263   < clt 11264   ≤ cle 11265   − cmin 11460  ℝ⁺crp 12992  ℑcim 15063  abscabs 15199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9451  df-inf 9452  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-nn 12229  df-2 12291  df-3 12292  df-n0 12489  df-z 12575  df-uz 12839  df-rp 12993  df-seq 13985  df-exp 14045  df-cj 15064  df-re 15065  df-im 15066  df-sqrt 15200  df-abs 15201

This theorem is referenced by:  cnrefiisp  45131