Theorem cnrefiisplem 42130

Description: Lemma for cnrefiisp 42131 (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 487	. . . . . . 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 41773	. . . . . . . 8 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
5	4	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
6	1, 5	eqeltrd 2913	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
7	6	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐷) ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
8		simpll 765	. . . . . 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 2904	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐷 ↔ 𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}))
11	10	biimpi 218	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝐷 → 𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}))
12		nelsn 4605	. . . . . . . . . 10 ⊢ (𝑤 ≠ (abs‘(ℑ‘𝐴)) → ¬ 𝑤 ∈ {(abs‘(ℑ‘𝐴))})
13		elunnel1 4126	. . . . . . . . . 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 597	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐷 ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
15		eliun 4923	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ↔ ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦 − 𝐴))})
16	14, 15	sylib 220	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐷 ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦 − 𝐴))})
17		velsn 4583	. . . . . . . . 9 ⊢ (𝑤 ∈ {(abs‘(𝑦 − 𝐴))} ↔ 𝑤 = (abs‘(𝑦 − 𝐴)))
18	17	rexbii 3247	. . . . . . . 8 ⊢ (∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦 − 𝐴))} ↔ ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦 − 𝐴)))
19	16, 18	sylib 220	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐷 ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦 − 𝐴)))
20	19	adantll 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐷) ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦 − 𝐴)))
21		simpr 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑤 = (abs‘(𝑦 − 𝐴)))
22		eldifi 4103	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ (𝐵 ∩ ℂ))
23	22	elin2d 4176	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ ℂ)
24	23	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑦 ∈ ℂ)
25	2	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝐴 ∈ ℂ)
26	24, 25	subcld 10997	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (𝑦 − 𝐴) ∈ ℂ)
27		eldifsni 4722	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ≠ 𝐴)
28	27	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑦 ≠ 𝐴)
29	24, 25, 28	subne0d 11006	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (𝑦 − 𝐴) ≠ 0)
30	26, 29	absrpcld 14808	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (abs‘(𝑦 − 𝐴)) ∈ ℝ+)
31	21, 30	eqeltrd 2913	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑤 ∈ ℝ+)
32	31	rexlimdva2 3287	. . . . . 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 3104	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ ℝ+)
35	34	ssd 41364	. . 3 ⊢ (𝜑 → 𝐷 ⊆ ℝ+)
36		cnrefiisplem.x	. . . 4 ⊢ 𝑋 = inf(𝐷, ℝ*, < )
37		xrltso 12535	. . . . . 6 ⊢ < Or ℝ*
38	37	a1i 11	. . . . 5 ⊢ (𝜑 → < Or ℝ*)
39		snfi 8594	. . . . . . . 8 ⊢ {(abs‘(ℑ‘𝐴))} ∈ Fin
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → {(abs‘(ℑ‘𝐴))} ∈ Fin)
41		cnrefiisplem.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin)
42		inss1 4205	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ℂ) ⊆ 𝐵
43	42	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ ℂ) ⊆ 𝐵)
44	43	ssdifssd 4119	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ⊆ 𝐵)
45	41, 44	ssfid 8741	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin)
46		snfi 8594	. . . . . . . . 9 ⊢ {(abs‘(𝑦 − 𝐴))} ∈ Fin
47	46	rgenw 3150	. . . . . . . 8 ⊢ ∀𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ∈ Fin
48		iunfi 8812	. . . . . . . 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 588	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ∈ Fin)
50	40, 49	unfid 41442	. . . . . 6 ⊢ (𝜑 → ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}) ∈ Fin)
51	9, 50	eqeltrid 2917	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Fin)
52		fvex 6683	. . . . . . . . . 10 ⊢ (abs‘(ℑ‘𝐴)) ∈ V
53	52	snid 4601	. . . . . . . . 9 ⊢ (abs‘(ℑ‘𝐴)) ∈ {(abs‘(ℑ‘𝐴))}
54		elun1 4152	. . . . . . . . 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 2912	. . . . . . 7 ⊢ (abs‘(ℑ‘𝐴)) ∈ 𝐷
57	56	a1i 11	. . . . . 6 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ 𝐷)
58	57	ne0d 4301	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ ∅)
59		rpssxr 41777	. . . . . 6 ⊢ ℝ+ ⊆ ℝ*
60	35, 59	sstrdi 3979	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ℝ*)
61		fiinfcl 8965	. . . . 5 ⊢ (( < Or ℝ* ∧ (𝐷 ∈ Fin ∧ 𝐷 ≠ ∅ ∧ 𝐷 ⊆ ℝ*)) → inf(𝐷, ℝ*, < ) ∈ 𝐷)
62	38, 51, 58, 60, 61	syl13anc 1368	. . . 4 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ 𝐷)
63	36, 62	eqeltrid 2917	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
64	35, 63	sseldd 3968	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ+)
65	35, 62	sseldd 3968	. . . . . . . . . 10 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ+)
66	65	rpred 12432	. . . . . . . . 9 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ)
67	66	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ∈ ℝ)
68	2	imcld 14554	. . . . . . . . . . 11 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ)
69	68	recnd 10669	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℂ)
70	69	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ)
71	70	abscld 14796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
72		recn 10627	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
73	72	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
74	2	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐴 ∈ ℂ)
75	73, 74	subcld 10997	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 − 𝐴) ∈ ℂ)
76	75	abscld 14796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(𝑦 − 𝐴)) ∈ ℝ)
77	60	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐷 ⊆ ℝ*)
78		infxrlb 12728	. . . . . . . . 9 ⊢ ((𝐷 ⊆ ℝ* ∧ (abs‘(ℑ‘𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
79	77, 56, 78	sylancl 588	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
80		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
81	74, 80	absimlere 41776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝑦 − 𝐴)))
82	67, 71, 76, 79, 81	letrd 10797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
83	36, 82	eqbrtrid 5101	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
84	83	ad4ant14 750	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
85		cnrefiisplem.c	. . . . . . . . 9 ⊢ 𝐶 = (ℝ ∪ 𝐵)
86	85	eleq2i 2904	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ (ℝ ∪ 𝐵))
87		elunnel1 4126	. . . . . . . 8 ⊢ ((𝑦 ∈ (ℝ ∪ 𝐵) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
88	86, 87	sylanb 583	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐶 ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
89	88	ad4ant24 752	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ 𝐵)
90	60	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝐷 ⊆ ℝ*)
91		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
92		simpll 765	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ℂ)
93	91, 92	elind 4171	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐵 ∩ ℂ))
94		nelsn 4605	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ 𝐴 → ¬ 𝑦 ∈ {𝐴})
95	94	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦 ∈ {𝐴})
96	93, 95	eldifd 3947	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}))
97		fvex 6683	. . . . . . . . . . . . . . 15 ⊢ (abs‘(𝑦 − 𝐴)) ∈ V
98	97	snid 4601	. . . . . . . . . . . . . 14 ⊢ (abs‘(𝑦 − 𝐴)) ∈ {(abs‘(𝑦 − 𝐴))}
99		fvoveq1 7179	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → (abs‘(𝑤 − 𝐴)) = (abs‘(𝑦 − 𝐴)))
100	99	sneqd 4579	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → {(abs‘(𝑤 − 𝐴))} = {(abs‘(𝑦 − 𝐴))})
101	100	eliuni 4925	. . . . . . . . . . . . . 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 588	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ ∪ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤 − 𝐴))})
103	100	cbviunv 4965	. . . . . . . . . . . . 13 ⊢ ∪ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤 − 𝐴))} = ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}
104	102, 103	eleqtrdi 2923	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
105		elun2 4153	. . . . . . . . . . . 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 2924	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ 𝐷)
108	107	adantll 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ 𝐷)
109		infxrlb 12728	. . . . . . . . 9 ⊢ ((𝐷 ⊆ ℝ* ∧ (abs‘(𝑦 − 𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
110	90, 108, 109	syl2anc 586	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
111	36, 110	eqbrtrid 5101	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
112	111	adantllr 717	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
113	89, 112	syldan 593	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
114	84, 113	pm2.61dan 811	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴)) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))
115	114	ex 415	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴))))
116	115	ralrimiva 3182	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴))))
117		breq1 5069	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ (abs‘(𝑦 − 𝐴)) ↔ 𝑋 ≤ (abs‘(𝑦 − 𝐴))))
118	117	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))) ↔ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴)))))
119	118	ralbidv 3197	. . 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 3623	. 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 586	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  {csn 4567  ∪ ciun 4919   class class class wbr 5066   Or wor 5473  ‘cfv 6355  (class class class)co 7156  Fincfn 8509  infcinf 8905  ℂcc 10535  ℝcr 10536  ℝ^*cxr 10674   < clt 10675   ≤ cle 10676   − cmin 10870  ℝ⁺crp 12390  ℑcim 14457  abscabs 14593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-inf 8907  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-n0 11899  df-z 11983  df-uz 12245  df-rp 12391  df-seq 13371  df-exp 13431  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595

This theorem is referenced by:  cnrefiisp  42131