Theorem cnrefiisplem 45844

Description: Lemma for cnrefiisp 45845 (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 45487	. . . . . . . 8 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
6	1, 5	eqeltrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
7	6	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐷) ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
8		simpll 767	. . . . . 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 2833	. . . . . . . . . . 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 4666	. . . . . . . . . 10 ⊢ (𝑤 ≠ (abs‘(ℑ‘𝐴)) → ¬ 𝑤 ∈ {(abs‘(ℑ‘𝐴))})
13		elunnel1 4154	. . . . . . . . . 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 4995	. . . . . . . . 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 4642	. . . . . . . . 9 ⊢ (𝑤 ∈ {(abs‘(𝑦 − 𝐴))} ↔ 𝑤 = (abs‘(𝑦 − 𝐴)))
18	17	rexbii 3094	. . . . . . . 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 4131	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ (𝐵 ∩ ℂ))
23	22	elin2d 4205	. . . . . . . . . . 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 11620	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (𝑦 − 𝐴) ∈ ℂ)
27		eldifsni 4790	. . . . . . . . . . 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 11629	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (𝑦 − 𝐴) ≠ 0)
30	26, 29	absrpcld 15487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → (abs‘(𝑦 − 𝐴)) ∈ ℝ+)
31	21, 30	eqeltrd 2841	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦 − 𝐴))) → 𝑤 ∈ ℝ+)
32	31	rexlimdva2 3157	. . . . . 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 3029	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ ℝ+)
35	34	ssd 45085	. . 3 ⊢ (𝜑 → 𝐷 ⊆ ℝ+)
36		cnrefiisplem.x	. . . 4 ⊢ 𝑋 = inf(𝐷, ℝ*, < )
37		xrltso 13183	. . . . . 6 ⊢ < Or ℝ*
38	37	a1i 11	. . . . 5 ⊢ (𝜑 → < Or ℝ*)
39		snfi 9083	. . . . . . . 8 ⊢ {(abs‘(ℑ‘𝐴))} ∈ Fin
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → {(abs‘(ℑ‘𝐴))} ∈ Fin)
41		cnrefiisplem.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin)
42		inss1 4237	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ℂ) ⊆ 𝐵
43	42	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ ℂ) ⊆ 𝐵)
44	43	ssdifssd 4147	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ⊆ 𝐵)
45	41, 44	ssfid 9301	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin)
46		snfi 9083	. . . . . . . . 9 ⊢ {(abs‘(𝑦 − 𝐴))} ∈ Fin
47	46	rgenw 3065	. . . . . . . 8 ⊢ ∀𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))} ∈ Fin
48		iunfi 9383	. . . . . . . 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 9212	. . . . . 6 ⊢ (𝜑 → ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}) ∈ Fin)
51	9, 50	eqeltrid 2845	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Fin)
52		fvex 6919	. . . . . . . . . 10 ⊢ (abs‘(ℑ‘𝐴)) ∈ V
53	52	snid 4662	. . . . . . . . 9 ⊢ (abs‘(ℑ‘𝐴)) ∈ {(abs‘(ℑ‘𝐴))}
54		elun1 4182	. . . . . . . . 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 2840	. . . . . . 7 ⊢ (abs‘(ℑ‘𝐴)) ∈ 𝐷
57	56	a1i 11	. . . . . 6 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ 𝐷)
58	57	ne0d 4342	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ ∅)
59		rpssxr 45491	. . . . . 6 ⊢ ℝ+ ⊆ ℝ*
60	35, 59	sstrdi 3996	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ℝ*)
61		fiinfcl 9541	. . . . 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 2845	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
64	35, 63	sseldd 3984	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ+)
65	35, 62	sseldd 3984	. . . . . . . . . 10 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ+)
66	65	rpred 13077	. . . . . . . . 9 ⊢ (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ)
67	66	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ∈ ℝ)
68	2	imcld 15234	. . . . . . . . . . 11 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ)
69	68	recnd 11289	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℂ)
70	69	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ)
71	70	abscld 15475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
72		recn 11245	. . . . . . . . . . 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 11620	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 − 𝐴) ∈ ℂ)
76	75	abscld 15475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(𝑦 − 𝐴)) ∈ ℝ)
77	60	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐷 ⊆ ℝ*)
78		infxrlb 13376	. . . . . . . . 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 45490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝑦 − 𝐴)))
82	67, 71, 76, 79, 81	letrd 11418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦 − 𝐴)))
83	36, 82	eqbrtrid 5178	. . . . . 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 2833	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ (ℝ ∪ 𝐵))
87		elunnel1 4154	. . . . . . . 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 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ℂ)
93	91, 92	elind 4200	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐵 ∩ ℂ))
94		nelsn 4666	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ≠ 𝐴 → ¬ 𝑦 ∈ {𝐴})
95	94	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦 ∈ {𝐴})
96	93, 95	eldifd 3962	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}))
97		fvex 6919	. . . . . . . . . . . . . . 15 ⊢ (abs‘(𝑦 − 𝐴)) ∈ V
98	97	snid 4662	. . . . . . . . . . . . . 14 ⊢ (abs‘(𝑦 − 𝐴)) ∈ {(abs‘(𝑦 − 𝐴))}
99		fvoveq1 7454	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → (abs‘(𝑤 − 𝐴)) = (abs‘(𝑦 − 𝐴)))
100	99	sneqd 4638	. . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ ∪ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤 − 𝐴))})
103	100	cbviunv 5040	. . . . . . . . . . . . 13 ⊢ ∪ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤 − 𝐴))} = ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}
104	102, 103	eleqtrdi 2851	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ∧ 𝑦 ∈ 𝐵) → (abs‘(𝑦 − 𝐴)) ∈ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))})
105		elun2 4183	. . . . . . . . . . . 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 2852	. . . . . . . . . 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 13376	. . . . . . . . 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 5178	. . . . . . 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 813	. . . 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 3146	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑋 ≤ (abs‘(𝑦 − 𝐴))))
117		breq1 5146	. . . . 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 3178	. . 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 3622	. 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 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626  ∪ ciun 4991   class class class wbr 5143   Or wor 5591  ‘cfv 6561  (class class class)co 7431  Fincfn 8985  infcinf 9481  ℂcc 11153  ℝcr 11154  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296   − cmin 11492  ℝ⁺crp 13034  ℑcim 15137  abscabs 15273

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275

This theorem is referenced by:  cnrefiisp  45845