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Theorem aalioulem2 26299

Description: Lemma for aaliou 26304. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Proof shortened by AV, 28-Sep-2020.)

**Hypotheses**
Ref	Expression
aalioulem2.a	⊢ 𝑁 = (deg‘𝐹)
aalioulem2.b	⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))
aalioulem2.c	⊢ (𝜑 → 𝑁 ∈ ℕ)
aalioulem2.d	⊢ (𝜑 → 𝐴 ∈ ℝ)

**Assertion**
Ref	Expression
aalioulem2	⊢ (𝜑 → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))

Distinct variable groups: 𝜑,𝑥,𝑝,𝑞 𝑥,𝐴,𝑝,𝑞 𝑥,𝐹,𝑝,𝑞 Allowed substitution hints: 𝑁(𝑥,𝑞,𝑝)

Proof of Theorem aalioulem2 Dummy variables 𝑟 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1rp 12946	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
2		snssi 4729	. . . . . . 7 ⊢ (1 ∈ ℝ⁺ → {1} ⊆ ℝ⁺)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ {1} ⊆ ℝ⁺
4		ssrab2 4020	. . . . . 6 ⊢ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ⊆ ℝ⁺
5	3, 4	unssi 4131	. . . . 5 ⊢ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ⊆ ℝ⁺
6		ltso 11226	. . . . . . 7 ⊢ < Or ℝ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → < Or ℝ)
8		snfi 8990	. . . . . . 7 ⊢ {1} ∈ Fin
9		aalioulem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))
10		aalioulem2.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	10	nnne0d 12227	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ≠ 0)
12		aalioulem2.a	. . . . . . . . . . . . . 14 ⊢ 𝑁 = (deg‘𝐹)
13	12	eqcomi 2745	. . . . . . . . . . . . 13 ⊢ (deg‘𝐹) = 𝑁
14		dgr0 26227	. . . . . . . . . . . . 13 ⊢ (deg‘0_𝑝) = 0
15	11, 13, 14	3netr4g 3011	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐹) ≠ (deg‘0_𝑝))
16		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
17	16	necon3i 2964	. . . . . . . . . . . 12 ⊢ ((deg‘𝐹) ≠ (deg‘0_𝑝) → 𝐹 ≠ 0_𝑝)
18	15, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ≠ 0_𝑝)
19		eqid 2736	. . . . . . . . . . . 12 ⊢ (^◡𝐹 “ {0}) = (^◡𝐹 “ {0})
20	19	fta1 26274	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℤ) ∧ 𝐹 ≠ 0_𝑝) → ((^◡𝐹 “ {0}) ∈ Fin ∧ (♯‘(^◡𝐹 “ {0})) ≤ (deg‘𝐹)))
21	9, 18, 20	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((^◡𝐹 “ {0}) ∈ Fin ∧ (♯‘(^◡𝐹 “ {0})) ≤ (deg‘𝐹)))
22	21	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 “ {0}) ∈ Fin)
23		abrexfi 9262	. . . . . . . . 9 ⊢ ((^◡𝐹 “ {0}) ∈ Fin → {𝑎 ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ∈ Fin)
24	22, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → {𝑎 ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ∈ Fin)
25		rabssab 4025	. . . . . . . 8 ⊢ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ⊆ {𝑎 ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}
26		ssfi 9107	. . . . . . . 8 ⊢ (({𝑎 ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ∈ Fin ∧ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ⊆ {𝑎 ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) → {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ∈ Fin)
27	24, 25, 26	sylancl 587	. . . . . . 7 ⊢ (𝜑 → {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ∈ Fin)
28		unfi 9105	. . . . . . 7 ⊢ (({1} ∈ Fin ∧ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ∈ Fin) → ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ∈ Fin)
29	8, 27, 28	sylancr 588	. . . . . 6 ⊢ (𝜑 → ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ∈ Fin)
30		1ex 11140	. . . . . . . . 9 ⊢ 1 ∈ V
31	30	snid 4606	. . . . . . . 8 ⊢ 1 ∈ {1}
32		elun1 4122	. . . . . . . 8 ⊢ (1 ∈ {1} → 1 ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}))
33		ne0i 4281	. . . . . . . 8 ⊢ (1 ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) → ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ≠ ∅)
34	31, 32, 33	mp2b 10	. . . . . . 7 ⊢ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ≠ ∅
35	34	a1i 11	. . . . . 6 ⊢ (𝜑 → ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ≠ ∅)
36		rpssre 12950	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
37	5, 36	sstri 3931	. . . . . . 7 ⊢ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ⊆ ℝ
38	37	a1i 11	. . . . . 6 ⊢ (𝜑 → ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ⊆ ℝ)
39		fiinfcl 9416	. . . . . 6 ⊢ (( < Or ℝ ∧ (({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ∈ Fin ∧ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ≠ ∅ ∧ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ⊆ ℝ)) → inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}))
40	7, 29, 35, 38, 39	syl13anc 1375	. . . . 5 ⊢ (𝜑 → inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}))
41	5, 40	sselid 3919	. . . 4 ⊢ (𝜑 → inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ∈ ℝ⁺)
42		0re 11146	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
43		rpge0 12956	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℝ⁺ → 0 ≤ 𝑑)
44	43	rgen 3053	. . . . . . . . . . 11 ⊢ ∀𝑑 ∈ ℝ⁺ 0 ≤ 𝑑
45		breq1 5088	. . . . . . . . . . . . 13 ⊢ (𝑐 = 0 → (𝑐 ≤ 𝑑 ↔ 0 ≤ 𝑑))
46	45	ralbidv 3160	. . . . . . . . . . . 12 ⊢ (𝑐 = 0 → (∀𝑑 ∈ ℝ⁺ 𝑐 ≤ 𝑑 ↔ ∀𝑑 ∈ ℝ⁺ 0 ≤ 𝑑))
47	46	rspcev 3564	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ ∀𝑑 ∈ ℝ⁺ 0 ≤ 𝑑) → ∃𝑐 ∈ ℝ ∀𝑑 ∈ ℝ⁺ 𝑐 ≤ 𝑑)
48	42, 44, 47	mp2an 693	. . . . . . . . . 10 ⊢ ∃𝑐 ∈ ℝ ∀𝑑 ∈ ℝ⁺ 𝑐 ≤ 𝑑
49		ssralv 3990	. . . . . . . . . . . 12 ⊢ (({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ⊆ ℝ⁺ → (∀𝑑 ∈ ℝ⁺ 𝑐 ≤ 𝑑 → ∀𝑑 ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))})𝑐 ≤ 𝑑))
50	5, 49	ax-mp 5	. . . . . . . . . . 11 ⊢ (∀𝑑 ∈ ℝ⁺ 𝑐 ≤ 𝑑 → ∀𝑑 ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))})𝑐 ≤ 𝑑)
51	50	reximi 3075	. . . . . . . . . 10 ⊢ (∃𝑐 ∈ ℝ ∀𝑑 ∈ ℝ⁺ 𝑐 ≤ 𝑑 → ∃𝑐 ∈ ℝ ∀𝑑 ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))})𝑐 ≤ 𝑑)
52	48, 51	ax-mp 5	. . . . . . . . 9 ⊢ ∃𝑐 ∈ ℝ ∀𝑑 ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))})𝑐 ≤ 𝑑
53		eqeq1 2740	. . . . . . . . . . . 12 ⊢ (𝑎 = (abs‘(𝐴 − 𝑟)) → (𝑎 = (abs‘(𝐴 − 𝑏)) ↔ (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑏))))
54	53	rexbidv 3161	. . . . . . . . . . 11 ⊢ (𝑎 = (abs‘(𝐴 − 𝑟)) → (∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏)) ↔ ∃𝑏 ∈ (^◡𝐹 “ {0})(abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑏))))
55		aalioulem2.d	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ)
56	55	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → 𝐴 ∈ ℝ)
57		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → 𝑟 ∈ ℝ)
58	56, 57	resubcld 11578	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (𝐴 − 𝑟) ∈ ℝ)
59	58	recnd 11173	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (𝐴 − 𝑟) ∈ ℂ)
60	55	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → 𝐴 ∈ ℝ)
61	60	recnd 11173	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → 𝐴 ∈ ℂ)
62		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → 𝑟 ∈ ℝ)
63	62	recnd 11173	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → 𝑟 ∈ ℂ)
64	61, 63	subeq0ad 11515	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → ((𝐴 − 𝑟) = 0 ↔ 𝐴 = 𝑟))
65	64	necon3abid 2968	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → ((𝐴 − 𝑟) ≠ 0 ↔ ¬ 𝐴 = 𝑟))
66	65	biimprd 248	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → (¬ 𝐴 = 𝑟 → (𝐴 − 𝑟) ≠ 0))
67	66	impr 454	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (𝐴 − 𝑟) ≠ 0)
68	59, 67	absrpcld 15413	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (abs‘(𝐴 − 𝑟)) ∈ ℝ⁺)
69	57	recnd 11173	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → 𝑟 ∈ ℂ)
70		simprl 771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (𝐹‘𝑟) = 0)
71		plyf 26163	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (Poly‘ℤ) → 𝐹:ℂ⟶ℂ)
72	9, 71	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
73	72	ffnd 6669	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn ℂ)
74	73	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → 𝐹 Fn ℂ)
75		fniniseg 7012	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn ℂ → (𝑟 ∈ (^◡𝐹 “ {0}) ↔ (𝑟 ∈ ℂ ∧ (𝐹‘𝑟) = 0)))
76	74, 75	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (𝑟 ∈ (^◡𝐹 “ {0}) ↔ (𝑟 ∈ ℂ ∧ (𝐹‘𝑟) = 0)))
77	69, 70, 76	mpbir2and 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → 𝑟 ∈ (^◡𝐹 “ {0}))
78		eqid 2736	. . . . . . . . . . . 12 ⊢ (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑟))
79		oveq2 7375	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑟 → (𝐴 − 𝑏) = (𝐴 − 𝑟))
80	79	fveq2d 6844	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑟 → (abs‘(𝐴 − 𝑏)) = (abs‘(𝐴 − 𝑟)))
81	80	rspceeqv 3587	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ (^◡𝐹 “ {0}) ∧ (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑟))) → ∃𝑏 ∈ (^◡𝐹 “ {0})(abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑏)))
82	77, 78, 81	sylancl 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → ∃𝑏 ∈ (^◡𝐹 “ {0})(abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑏)))
83	54, 68, 82	elrabd 3636	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (abs‘(𝐴 − 𝑟)) ∈ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))})
84		elun2 4123	. . . . . . . . . 10 ⊢ ((abs‘(𝐴 − 𝑟)) ∈ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} → (abs‘(𝐴 − 𝑟)) ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}))
85	83, 84	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (abs‘(𝐴 − 𝑟)) ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}))
86		infrelb 12141	. . . . . . . . 9 ⊢ ((({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ⊆ ℝ ∧ ∃𝑐 ∈ ℝ ∀𝑑 ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))})𝑐 ≤ 𝑑 ∧ (abs‘(𝐴 − 𝑟)) ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))})) → inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟)))
87	37, 52, 85, 86	mp3an12i 1468	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟)))
88	87	expr 456	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → (¬ 𝐴 = 𝑟 → inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟))))
89	88	orrd 864	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → (𝐴 = 𝑟 ∨ inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟))))
90	89	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟)))))
91	90	ralrimiva 3129	. . . 4 ⊢ (𝜑 → ∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟)))))
92		breq1 5088	. . . . . . . 8 ⊢ (𝑥 = inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) → (𝑥 ≤ (abs‘(𝐴 − 𝑟)) ↔ inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟))))
93	92	orbi2d 916	. . . . . . 7 ⊢ (𝑥 = inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) → ((𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟))) ↔ (𝐴 = 𝑟 ∨ inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟)))))
94	93	imbi2d 340	. . . . . 6 ⊢ (𝑥 = inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) → (((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟)))) ↔ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟))))))
95	94	ralbidv 3160	. . . . 5 ⊢ (𝑥 = inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) → (∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟)))) ↔ ∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟))))))
96	95	rspcev 3564	. . . 4 ⊢ ((inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ∈ ℝ⁺ ∧ ∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ≤ (abs‘(𝐴 − 𝑟))))) → ∃𝑥 ∈ ℝ⁺ ∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟)))))
97	41, 91, 96	syl2anc 585	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ⁺ ∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟)))))
98		fveqeq2 6849	. . . . . . . 8 ⊢ (𝑟 = (𝑝 / 𝑞) → ((𝐹‘𝑟) = 0 ↔ (𝐹‘(𝑝 / 𝑞)) = 0))
99		eqeq2 2748	. . . . . . . . 9 ⊢ (𝑟 = (𝑝 / 𝑞) → (𝐴 = 𝑟 ↔ 𝐴 = (𝑝 / 𝑞)))
100		oveq2 7375	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑝 / 𝑞) → (𝐴 − 𝑟) = (𝐴 − (𝑝 / 𝑞)))
101	100	fveq2d 6844	. . . . . . . . . 10 ⊢ (𝑟 = (𝑝 / 𝑞) → (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − (𝑝 / 𝑞))))
102	101	breq2d 5097	. . . . . . . . 9 ⊢ (𝑟 = (𝑝 / 𝑞) → (𝑥 ≤ (abs‘(𝐴 − 𝑟)) ↔ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
103	99, 102	orbi12d 919	. . . . . . . 8 ⊢ (𝑟 = (𝑝 / 𝑞) → ((𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟))) ↔ (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
104	98, 103	imbi12d 344	. . . . . . 7 ⊢ (𝑟 = (𝑝 / 𝑞) → (((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟)))) ↔ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))))
105	104	rspcv 3560	. . . . . 6 ⊢ ((𝑝 / 𝑞) ∈ ℝ → (∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟)))) → ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))))
106		znq 12902	. . . . . . 7 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℚ)
107		qre 12903	. . . . . . 7 ⊢ ((𝑝 / 𝑞) ∈ ℚ → (𝑝 / 𝑞) ∈ ℝ)
108	106, 107	syl 17	. . . . . 6 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℝ)
109	105, 108	syl11 33	. . . . 5 ⊢ (∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟)))) → ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))))
110	109	ralrimivv 3178	. . . 4 ⊢ (∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟)))) → ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
111	110	reximi 3075	. . 3 ⊢ (∃𝑥 ∈ ℝ⁺ ∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟)))) → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
112	97, 111	syl 17	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
113		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑥 ∈ ℝ⁺)
114		simprr 773	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑞 ∈ ℕ)
115	10	nnnn0d 12498	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
116	115	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℕ₀)
117	114, 116	nnexpcld 14207	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℕ)
118	117	nnrpd 12984	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℝ⁺)
119	113, 118	rpdivcld 13003	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 / (𝑞↑𝑁)) ∈ ℝ⁺)
120	119	rpred 12986	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 / (𝑞↑𝑁)) ∈ ℝ)
121	120	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (𝑥 / (𝑞↑𝑁)) ∈ ℝ)
122		simpllr 776	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → 𝑥 ∈ ℝ⁺)
123	122	rpred 12986	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → 𝑥 ∈ ℝ)
124	55	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝐴 ∈ ℝ)
125	108	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑝 / 𝑞) ∈ ℝ)
126	124, 125	resubcld 11578	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ)
127	126	recnd 11173	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ)
128	127	abscld 15401	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ)
129	128	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ)
130		rpre 12951	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
131	130	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑥 ∈ ℝ)
132	113	rpcnne0d 12995	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
133		divid 11840	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (𝑥 / 𝑥) = 1)
134	132, 133	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 / 𝑥) = 1)
135	117	nnge1d 12225	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 1 ≤ (𝑞↑𝑁))
136	134, 135	eqbrtrd 5107	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 / 𝑥) ≤ (𝑞↑𝑁))
137	131, 113, 118, 136	lediv23d 13054	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 / (𝑞↑𝑁)) ≤ 𝑥)
138	137	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (𝑥 / (𝑞↑𝑁)) ≤ 𝑥)
139		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))
140	121, 123, 129, 138, 139	letrd 11303	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))
141	140	ex 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))) → (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
142	141	orim2d 969	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
143	142	imim2d 57	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) → ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))))
144	143	ralimdvva 3184	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) → ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))))
145	144	reximdva 3150	. 2 ⊢ (𝜑 → (∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))))
146	112, 145	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 {cab 2714 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 {crab 3389 ∪ cun 3887 ⊆ wss 3889 ∅c0 4273 {csn 4567 class class class wbr 5085 Or wor 5538 ^◡ccnv 5630 “ cima 5634 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 Fincfn 8893 infcinf 9354 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 ℕcn 12174 ℕ₀cn0 12437 ℤcz 12524 ℚcq 12898 ℝ⁺crp 12942 ↑cexp 14023 ♯chash 14292 abscabs 15196 0_𝑝c0p 25636 Polycply 26149 degcdgr 26152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-oadd 8409 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-xnn0 12511 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-0p 25637 df-ply 26153 df-idp 26154 df-coe 26155 df-dgr 26156 df-quot 26257

This theorem is referenced by: aalioulem6 26303

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