aalioulem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > aalioulem2		Structured version Visualization version GIF version

Theorem aalioulem2 26496

Description: Lemma for aaliou 26501. (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 13015	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
2		snssi 4751	. . . . . . 7 ⊢ (1 ∈ ℝ⁺ → {1} ⊆ ℝ⁺)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ {1} ⊆ ℝ⁺
4		ssrab2 4034	. . . . . 6 ⊢ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ⊆ ℝ⁺
5	3, 4	unssi 4144	. . . . 5 ⊢ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ⊆ ℝ⁺
6		ltso 11285	. . . . . . 7 ⊢ < Or ℝ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → < Or ℝ)
8		snfi 9036	. . . . . . 7 ⊢ {1} ∈ Fin
9		aalioulem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))
10		aalioulem2.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	10	nnne0d 12281	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ≠ 0)
12		aalioulem2.a	. . . . . . . . . . . . . 14 ⊢ 𝑁 = (deg‘𝐹)
13	12	eqcomi 2772	. . . . . . . . . . . . 13 ⊢ (deg‘𝐹) = 𝑁
14		dgr0 26419	. . . . . . . . . . . . 13 ⊢ (deg‘0_𝑝) = 0
15	11, 13, 14	3netr4g 3037	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐹) ≠ (deg‘0_𝑝))
16		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
17	16	necon3i 2990	. . . . . . . . . . . 12 ⊢ ((deg‘𝐹) ≠ (deg‘0_𝑝) → 𝐹 ≠ 0_𝑝)
18	15, 17	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ≠ 0_𝑝)
19		eqid 2763	. . . . . . . . . . . 12 ⊢ (^◡𝐹 “ {0}) = (^◡𝐹 “ {0})
20	19	fta1 26469	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℤ) ∧ 𝐹 ≠ 0_𝑝) → ((^◡𝐹 “ {0}) ∈ Fin ∧ (♯‘(^◡𝐹 “ {0})) ≤ (deg‘𝐹)))
21	9, 18, 20	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → ((^◡𝐹 “ {0}) ∈ Fin ∧ (♯‘(^◡𝐹 “ {0})) ≤ (deg‘𝐹)))
22	21	simpld 499	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 “ {0}) ∈ Fin)
23		abrexfi 9305	. . . . . . . . 9 ⊢ ((^◡𝐹 “ {0}) ∈ Fin → {𝑎 ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ∈ Fin)
24	22, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → {𝑎 ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ∈ Fin)
25		rabssab 4039	. . . . . . . 8 ⊢ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ⊆ {𝑎 ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}
26		ssfi 9153	. . . . . . . 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 597	. . . . . . 7 ⊢ (𝜑 → {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))} ∈ Fin)
28		unfi 9151	. . . . . . 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 598	. . . . . 6 ⊢ (𝜑 → ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}) ∈ Fin)
30		1ex 11198	. . . . . . . . 9 ⊢ 1 ∈ V
31	30	snid 4628	. . . . . . . 8 ⊢ 1 ∈ {1}
32		elun1 4135	. . . . . . . 8 ⊢ (1 ∈ {1} → 1 ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}))
33		ne0i 4294	. . . . . . . 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 13019	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
37	5, 36	sstri 3946	. . . . . . 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 9459	. . . . . 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 1399	. . . . 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 3935	. . . 4 ⊢ (𝜑 → inf(({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}), ℝ, < ) ∈ ℝ⁺)
42		0re 11205	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
43		rpge0 13025	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℝ⁺ → 0 ≤ 𝑑)
44	43	rgen 3081	. . . . . . . . . . 11 ⊢ ∀𝑑 ∈ ℝ⁺ 0 ≤ 𝑑
45		breq1 5112	. . . . . . . . . . . . 13 ⊢ (𝑐 = 0 → (𝑐 ≤ 𝑑 ↔ 0 ≤ 𝑑))
46	45	ralbidv 3188	. . . . . . . . . . . 12 ⊢ (𝑐 = 0 → (∀𝑑 ∈ ℝ⁺ 𝑐 ≤ 𝑑 ↔ ∀𝑑 ∈ ℝ⁺ 0 ≤ 𝑑))
47	46	rspcev 3581	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ ∀𝑑 ∈ ℝ⁺ 0 ≤ 𝑑) → ∃𝑐 ∈ ℝ ∀𝑑 ∈ ℝ⁺ 𝑐 ≤ 𝑑)
48	42, 44, 47	mp2an 704	. . . . . . . . . 10 ⊢ ∃𝑐 ∈ ℝ ∀𝑑 ∈ ℝ⁺ 𝑐 ≤ 𝑑
49		ssralv 4006	. . . . . . . . . . . 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 3103	. . . . . . . . . 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 2767	. . . . . . . . . . . 12 ⊢ (𝑎 = (abs‘(𝐴 − 𝑟)) → (𝑎 = (abs‘(𝐴 − 𝑏)) ↔ (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑏))))
54	53	rexbidv 3189	. . . . . . . . . . 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 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → 𝐴 ∈ ℝ)
57		simplr 780	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → 𝑟 ∈ ℝ)
58	56, 57	resubcld 11637	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (𝐴 − 𝑟) ∈ ℝ)
59	58	recnd 11232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (𝐴 − 𝑟) ∈ ℂ)
60	55	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → 𝐴 ∈ ℝ)
61	60	recnd 11232	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → 𝐴 ∈ ℂ)
62		simplr 780	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → 𝑟 ∈ ℝ)
63	62	recnd 11232	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → 𝑟 ∈ ℂ)
64	61, 63	subeq0ad 11574	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → ((𝐴 − 𝑟) = 0 ↔ 𝐴 = 𝑟))
65	64	necon3abid 2994	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → ((𝐴 − 𝑟) ≠ 0 ↔ ¬ 𝐴 = 𝑟))
66	65	biimprd 251	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝐹‘𝑟) = 0) → (¬ 𝐴 = 𝑟 → (𝐴 − 𝑟) ≠ 0))
67	66	impr 459	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (𝐴 − 𝑟) ≠ 0)
68	59, 67	absrpcld 15498	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (abs‘(𝐴 − 𝑟)) ∈ ℝ⁺)
69	57	recnd 11232	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → 𝑟 ∈ ℂ)
70		simprl 782	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (𝐹‘𝑟) = 0)
71		plyf 26355	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (Poly‘ℤ) → 𝐹:ℂ⟶ℂ)
72	9, 71	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
73	72	ffnd 6706	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn ℂ)
74	73	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → 𝐹 Fn ℂ)
75		fniniseg 7055	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn ℂ → (𝑟 ∈ (^◡𝐹 “ {0}) ↔ (𝑟 ∈ ℂ ∧ (𝐹‘𝑟) = 0)))
76	74, 75	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (𝑟 ∈ (^◡𝐹 “ {0}) ↔ (𝑟 ∈ ℂ ∧ (𝐹‘𝑟) = 0)))
77	69, 70, 76	mpbir2and 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → 𝑟 ∈ (^◡𝐹 “ {0}))
78		eqid 2763	. . . . . . . . . . . 12 ⊢ (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑟))
79		oveq2 7418	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑟 → (𝐴 − 𝑏) = (𝐴 − 𝑟))
80	79	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑟 → (abs‘(𝐴 − 𝑏)) = (abs‘(𝐴 − 𝑟)))
81	80	rspceeqv 3604	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ (^◡𝐹 “ {0}) ∧ (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑟))) → ∃𝑏 ∈ (^◡𝐹 “ {0})(abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑏)))
82	77, 78, 81	sylancl 597	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → ∃𝑏 ∈ (^◡𝐹 “ {0})(abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − 𝑏)))
83	54, 68, 82	elrabd 3652	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (abs‘(𝐴 − 𝑟)) ∈ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))})
84		elun2 4136	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ ((𝐹‘𝑟) = 0 ∧ ¬ 𝐴 = 𝑟)) → (abs‘(𝐴 − 𝑟)) ∈ ({1} ∪ {𝑎 ∈ ℝ⁺ ∣ ∃𝑏 ∈ (^◡𝐹 “ {0})𝑎 = (abs‘(𝐴 − 𝑏))}))
86		infrelb 12195	. . . . . . . . 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 1494	. . . . . . . 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 461	. . . . . . 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 876	. . . . . 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 417	. . . . 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 3157	. . . 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 5112	. . . . . . . 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 928	. . . . . . 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 343	. . . . . 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 3188	. . . . 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 3581	. . . 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 595	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ⁺ ∀𝑟 ∈ ℝ ((𝐹‘𝑟) = 0 → (𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟)))))
98		fveqeq2 6890	. . . . . . . 8 ⊢ (𝑟 = (𝑝 / 𝑞) → ((𝐹‘𝑟) = 0 ↔ (𝐹‘(𝑝 / 𝑞)) = 0))
99		eqeq2 2775	. . . . . . . . 9 ⊢ (𝑟 = (𝑝 / 𝑞) → (𝐴 = 𝑟 ↔ 𝐴 = (𝑝 / 𝑞)))
100		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑝 / 𝑞) → (𝐴 − 𝑟) = (𝐴 − (𝑝 / 𝑞)))
101	100	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝑟 = (𝑝 / 𝑞) → (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − (𝑝 / 𝑞))))
102	101	breq2d 5121	. . . . . . . . 9 ⊢ (𝑟 = (𝑝 / 𝑞) → (𝑥 ≤ (abs‘(𝐴 − 𝑟)) ↔ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
103	99, 102	orbi12d 931	. . . . . . . 8 ⊢ (𝑟 = (𝑝 / 𝑞) → ((𝐴 = 𝑟 ∨ 𝑥 ≤ (abs‘(𝐴 − 𝑟))) ↔ (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
104	98, 103	imbi12d 347	. . . . . . 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 3577	. . . . . 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 12971	. . . . . . 7 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℚ)
107		qre 12972	. . . . . . 7 ⊢ ((𝑝 / 𝑞) ∈ ℚ → (𝑝 / 𝑞) ∈ ℝ)
108	106, 107	syl 18	. . . . . 6 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℝ)
109	105, 108	syl11 34	. . . . 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 3206	. . . 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 3103	. . 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 18	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
113		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑥 ∈ ℝ⁺)
114		simprr 784	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑞 ∈ ℕ)
115	10	nnnn0d 12560	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
116	115	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℕ₀)
117	114, 116	nnexpcld 14277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℕ)
118	117	nnrpd 13053	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℝ⁺)
119	113, 118	rpdivcld 13072	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 / (𝑞↑𝑁)) ∈ ℝ⁺)
120	119	rpred 13055	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 / (𝑞↑𝑁)) ∈ ℝ)
121	120	adantr 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (𝑥 / (𝑞↑𝑁)) ∈ ℝ)
122		simpllr 787	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → 𝑥 ∈ ℝ⁺)
123	122	rpred 13055	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → 𝑥 ∈ ℝ)
124	55	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝐴 ∈ ℝ)
125	108	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑝 / 𝑞) ∈ ℝ)
126	124, 125	resubcld 11637	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ)
127	126	recnd 11232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ)
128	127	abscld 15486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ)
129	128	adantr 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ)
130		rpre 13020	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
131	130	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑥 ∈ ℝ)
132	113	rpcnne0d 13064	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
133		divid 11897	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (𝑥 / 𝑥) = 1)
134	132, 133	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 / 𝑥) = 1)
135	117	nnge1d 12279	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 1 ≤ (𝑞↑𝑁))
136	134, 135	eqbrtrd 5133	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 / 𝑥) ≤ (𝑞↑𝑁))
137	131, 113, 118, 136	lediv23d 13123	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 / (𝑞↑𝑁)) ≤ 𝑥)
138	137	adantr 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (𝑥 / (𝑞↑𝑁)) ≤ 𝑥)
139		simpr 489	. . . . . . . 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 11362	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ 𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))
141	140	ex 417	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑥 ≤ (abs‘(𝐴 − (𝑝 / 𝑞))) → (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
142	141	orim2d 982	. . . . 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 58	. . . 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 3212	. . 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 3178	. 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 16	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 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 {cab 2741 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 {crab 3416 ∪ cun 3903 ⊆ wss 3905 ∅c0 4286 {csn 4589 class class class wbr 5109 Or wor 5568 ^◡ccnv 5660 “ cima 5664 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 Fincfn 8939 infcinf 9397 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 < clt 11238 ≤ cle 11239 − cmin 11436 / cdiv 11866 ℕcn 12228 ℕ₀cn0 12499 ℤcz 12586 ℚcq 12967 ℝ⁺crp 13011 ↑cexp 14093 ♯chash 14362 abscabs 15281 0_𝑝c0p 25828 Polycply 26341 degcdgr 26344

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 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 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-oi 9468 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-n0 12500 df-xnn0 12573 df-z 12587 df-uz 12858 df-q 12968 df-rp 13012 df-fz 13531 df-fzo 13679 df-fl 13821 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-rlim 15536 df-sum 15734 df-0p 25829 df-ply 26345 df-idp 26346 df-coe 26347 df-dgr 26348 df-quot 26452

This theorem is referenced by: aalioulem6 26500

W3C validator