aalioulem6 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > aalioulem6		Structured version Visualization version GIF version

Theorem aalioulem6 24433

Description: Lemma for aaliou 24434. (Contributed by Stefan O'Rear, 16-Nov-2014.)

**Hypotheses**
Ref	Expression
aalioulem2.a	⊢ 𝑁 = (deg‘𝐹)
aalioulem2.b	⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))
aalioulem2.c	⊢ (𝜑 → 𝑁 ∈ ℕ)
aalioulem2.d	⊢ (𝜑 → 𝐴 ∈ ℝ)
aalioulem3.e	⊢ (𝜑 → (𝐹‘𝐴) = 0)

**Assertion**
Ref	Expression
aalioulem6	⊢ (𝜑 → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (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 𝑥,𝑁 Allowed substitution hints: 𝑁(𝑞,𝑝)

Proof of Theorem aalioulem6 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aalioulem2.a	. . . 4 ⊢ 𝑁 = (deg‘𝐹)
2		aalioulem2.b	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))
3		aalioulem2.c	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4		aalioulem2.d	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	1, 2, 3, 4	aalioulem2 24429	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
6		aalioulem3.e	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
7	1, 2, 3, 4, 6	aalioulem5 24432	. . 3 ⊢ (𝜑 → ∃𝑏 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
8		reeanv 3288	. . 3 ⊢ (∃𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) ↔ (∃𝑎 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ∃𝑏 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))))
9	5, 7, 8	sylanbrc 579	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))))
10		r19.26-2 3246	. . . 4 ⊢ (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) ↔ (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))))
11		ifcl 4321	. . . . . 6 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
12	11	adantl 474	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
13		simpr 478	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) = 0) → (𝐹‘(𝑝 / 𝑞)) = 0)
14	11	ad2antlr 719	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
15		nnrp 12087	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℝ⁺)
16	15	ad2antll 721	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑞 ∈ ℝ⁺)
17	3	ad2antrr 718	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℕ)
18	17	nnzd 11771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℤ)
19	16, 18	rpexpcld 13288	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℝ⁺)
20	14, 19	rpdivcld 12134	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ⁺)
21	20	rpred 12117	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ)
22		simplrl 796	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ⁺)
23	22, 19	rpdivcld 12134	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ⁺)
24	23	rpred 12117	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ)
25	4	ad2antrr 718	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝐴 ∈ ℝ)
26		znq 12037	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℚ)
27		qre 12038	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 / 𝑞) ∈ ℚ → (𝑝 / 𝑞) ∈ ℝ)
28	26, 27	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℝ)
29	28	adantl 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑝 / 𝑞) ∈ ℝ)
30	25, 29	resubcld 10750	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ)
31	30	recnd 10357	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ)
32	31	abscld 14516	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ)
33	21, 24, 32	3jca 1159	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ ∧ (𝑎 / (𝑞↑𝑁)) ∈ ℝ ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ))
34	33	adantr 473	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → ((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ ∧ (𝑎 / (𝑞↑𝑁)) ∈ ℝ ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ))
35	14	rpred 12117	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ)
36	22	rpred 12117	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ)
37		simplrr 797	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑏 ∈ ℝ⁺)
38	37	rpred 12117	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑏 ∈ ℝ)
39		min1 12269	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
40	36, 38, 39	syl2anc 580	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
41	35, 36, 19, 40	lediv1dd 12175	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (𝑎 / (𝑞↑𝑁)))
42	41	anim1i 609	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → ((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (𝑎 / (𝑞↑𝑁)) ∧ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
43		letr 10421	. . . . . . . . . . . . 13 ⊢ (((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ ∧ (𝑎 / (𝑞↑𝑁)) ∈ ℝ ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ) → (((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (𝑎 / (𝑞↑𝑁)) ∧ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
44	34, 42, 43	sylc 65	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))
45	44	ex 402	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
46	45	adantr 473	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) = 0) → ((𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
47	46	orim2d 990	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) = 0) → ((𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
48	13, 47	embantd 59	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) = 0) → (((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) → (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
49	48	adantrd 486	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) = 0) → ((((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) → (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
50		simpr 478	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) ≠ 0) → (𝐹‘(𝑝 / 𝑞)) ≠ 0)
51	37, 19	rpdivcld 12134	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑏 / (𝑞↑𝑁)) ∈ ℝ⁺)
52	51	rpred 12117	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑏 / (𝑞↑𝑁)) ∈ ℝ)
53	21, 52, 32	3jca 1159	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ ∧ (𝑏 / (𝑞↑𝑁)) ∈ ℝ ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ))
54	53	adantr 473	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → ((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ ∧ (𝑏 / (𝑞↑𝑁)) ∈ ℝ ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ))
55		min2 12270	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
56	36, 38, 55	syl2anc 580	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
57	35, 38, 19, 56	lediv1dd 12175	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (𝑏 / (𝑞↑𝑁)))
58	57	anim1i 609	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → ((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (𝑏 / (𝑞↑𝑁)) ∧ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
59		letr 10421	. . . . . . . . . . . . 13 ⊢ (((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ ∧ (𝑏 / (𝑞↑𝑁)) ∈ ℝ ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ) → (((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (𝑏 / (𝑞↑𝑁)) ∧ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
60	54, 58, 59	sylc 65	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))
61	60	ex 402	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
62	61	adantr 473	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) ≠ 0) → ((𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
63	62	orim2d 990	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) ≠ 0) → ((𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
64	50, 63	embantd 59	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) ≠ 0) → (((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) → (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
65	64	adantld 485	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) ≠ 0) → ((((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) → (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
66	49, 65	pm2.61dane 3058	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) → (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
67	66	ralimdvva 3145	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) → (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) → ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
68		oveq1 6885	. . . . . . . . 9 ⊢ (𝑥 = if(𝑎 ≤ 𝑏, 𝑎, 𝑏) → (𝑥 / (𝑞↑𝑁)) = (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)))
69	68	breq1d 4853	. . . . . . . 8 ⊢ (𝑥 = if(𝑎 ≤ 𝑏, 𝑎, 𝑏) → ((𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))) ↔ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
70	69	orbi2d 940	. . . . . . 7 ⊢ (𝑥 = if(𝑎 ≤ 𝑏, 𝑎, 𝑏) → ((𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
71	70	2ralbidv 3170	. . . . . 6 ⊢ (𝑥 = if(𝑎 ≤ 𝑏, 𝑎, 𝑏) → (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
72	71	rspcev 3497	. . . . 5 ⊢ ((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺ ∧ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))
73	12, 67, 72	syl6an 675	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) → (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
74	10, 73	syl5bir 235	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) → ((∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
75	74	rexlimdvva 3219	. 2 ⊢ (𝜑 → (∃𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) ∧ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
76	9, 75	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 385 ∨ wo 874 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ∃wrex 3090 ifcif 4277 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 ℝcr 10223 0cc0 10224 ≤ cle 10364 − cmin 10556 / cdiv 10976 ℕcn 11312 ℤcz 11666 ℚcq 12033 ℝ⁺crp 12074 ↑cexp 13114 abscabs 14315 Polycply 24281 degcdgr 24284

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-xnn0 11653 df-z 11667 df-dec 11784 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-clim 14560 df-rlim 14561 df-sum 14758 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-hom 16291 df-cco 16292 df-rest 16398 df-topn 16399 df-0g 16417 df-gsum 16418 df-topgen 16419 df-pt 16420 df-prds 16423 df-xrs 16477 df-qtop 16482 df-imas 16483 df-xps 16485 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-grp 17741 df-minusg 17742 df-mulg 17857 df-subg 17904 df-cntz 18062 df-cmn 18510 df-mgp 18806 df-ur 18818 df-ring 18865 df-cring 18866 df-subrg 19096 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-fbas 20065 df-fg 20066 df-cnfld 20069 df-top 21027 df-topon 21044 df-topsp 21066 df-bases 21079 df-cld 21152 df-ntr 21153 df-cls 21154 df-nei 21231 df-lp 21269 df-perf 21270 df-cn 21360 df-cnp 21361 df-haus 21448 df-cmp 21519 df-tx 21694 df-hmeo 21887 df-fil 21978 df-fm 22070 df-flim 22071 df-flf 22072 df-xms 22453 df-ms 22454 df-tms 22455 df-cncf 23009 df-0p 23778 df-limc 23971 df-dv 23972 df-dvn 23973 df-cpn 23974 df-ply 24285 df-idp 24286 df-coe 24287 df-dgr 24288 df-quot 24387

This theorem is referenced by: aaliou 24434

W3C validator