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Theorem aalioulem6 24928

Description: Lemma for aaliou 24929. (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 24924	. . 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 24927	. . 3 ⊢ (𝜑 → ∃𝑏 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
8		reeanv 3369	. . 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 585	. 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 3173	. . . 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 4513	. . . . . 6 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
12	11	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
13		simpr 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) = 0) → (𝐹‘(𝑝 / 𝑞)) = 0)
14	11	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
15		nnrp 12403	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℝ⁺)
16	15	ad2antll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑞 ∈ ℝ⁺)
17	3	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℕ)
18	17	nnzd 12089	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℤ)
19	16, 18	rpexpcld 13611	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℝ⁺)
20	14, 19	rpdivcld 12451	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ⁺)
21	20	rpred 12434	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ)
22		simplrl 775	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ⁺)
23	22, 19	rpdivcld 12451	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ⁺)
24	23	rpred 12434	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ)
25	4	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝐴 ∈ ℝ)
26		znq 12355	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℚ)
27		qre 12356	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 / 𝑞) ∈ ℚ → (𝑝 / 𝑞) ∈ ℝ)
28	26, 27	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℝ)
29	28	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑝 / 𝑞) ∈ ℝ)
30	25, 29	resubcld 11070	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ)
31	30	recnd 10671	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ)
32	31	abscld 14798	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ)
33	21, 24, 32	3jca 1124	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 12434	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ)
36	22	rpred 12434	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ)
37		simplrr 776	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑏 ∈ ℝ⁺)
38	37	rpred 12434	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑏 ∈ ℝ)
39		min1 12585	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
40	36, 38, 39	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
41	35, 36, 19, 40	lediv1dd 12492	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (𝑎 / (𝑞↑𝑁)))
42	41	anim1i 616	. . . . . . . . . . . . 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 10736	. . . . . . . . . . . . 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 415	. . . . . . . . . . 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 483	. . . . . . . . . 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 963	. . . . . . . . 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 494	. . . . . . 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 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) ≠ 0) → (𝐹‘(𝑝 / 𝑞)) ≠ 0)
51	37, 19	rpdivcld 12451	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑏 / (𝑞↑𝑁)) ∈ ℝ⁺)
52	51	rpred 12434	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑏 / (𝑞↑𝑁)) ∈ ℝ)
53	21, 52, 32	3jca 1124	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 12586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
56	36, 38, 55	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
57	35, 38, 19, 56	lediv1dd 12492	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (𝑏 / (𝑞↑𝑁)))
58	57	anim1i 616	. . . . . . . . . . . . 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 10736	. . . . . . . . . . . . 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 415	. . . . . . . . . . 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 483	. . . . . . . . . 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 963	. . . . . . . . 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 493	. . . . . . 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 3106	. . . . . 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 3181	. . . . 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 7165	. . . . . . . . 9 ⊢ (𝑥 = if(𝑎 ≤ 𝑏, 𝑎, 𝑏) → (𝑥 / (𝑞↑𝑁)) = (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)))
69	68	breq1d 5078	. . . . . . . 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 912	. . . . . . 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 3201	. . . . . 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 3625	. . . . 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 682	. . . 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 245	. . 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 3296	. 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 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ifcif 4469 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℕcn 11640 ℤcz 11984 ℚcq 12351 ℝ⁺crp 12392 ↑cexp 13432 abscabs 14595 Polycply 24776 degcdgr 24779

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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-cntz 18449 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-0p 24273 df-limc 24466 df-dv 24467 df-dvn 24468 df-cpn 24469 df-ply 24780 df-idp 24781 df-coe 24782 df-dgr 24783 df-quot 24882

This theorem is referenced by: aaliou 24929

W3C validator