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Theorem aalioulem6 26299

Description: Lemma for aaliou 26300. (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 26295	. . 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 26298	. . 3 ⊢ (𝜑 → ∃𝑏 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑏 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
8		reeanv 3206	. . 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 583	. 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 3119	. . . 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 4523	. . . . . 6 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
12	11	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
13		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) = 0) → (𝐹‘(𝑝 / 𝑞)) = 0)
14	11	ad2antlr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
15		nnrp 12915	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℝ⁺)
16	15	ad2antll 729	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑞 ∈ ℝ⁺)
17	3	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℕ)
18	17	nnzd 12512	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℤ)
19	16, 18	rpexpcld 14168	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℝ⁺)
20	14, 19	rpdivcld 12964	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ⁺)
21	20	rpred 12947	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ∈ ℝ)
22		simplrl 776	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ⁺)
23	22, 19	rpdivcld 12964	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ⁺)
24	23	rpred 12947	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ)
25	4	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝐴 ∈ ℝ)
26		znq 12863	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℚ)
27		qre 12864	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 / 𝑞) ∈ ℚ → (𝑝 / 𝑞) ∈ ℝ)
28	26, 27	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℝ)
29	28	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑝 / 𝑞) ∈ ℝ)
30	25, 29	resubcld 11563	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ)
31	30	recnd 11158	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ)
32	31	abscld 15360	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ)
33	21, 24, 32	3jca 1128	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 12947	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ)
36	22	rpred 12947	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ)
37		simplrr 777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑏 ∈ ℝ⁺)
38	37	rpred 12947	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑏 ∈ ℝ)
39		min1 13102	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
40	36, 38, 39	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
41	35, 36, 19, 40	lediv1dd 13005	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (𝑎 / (𝑞↑𝑁)))
42	41	anim1i 615	. . . . . . . . . . . . 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 11225	. . . . . . . . . . . . 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 412	. . . . . . . . . . 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 480	. . . . . . . . . 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 968	. . . . . . . . 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 491	. . . . . . 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 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝐹‘(𝑝 / 𝑞)) ≠ 0) → (𝐹‘(𝑝 / 𝑞)) ≠ 0)
51	37, 19	rpdivcld 12964	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑏 / (𝑞↑𝑁)) ∈ ℝ⁺)
52	51	rpred 12947	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑏 / (𝑞↑𝑁)) ∈ ℝ)
53	21, 52, 32	3jca 1128	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 13103	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
56	36, 38, 55	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
57	35, 38, 19, 56	lediv1dd 13005	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)) ≤ (𝑏 / (𝑞↑𝑁)))
58	57	anim1i 615	. . . . . . . . . . . . 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 11225	. . . . . . . . . . . . 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 412	. . . . . . . . . . 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 480	. . . . . . . . . 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 968	. . . . . . . . 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 490	. . . . . . 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 3017	. . . . . 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 7363	. . . . . . . . 9 ⊢ (𝑥 = if(𝑎 ≤ 𝑏, 𝑎, 𝑏) → (𝑥 / (𝑞↑𝑁)) = (if(𝑎 ≤ 𝑏, 𝑎, 𝑏) / (𝑞↑𝑁)))
69	68	breq1d 5106	. . . . . . . 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 915	. . . . . . 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 3198	. . . . . 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 3574	. . . . 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 684	. . . 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	biimtrrid 243	. . 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 3191	. 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 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 ∃wrex 3058 ifcif 4477 class class class wbr 5096 ‘cfv 6490 (class class class)co 7356 ℝcr 11023 0cc0 11024 ≤ cle 11165 − cmin 11362 / cdiv 11792 ℕcn 12143 ℤcz 12486 ℚcq 12859 ℝ⁺crp 12903 ↑cexp 13982 abscabs 15155 Polycply 26143 degcdgr 26146

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-xnn0 12473 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-clim 15409 df-rlim 15410 df-sum 15608 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-grp 18864 df-minusg 18865 df-mulg 18996 df-subg 19051 df-cntz 19244 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-subrng 20477 df-subrg 20501 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-cmp 23329 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-0p 25625 df-limc 25821 df-dv 25822 df-dvn 25823 df-cpn 25824 df-ply 26147 df-idp 26148 df-coe 26149 df-dgr 26150 df-quot 26253

This theorem is referenced by: aaliou 26300

W3C validator