Theorem resqrex 15397

Description: Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.)

Assertion

Ref	Expression
resqrex	⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem resqrex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0re 11291	. . . . 5 ⊢ 0 ∈ ℝ
2		leloe 11377	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
3	1, 2	mpan 703	. . . 4 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
4		elrp 13103	. . . . . . 7 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
5		01sqrex 15396	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ+ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴))
6		rprege0 13117	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
7	6	anim1i 627	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥↑2) = 𝐴) → ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴))
8		anass 474	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴) ↔ (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
9	7, 8	sylib 221	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥↑2) = 𝐴) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
10	9	adantrl 729	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴)) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
11	10	reximi2 3096	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ+ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
12	5, 11	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
13	4, 12	sylanbr 594	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
14	13	exp31 425	. . . . 5 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
15		sq0 14315	. . . . . . . . 9 ⊢ (0↑2) = 0
16		id 23	. . . . . . . . 9 ⊢ (0 = 𝐴 → 0 = 𝐴)
17	15, 16	eqtrid 2808	. . . . . . . 8 ⊢ (0 = 𝐴 → (0↑2) = 𝐴)
18		0le0 12425	. . . . . . . 8 ⊢ 0 ≤ 0
19	17, 18	jctil 529	. . . . . . 7 ⊢ (0 = 𝐴 → (0 ≤ 0 ∧ (0↑2) = 𝐴))
20		breq2 5107	. . . . . . . . 9 ⊢ (𝑥 = 0 → (0 ≤ 𝑥 ↔ 0 ≤ 0))
21		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥↑2) = (0↑2))
22	21	eqeq1d 2763	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥↑2) = 𝐴 ↔ (0↑2) = 𝐴))
23	20, 22	anbi12d 644	. . . . . . . 8 ⊢ (𝑥 = 0 → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ 0 ∧ (0↑2) = 𝐴)))
24	23	rspcev 3577	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (0 ≤ 0 ∧ (0↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
25	1, 19, 24	sylancr 599	. . . . . 6 ⊢ (0 = 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
26	25	a1i13 28	. . . . 5 ⊢ (𝐴 ∈ ℝ → (0 = 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
27	14, 26	jaod 873	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0 < 𝐴 ∨ 0 = 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
28	3, 27	sylbid 243	. . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
29	28	imp 412	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
30		0lt1 11819	. . . . . . . . . 10 ⊢ 0 < 1
31		1re 11289	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
32		ltletr 11383	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴))
33	1, 31, 32	mp3an12 1480	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴))
34	30, 33	mpani 709	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (1 ≤ 𝐴 → 0 < 𝐴))
35	34	imp 412	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 𝐴)
36	4	biimpri 231	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ+)
37	35, 36	syldan 603	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+)
38	37	rpreccld 13155	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ∈ ℝ+)
39		simpr 490	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
40		lerec 12181	. . . . . . . . . 10 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
41	31, 30, 40	mpanl12 715	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
42	35, 41	syldan 603	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
43	39, 42	mpbid 235	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ≤ (1 / 1))
44		1div1e1 11988	. . . . . . 7 ⊢ (1 / 1) = 1
45	43, 44	breqtrdi 5146	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ≤ 1)
46		01sqrex 15396	. . . . . 6 ⊢ (((1 / 𝐴) ∈ ℝ+ ∧ (1 / 𝐴) ≤ 1) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)))
47	38, 45, 46	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)))
48		rpre 13110	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
49	48	3ad2ant2 1152	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 𝑦 ∈ ℝ)
50		rpgt0 13114	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 0 < 𝑦)
51	50	3ad2ant2 1152	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 < 𝑦)
52		gt0ne0 11762	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 𝑦 ≠ 0)
53		rereccl 12016	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 𝑦 ≠ 0) → (1 / 𝑦) ∈ ℝ)
54	52, 53	syldan 603	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → (1 / 𝑦) ∈ ℝ)
55	49, 51, 54	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / 𝑦) ∈ ℝ)
56		recgt0 12144	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 0 < (1 / 𝑦))
57		ltle 11379	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / 𝑦) ∈ ℝ) → (0 < (1 / 𝑦) → 0 ≤ (1 / 𝑦)))
58	1, 57	mpan 703	. . . . . . . . 9 ⊢ ((1 / 𝑦) ∈ ℝ → (0 < (1 / 𝑦) → 0 ≤ (1 / 𝑦)))
59	54, 56, 58	sylc 66	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 0 ≤ (1 / 𝑦))
60	49, 51, 59	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 ≤ (1 / 𝑦))
61		recn 11271	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
62	61	adantr 486	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 𝑦 ∈ ℂ)
63	62, 52	sqrecd 14273	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2)))
64	49, 51, 63	syl2anc 596	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2)))
65		simp3r 1221	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (𝑦↑2) = (1 / 𝐴))
66	65	oveq2d 7428	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (𝑦↑2)) = (1 / (1 / 𝐴)))
67		recn 11271	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
68		gt0ne0 11762	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0)
69	35, 68	syldan 603	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ≠ 0)
70		recrec 11995	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / (1 / 𝐴)) = 𝐴)
71	67, 69, 70	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / (1 / 𝐴)) = 𝐴)
72	71	3ad2ant1 1151	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (1 / 𝐴)) = 𝐴)
73	64, 66, 72	3eqtrd 2800	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = 𝐴)
74		breq2 5107	. . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (1 / 𝑦)))
75		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑥 = (1 / 𝑦) → (𝑥↑2) = ((1 / 𝑦)↑2))
76	75	eqeq1d 2763	. . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑦) → ((𝑥↑2) = 𝐴 ↔ ((1 / 𝑦)↑2) = 𝐴))
77	74, 76	anbi12d 644	. . . . . . . 8 ⊢ (𝑥 = (1 / 𝑦) → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴)))
78	77	rspcev 3577	. . . . . . 7 ⊢ (((1 / 𝑦) ∈ ℝ ∧ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
79	55, 60, 73, 78	syl12anc 850	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
80	79	rexlimdv3a 3168	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
81	47, 80	mpd 16	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
82	81	ex 418	. . 3 ⊢ (𝐴 ∈ ℝ → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
83	82	adantr 486	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
84		simpl 488	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
85		letric 11391	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴))
86	84, 31, 85	sylancl 598	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴))
87	29, 83, 86	mpjaod 874	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   class class class wbr 5103  (class class class)co 7412  ℂcc 11179  ℝcr 11180  0cc0 11181  1c1 11182   < clt 11324   ≤ cle 11325   / cdiv 11954  2c2 12378  ℝ⁺crp 13101  ↑cexp 14184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9418  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-rp 13102  df-seq 14125  df-exp 14185

This theorem is used by:  resqreu  15399  resqrtcl  15400