Theorem resqrex 15299

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 11292	. . . . 5 ⊢ 0 ∈ ℝ
2		leloe 11376	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
3	1, 2	mpan 689	. . . 4 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
4		elrp 13059	. . . . . . 7 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
5		01sqrex 15298	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ+ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴))
6		rprege0 13072	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
7	6	anim1i 614	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥↑2) = 𝐴) → ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴))
8		anass 468	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴) ↔ (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
9	7, 8	sylib 218	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥↑2) = 𝐴) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
10	9	adantrl 715	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴)) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
11	10	reximi2 3085	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ+ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
12	5, 11	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
13	4, 12	sylanbr 581	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
14	13	exp31 419	. . . . 5 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
15		sq0 14241	. . . . . . . . 9 ⊢ (0↑2) = 0
16		id 22	. . . . . . . . 9 ⊢ (0 = 𝐴 → 0 = 𝐴)
17	15, 16	eqtrid 2792	. . . . . . . 8 ⊢ (0 = 𝐴 → (0↑2) = 𝐴)
18		0le0 12394	. . . . . . . 8 ⊢ 0 ≤ 0
19	17, 18	jctil 519	. . . . . . 7 ⊢ (0 = 𝐴 → (0 ≤ 0 ∧ (0↑2) = 𝐴))
20		breq2 5170	. . . . . . . . 9 ⊢ (𝑥 = 0 → (0 ≤ 𝑥 ↔ 0 ≤ 0))
21		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥↑2) = (0↑2))
22	21	eqeq1d 2742	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥↑2) = 𝐴 ↔ (0↑2) = 𝐴))
23	20, 22	anbi12d 631	. . . . . . . 8 ⊢ (𝑥 = 0 → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ 0 ∧ (0↑2) = 𝐴)))
24	23	rspcev 3635	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (0 ≤ 0 ∧ (0↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
25	1, 19, 24	sylancr 586	. . . . . 6 ⊢ (0 = 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
26	25	a1i13 27	. . . . 5 ⊢ (𝐴 ∈ ℝ → (0 = 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
27	14, 26	jaod 858	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0 < 𝐴 ∨ 0 = 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
28	3, 27	sylbid 240	. . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
29	28	imp 406	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
30		0lt1 11812	. . . . . . . . . 10 ⊢ 0 < 1
31		1re 11290	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
32		ltletr 11382	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴))
33	1, 31, 32	mp3an12 1451	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴))
34	30, 33	mpani 695	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (1 ≤ 𝐴 → 0 < 𝐴))
35	34	imp 406	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 𝐴)
36	4	biimpri 228	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ+)
37	35, 36	syldan 590	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+)
38	37	rpreccld 13109	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ∈ ℝ+)
39		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
40		lerec 12178	. . . . . . . . . 10 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
41	31, 30, 40	mpanl12 701	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
42	35, 41	syldan 590	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
43	39, 42	mpbid 232	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ≤ (1 / 1))
44		1div1e1 11985	. . . . . . 7 ⊢ (1 / 1) = 1
45	43, 44	breqtrdi 5207	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ≤ 1)
46		01sqrex 15298	. . . . . 6 ⊢ (((1 / 𝐴) ∈ ℝ+ ∧ (1 / 𝐴) ≤ 1) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)))
47	38, 45, 46	syl2anc 583	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)))
48		rpre 13065	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
49	48	3ad2ant2 1134	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 𝑦 ∈ ℝ)
50		rpgt0 13069	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 0 < 𝑦)
51	50	3ad2ant2 1134	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 < 𝑦)
52		gt0ne0 11755	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 𝑦 ≠ 0)
53		rereccl 12012	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 𝑦 ≠ 0) → (1 / 𝑦) ∈ ℝ)
54	52, 53	syldan 590	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → (1 / 𝑦) ∈ ℝ)
55	49, 51, 54	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / 𝑦) ∈ ℝ)
56		recgt0 12140	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 0 < (1 / 𝑦))
57		ltle 11378	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / 𝑦) ∈ ℝ) → (0 < (1 / 𝑦) → 0 ≤ (1 / 𝑦)))
58	1, 57	mpan 689	. . . . . . . . 9 ⊢ ((1 / 𝑦) ∈ ℝ → (0 < (1 / 𝑦) → 0 ≤ (1 / 𝑦)))
59	54, 56, 58	sylc 65	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 0 ≤ (1 / 𝑦))
60	49, 51, 59	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 ≤ (1 / 𝑦))
61		recn 11274	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
62	61	adantr 480	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 𝑦 ∈ ℂ)
63	62, 52	sqrecd 14200	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2)))
64	49, 51, 63	syl2anc 583	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2)))
65		simp3r 1202	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (𝑦↑2) = (1 / 𝐴))
66	65	oveq2d 7464	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (𝑦↑2)) = (1 / (1 / 𝐴)))
67		recn 11274	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
68		gt0ne0 11755	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0)
69	35, 68	syldan 590	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ≠ 0)
70		recrec 11991	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / (1 / 𝐴)) = 𝐴)
71	67, 69, 70	syl2an2r 684	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / (1 / 𝐴)) = 𝐴)
72	71	3ad2ant1 1133	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (1 / 𝐴)) = 𝐴)
73	64, 66, 72	3eqtrd 2784	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = 𝐴)
74		breq2 5170	. . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (1 / 𝑦)))
75		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑥 = (1 / 𝑦) → (𝑥↑2) = ((1 / 𝑦)↑2))
76	75	eqeq1d 2742	. . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑦) → ((𝑥↑2) = 𝐴 ↔ ((1 / 𝑦)↑2) = 𝐴))
77	74, 76	anbi12d 631	. . . . . . . 8 ⊢ (𝑥 = (1 / 𝑦) → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴)))
78	77	rspcev 3635	. . . . . . 7 ⊢ (((1 / 𝑦) ∈ ℝ ∧ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
79	55, 60, 73, 78	syl12anc 836	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
80	79	rexlimdv3a 3165	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
81	47, 80	mpd 15	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
82	81	ex 412	. . 3 ⊢ (𝐴 ∈ ℝ → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
83	82	adantr 480	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
84		simpl 482	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
85		letric 11390	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴))
86	84, 31, 85	sylancl 585	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴))
87	29, 83, 86	mpjaod 859	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076   class class class wbr 5166  (class class class)co 7448  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   < clt 11324   ≤ cle 11325   / cdiv 11947  2c2 12348  ℝ⁺crp 13057  ↑cexp 14112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-seq 14053  df-exp 14113

This theorem is referenced by:  resqreu  15301  resqrtcl  15302