Theorem resqrex 15267

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 11176	. . . . 5 ⊢ 0 ∈ ℝ
2		leloe 11262	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
3	1, 2	mpan 700	. . . 4 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
4		elrp 12988	. . . . . . 7 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
5		01sqrex 15266	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ+ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴))
6		rprege0 13002	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
7	6	anim1i 624	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥↑2) = 𝐴) → ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴))
8		anass 472	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴) ↔ (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
9	7, 8	sylib 220	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥↑2) = 𝐴) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
10	9	adantrl 726	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴)) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
11	10	reximi2 3094	. . . . . . . 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 591	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
14	13	exp31 423	. . . . 5 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
15		sq0 14198	. . . . . . . . 9 ⊢ (0↑2) = 0
16		id 22	. . . . . . . . 9 ⊢ (0 = 𝐴 → 0 = 𝐴)
17	15, 16	eqtrid 2808	. . . . . . . 8 ⊢ (0 = 𝐴 → (0↑2) = 𝐴)
18		0le0 12312	. . . . . . . 8 ⊢ 0 ≤ 0
19	17, 18	jctil 527	. . . . . . 7 ⊢ (0 = 𝐴 → (0 ≤ 0 ∧ (0↑2) = 𝐴))
20		breq2 5101	. . . . . . . . 9 ⊢ (𝑥 = 0 → (0 ≤ 𝑥 ↔ 0 ≤ 0))
21		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥↑2) = (0↑2))
22	21	eqeq1d 2763	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥↑2) = 𝐴 ↔ (0↑2) = 𝐴))
23	20, 22	anbi12d 641	. . . . . . . 8 ⊢ (𝑥 = 0 → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ 0 ∧ (0↑2) = 𝐴)))
24	23	rspcev 3580	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (0 ≤ 0 ∧ (0↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
25	1, 19, 24	sylancr 596	. . . . . 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 870	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0 < 𝐴 ∨ 0 = 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
28	3, 27	sylbid 242	. . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
29	28	imp 410	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
30		0lt1 11702	. . . . . . . . . 10 ⊢ 0 < 1
31		1re 11174	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
32		ltletr 11268	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴))
33	1, 31, 32	mp3an12 1471	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴))
34	30, 33	mpani 706	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (1 ≤ 𝐴 → 0 < 𝐴))
35	34	imp 410	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 𝐴)
36	4	biimpri 230	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ+)
37	35, 36	syldan 600	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+)
38	37	rpreccld 13040	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ∈ ℝ+)
39		simpr 488	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
40		lerec 12068	. . . . . . . . . 10 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
41	31, 30, 40	mpanl12 712	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
42	35, 41	syldan 600	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
43	39, 42	mpbid 234	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ≤ (1 / 1))
44		1div1e1 11874	. . . . . . 7 ⊢ (1 / 1) = 1
45	43, 44	breqtrdi 5138	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ≤ 1)
46		01sqrex 15266	. . . . . 6 ⊢ (((1 / 𝐴) ∈ ℝ+ ∧ (1 / 𝐴) ≤ 1) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)))
47	38, 45, 46	syl2anc 593	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)))
48		rpre 12995	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
49	48	3ad2ant2 1146	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 𝑦 ∈ ℝ)
50		rpgt0 12999	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 0 < 𝑦)
51	50	3ad2ant2 1146	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 < 𝑦)
52		gt0ne0 11645	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 𝑦 ≠ 0)
53		rereccl 11902	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 𝑦 ≠ 0) → (1 / 𝑦) ∈ ℝ)
54	52, 53	syldan 600	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → (1 / 𝑦) ∈ ℝ)
55	49, 51, 54	syl2anc 593	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / 𝑦) ∈ ℝ)
56		recgt0 12030	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 0 < (1 / 𝑦))
57		ltle 11264	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / 𝑦) ∈ ℝ) → (0 < (1 / 𝑦) → 0 ≤ (1 / 𝑦)))
58	1, 57	mpan 700	. . . . . . . . 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 593	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 ≤ (1 / 𝑦))
61		recn 11156	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
62	61	adantr 484	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 𝑦 ∈ ℂ)
63	62, 52	sqrecd 14156	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2)))
64	49, 51, 63	syl2anc 593	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2)))
65		simp3r 1215	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (𝑦↑2) = (1 / 𝐴))
66	65	oveq2d 7406	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (𝑦↑2)) = (1 / (1 / 𝐴)))
67		recn 11156	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
68		gt0ne0 11645	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0)
69	35, 68	syldan 600	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ≠ 0)
70		recrec 11881	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / (1 / 𝐴)) = 𝐴)
71	67, 69, 70	syl2an2r 695	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / (1 / 𝐴)) = 𝐴)
72	71	3ad2ant1 1145	. . . . . . . 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 5101	. . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (1 / 𝑦)))
75		oveq1 7397	. . . . . . . . . 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 641	. . . . . . . 8 ⊢ (𝑥 = (1 / 𝑦) → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴)))
78	77	rspcev 3580	. . . . . . 7 ⊢ (((1 / 𝑦) ∈ ℝ ∧ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
79	55, 60, 73, 78	syl12anc 847	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
80	79	rexlimdv3a 3166	. . . . 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 416	. . 3 ⊢ (𝐴 ∈ ℝ → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
83	82	adantr 484	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
84		simpl 486	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
85		letric 11276	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴))
86	84, 31, 85	sylancl 595	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴))
87	29, 83, 86	mpjaod 871	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085   class class class wbr 5097  (class class class)co 7390  ℂcc 11064  ℝcr 11065  0cc0 11066  1c1 11067   < clt 11209   ≤ cle 11210   / cdiv 11837  2c2 12265  ℝ⁺crp 12986  ↑cexp 14067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143  ax-pre-sup 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-er 8671  df-en 8921  df-dom 8922  df-sdom 8923  df-sup 9381  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-div 11838  df-nn 12204  df-2 12273  df-3 12274  df-n0 12475  df-z 12562  df-uz 12833  df-rp 12987  df-seq 14008  df-exp 14068

This theorem is referenced by:  resqreu  15269  resqrtcl  15270