Theorem resqrex 13839

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 9919	. . . . 5 ⊢ 0 ∈ ℝ
2		leloe 10003	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
3	1, 2	mpan 702	. . . 4 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
4		elrp 11710	. . . . . . 7 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
5		01sqrex 13838	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ+ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴))
6		rprege0 11723	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
7	6	anim1i 590	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥↑2) = 𝐴) → ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴))
8		anass 679	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴) ↔ (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
9	7, 8	sylib 207	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥↑2) = 𝐴) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
10	9	adantrl 748	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴)) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
11	10	reximi2 2993	. . . . . . . 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 489	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
14	13	exp31 628	. . . . 5 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
15		sq0 12817	. . . . . . . . . 10 ⊢ (0↑2) = 0
16		id 22	. . . . . . . . . 10 ⊢ (0 = 𝐴 → 0 = 𝐴)
17	15, 16	syl5eq 2656	. . . . . . . . 9 ⊢ (0 = 𝐴 → (0↑2) = 𝐴)
18		0le0 10987	. . . . . . . . 9 ⊢ 0 ≤ 0
19	17, 18	jctil 558	. . . . . . . 8 ⊢ (0 = 𝐴 → (0 ≤ 0 ∧ (0↑2) = 𝐴))
20		breq2 4587	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (0 ≤ 𝑥 ↔ 0 ≤ 0))
21		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥↑2) = (0↑2))
22	21	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ((𝑥↑2) = 𝐴 ↔ (0↑2) = 𝐴))
23	20, 22	anbi12d 743	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ 0 ∧ (0↑2) = 𝐴)))
24	23	rspcev 3282	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ (0 ≤ 0 ∧ (0↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
25	1, 19, 24	sylancr 694	. . . . . . 7 ⊢ (0 = 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
26	25	a1d 25	. . . . . 6 ⊢ (0 = 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
27	26	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ → (0 = 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
28	14, 27	jaod 394	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0 < 𝐴 ∨ 0 = 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
29	3, 28	sylbid 229	. . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
30	29	imp 444	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
31		0lt1 10429	. . . . . . . . . 10 ⊢ 0 < 1
32		1re 9918	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
33		ltletr 10008	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴))
34	1, 32, 33	mp3an12 1406	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴))
35	31, 34	mpani 708	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (1 ≤ 𝐴 → 0 < 𝐴))
36	35	imp 444	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 𝐴)
37	4	biimpri 217	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ+)
38	36, 37	syldan 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+)
39	38	rpreccld 11758	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ∈ ℝ+)
40		simpr 476	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
41		lerec 10785	. . . . . . . . . 10 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
42	32, 31, 41	mpanl12 714	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
43	36, 42	syldan 486	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
44	40, 43	mpbid 221	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ≤ (1 / 1))
45		1div1e1 10596	. . . . . . 7 ⊢ (1 / 1) = 1
46	44, 45	syl6breq 4624	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ≤ 1)
47		01sqrex 13838	. . . . . 6 ⊢ (((1 / 𝐴) ∈ ℝ+ ∧ (1 / 𝐴) ≤ 1) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)))
48	39, 46, 47	syl2anc 691	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)))
49		rpre 11715	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
50	49	3ad2ant2 1076	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 𝑦 ∈ ℝ)
51		rpgt0 11720	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 0 < 𝑦)
52	51	3ad2ant2 1076	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 < 𝑦)
53		gt0ne0 10372	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 𝑦 ≠ 0)
54		rereccl 10622	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 𝑦 ≠ 0) → (1 / 𝑦) ∈ ℝ)
55	53, 54	syldan 486	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → (1 / 𝑦) ∈ ℝ)
56	50, 52, 55	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / 𝑦) ∈ ℝ)
57		recgt0 10746	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 0 < (1 / 𝑦))
58		ltle 10005	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / 𝑦) ∈ ℝ) → (0 < (1 / 𝑦) → 0 ≤ (1 / 𝑦)))
59	1, 58	mpan 702	. . . . . . . . 9 ⊢ ((1 / 𝑦) ∈ ℝ → (0 < (1 / 𝑦) → 0 ≤ (1 / 𝑦)))
60	55, 57, 59	sylc 63	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 0 ≤ (1 / 𝑦))
61	50, 52, 60	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 ≤ (1 / 𝑦))
62		recn 9905	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
63	62	adantr 480	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 𝑦 ∈ ℂ)
64	63, 53	sqrecd 12874	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2)))
65	50, 52, 64	syl2anc 691	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2)))
66		simp3r 1083	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (𝑦↑2) = (1 / 𝐴))
67	66	oveq2d 6565	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (𝑦↑2)) = (1 / (1 / 𝐴)))
68		gt0ne0 10372	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0)
69	36, 68	syldan 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ≠ 0)
70		recn 9905	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
71		recrec 10601	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / (1 / 𝐴)) = 𝐴)
72	70, 71	sylan 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / (1 / 𝐴)) = 𝐴)
73	69, 72	syldan 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / (1 / 𝐴)) = 𝐴)
74	73	3ad2ant1 1075	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (1 / 𝐴)) = 𝐴)
75	65, 67, 74	3eqtrd 2648	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = 𝐴)
76		breq2 4587	. . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (1 / 𝑦)))
77		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑥 = (1 / 𝑦) → (𝑥↑2) = ((1 / 𝑦)↑2))
78	77	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑥 = (1 / 𝑦) → ((𝑥↑2) = 𝐴 ↔ ((1 / 𝑦)↑2) = 𝐴))
79	76, 78	anbi12d 743	. . . . . . . 8 ⊢ (𝑥 = (1 / 𝑦) → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴)))
80	79	rspcev 3282	. . . . . . 7 ⊢ (((1 / 𝑦) ∈ ℝ ∧ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
81	56, 61, 75, 80	syl12anc 1316	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
82	81	rexlimdv3a 3015	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
83	48, 82	mpd 15	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
84	83	ex 449	. . 3 ⊢ (𝐴 ∈ ℝ → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
85	84	adantr 480	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
86		simpl 472	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
87		letric 10016	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴))
88	86, 32, 87	sylancl 693	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴))
89	30, 85, 88	mpjaod 395	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   class class class wbr 4583  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   < clt 9953   ≤ cle 9954   / cdiv 10563  2c2 10947  ℝ⁺crp 11708  ↑cexp 12722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-seq 12664  df-exp 12723

This theorem is referenced by:  resqreu  13841  resqrtcl  13842