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Theorem resqrex 13787
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.
StepHypRef Expression
1 0re 9896 . . . . 5 0 ∈ ℝ
2 leloe 9975 . . . . 5 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
31, 2mpan 701 . . . 4 (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
4 elrp 11668 . . . . . . 7 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
5 01sqrex 13786 . . . . . . . 8 ((𝐴 ∈ ℝ+𝐴 ≤ 1) → ∃𝑥 ∈ ℝ+ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴))
6 rprege0 11681 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
76anim1i 589 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+ ∧ (𝑥↑2) = 𝐴) → ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴))
8 anass 678 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑥↑2) = 𝐴) ↔ (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
97, 8sylib 206 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ (𝑥↑2) = 𝐴) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
109adantrl 747 . . . . . . . . 9 ((𝑥 ∈ ℝ+ ∧ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴)) → (𝑥 ∈ ℝ ∧ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
1110reximi2 2992 . . . . . . . 8 (∃𝑥 ∈ ℝ+ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
125, 11syl 17 . . . . . . 7 ((𝐴 ∈ ℝ+𝐴 ≤ 1) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
134, 12sylanbr 488 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
1413exp31 627 . . . . 5 (𝐴 ∈ ℝ → (0 < 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
15 sq0 12774 . . . . . . . . . 10 (0↑2) = 0
16 id 22 . . . . . . . . . 10 (0 = 𝐴 → 0 = 𝐴)
1715, 16syl5eq 2655 . . . . . . . . 9 (0 = 𝐴 → (0↑2) = 𝐴)
18 0le0 10959 . . . . . . . . 9 0 ≤ 0
1917, 18jctil 557 . . . . . . . 8 (0 = 𝐴 → (0 ≤ 0 ∧ (0↑2) = 𝐴))
20 breq2 4581 . . . . . . . . . 10 (𝑥 = 0 → (0 ≤ 𝑥 ↔ 0 ≤ 0))
21 oveq1 6533 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥↑2) = (0↑2))
2221eqeq1d 2611 . . . . . . . . . 10 (𝑥 = 0 → ((𝑥↑2) = 𝐴 ↔ (0↑2) = 𝐴))
2320, 22anbi12d 742 . . . . . . . . 9 (𝑥 = 0 → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ 0 ∧ (0↑2) = 𝐴)))
2423rspcev 3281 . . . . . . . 8 ((0 ∈ ℝ ∧ (0 ≤ 0 ∧ (0↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
251, 19, 24sylancr 693 . . . . . . 7 (0 = 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
2625a1d 25 . . . . . 6 (0 = 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
2726a1i 11 . . . . 5 (𝐴 ∈ ℝ → (0 = 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
2814, 27jaod 393 . . . 4 (𝐴 ∈ ℝ → ((0 < 𝐴 ∨ 0 = 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
293, 28sylbid 228 . . 3 (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))))
3029imp 443 . 2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≤ 1 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
31 0lt1 10401 . . . . . . . . . 10 0 < 1
32 1re 9895 . . . . . . . . . . 11 1 ∈ ℝ
33 ltletr 9980 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴))
341, 32, 33mp3an12 1405 . . . . . . . . . 10 (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴))
3531, 34mpani 707 . . . . . . . . 9 (𝐴 ∈ ℝ → (1 ≤ 𝐴 → 0 < 𝐴))
3635imp 443 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 𝐴)
374biimpri 216 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ+)
3836, 37syldan 485 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+)
3938rpreccld 11716 . . . . . 6 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ∈ ℝ+)
40 simpr 475 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
41 lerec 10757 . . . . . . . . . 10 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
4232, 31, 41mpanl12 713 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
4336, 42syldan 485 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / 1)))
4440, 43mpbid 220 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ≤ (1 / 1))
45 1div1e1 10568 . . . . . . 7 (1 / 1) = 1
4644, 45syl6breq 4618 . . . . . 6 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / 𝐴) ≤ 1)
47 01sqrex 13786 . . . . . 6 (((1 / 𝐴) ∈ ℝ+ ∧ (1 / 𝐴) ≤ 1) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)))
4839, 46, 47syl2anc 690 . . . . 5 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)))
49 rpre 11673 . . . . . . . . 9 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
50493ad2ant2 1075 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 𝑦 ∈ ℝ)
51 rpgt0 11678 . . . . . . . . 9 (𝑦 ∈ ℝ+ → 0 < 𝑦)
52513ad2ant2 1075 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 < 𝑦)
53 gt0ne0 10344 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 𝑦 ≠ 0)
54 rereccl 10594 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝑦 ≠ 0) → (1 / 𝑦) ∈ ℝ)
5553, 54syldan 485 . . . . . . . 8 ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → (1 / 𝑦) ∈ ℝ)
5650, 52, 55syl2anc 690 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / 𝑦) ∈ ℝ)
57 recgt0 10718 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 0 < (1 / 𝑦))
58 ltle 9977 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / 𝑦) ∈ ℝ) → (0 < (1 / 𝑦) → 0 ≤ (1 / 𝑦)))
591, 58mpan 701 . . . . . . . . 9 ((1 / 𝑦) ∈ ℝ → (0 < (1 / 𝑦) → 0 ≤ (1 / 𝑦)))
6055, 57, 59sylc 62 . . . . . . . 8 ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 0 ≤ (1 / 𝑦))
6150, 52, 60syl2anc 690 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → 0 ≤ (1 / 𝑦))
62 recn 9882 . . . . . . . . . . 11 (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
6362adantr 479 . . . . . . . . . 10 ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → 𝑦 ∈ ℂ)
6463, 53sqrecd 12831 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 0 < 𝑦) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2)))
6550, 52, 64syl2anc 690 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = (1 / (𝑦↑2)))
66 simp3r 1082 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (𝑦↑2) = (1 / 𝐴))
6766oveq2d 6542 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (𝑦↑2)) = (1 / (1 / 𝐴)))
68 gt0ne0 10344 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0)
6936, 68syldan 485 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ≠ 0)
70 recn 9882 . . . . . . . . . . 11 (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
71 recrec 10573 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / (1 / 𝐴)) = 𝐴)
7270, 71sylan 486 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / (1 / 𝐴)) = 𝐴)
7369, 72syldan 485 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1 / (1 / 𝐴)) = 𝐴)
74733ad2ant1 1074 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → (1 / (1 / 𝐴)) = 𝐴)
7565, 67, 743eqtrd 2647 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ((1 / 𝑦)↑2) = 𝐴)
76 breq2 4581 . . . . . . . . 9 (𝑥 = (1 / 𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (1 / 𝑦)))
77 oveq1 6533 . . . . . . . . . 10 (𝑥 = (1 / 𝑦) → (𝑥↑2) = ((1 / 𝑦)↑2))
7877eqeq1d 2611 . . . . . . . . 9 (𝑥 = (1 / 𝑦) → ((𝑥↑2) = 𝐴 ↔ ((1 / 𝑦)↑2) = 𝐴))
7976, 78anbi12d 742 . . . . . . . 8 (𝑥 = (1 / 𝑦) → ((0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴) ↔ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴)))
8079rspcev 3281 . . . . . . 7 (((1 / 𝑦) ∈ ℝ ∧ (0 ≤ (1 / 𝑦) ∧ ((1 / 𝑦)↑2) = 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
8156, 61, 75, 80syl12anc 1315 . . . . . 6 (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℝ+ ∧ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴))) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
8281rexlimdv3a 3014 . . . . 5 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (∃𝑦 ∈ ℝ+ (𝑦 ≤ 1 ∧ (𝑦↑2) = (1 / 𝐴)) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
8348, 82mpd 15 . . . 4 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
8483ex 448 . . 3 (𝐴 ∈ ℝ → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
8584adantr 479 . 2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1 ≤ 𝐴 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)))
86 simpl 471 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
87 letric 9988 . . 3 ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴))
8886, 32, 87sylancl 692 . 2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≤ 1 ∨ 1 ≤ 𝐴))
8930, 85, 88mpjaod 394 1 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wrex 2896   class class class wbr 4577  (class class class)co 6526  cc 9790  cr 9791  0cc0 9792  1c1 9793   < clt 9930  cle 9931   / cdiv 10535  2c2 10919  +crp 11666  cexp 12679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-sup 8208  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-2 10928  df-3 10929  df-n0 11142  df-z 11213  df-uz 11522  df-rp 11667  df-seq 12621  df-exp 12680
This theorem is referenced by:  resqreu  13789  resqrtcl  13790
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