constrresqrtcl - Mathbox for Thierry Arnoux

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Theorem constrresqrtcl 34457

Description: If a positive real number 𝑋 is constructible, then, so is its square root. (Contributed by Thierry Arnoux, 5-Nov-2025.)

**Hypotheses**
Ref	Expression
constrresqrtcl.1	⊢ (𝜑 → 𝑋 ∈ Constr)
constrresqrtcl.2	⊢ (𝜑 → 𝑋 ∈ ℝ)
constrresqrtcl.3	⊢ (𝜑 → 0 ≤ 𝑋)

**Assertion**
Ref	Expression
constrresqrtcl	⊢ (𝜑 → (√‘𝑋) ∈ Constr)

**Proof of Theorem constrresqrtcl**
Step	Hyp	Ref	Expression
1		0zd 12710	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
2	1	zconstr 34444	. 2 ⊢ (𝜑 → 0 ∈ Constr)
3		1zzd 12732	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
4	3	zconstr 34444	. 2 ⊢ (𝜑 → 1 ∈ Constr)
5		iconstr 34446	. . . 4 ⊢ i ∈ Constr
6	5	a1i 11	. . 3 ⊢ (𝜑 → i ∈ Constr)
7		constrresqrtcl.2	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8	7	recnd 11342	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
9		1cnd 11307	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
10	8, 9	subcld 11674	. . . . 5 ⊢ (𝜑 → (𝑋 − 1) ∈ ℂ)
11		2cnd 12426	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
12		2ne0 12454	. . . . . 6 ⊢ 2 ≠ 0
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ≠ 0)
14	10, 11, 13	divrecd 12101	. . . 4 ⊢ (𝜑 → ((𝑋 − 1) / 2) = ((𝑋 − 1) · (1 / 2)))
15	8, 9	negsubd 11680	. . . . . 6 ⊢ (𝜑 → (𝑋 + -1) = (𝑋 − 1))
16		constrresqrtcl.1	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Constr)
17	4	constrnegcl 34443	. . . . . . 7 ⊢ (𝜑 → -1 ∈ Constr)
18	16, 17	constraddcl 34442	. . . . . 6 ⊢ (𝜑 → (𝑋 + -1) ∈ Constr)
19	15, 18	eqeltrrd 2861	. . . . 5 ⊢ (𝜑 → (𝑋 − 1) ∈ Constr)
20		2z 12733	. . . . . . . 8 ⊢ 2 ∈ ℤ
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ)
22	21	zconstr 34444	. . . . . 6 ⊢ (𝜑 → 2 ∈ Constr)
23	22, 13	constrinvcl 34453	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ Constr)
24	19, 23	constrmulcl 34451	. . . 4 ⊢ (𝜑 → ((𝑋 − 1) · (1 / 2)) ∈ Constr)
25	14, 24	eqeltrd 2860	. . 3 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ Constr)
26	6, 25	constrmulcl 34451	. 2 ⊢ (𝜑 → (i · ((𝑋 − 1) / 2)) ∈ Constr)
27	8, 9	addcld 11333	. . . 4 ⊢ (𝜑 → (𝑋 + 1) ∈ ℂ)
28	27, 11, 13	divrecd 12101	. . 3 ⊢ (𝜑 → ((𝑋 + 1) / 2) = ((𝑋 + 1) · (1 / 2)))
29	16, 4	constraddcl 34442	. . . 4 ⊢ (𝜑 → (𝑋 + 1) ∈ Constr)
30	29, 23	constrmulcl 34451	. . 3 ⊢ (𝜑 → ((𝑋 + 1) · (1 / 2)) ∈ Constr)
31	28, 30	eqeltrd 2860	. 2 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ Constr)
32		constrresqrtcl.3	. . 3 ⊢ (𝜑 → 0 ≤ 𝑋)
33	7, 32	resqrtcld 15592	. 2 ⊢ (𝜑 → (√‘𝑋) ∈ ℝ)
34	33	recnd 11342	. 2 ⊢ (𝜑 → (√‘𝑋) ∈ ℂ)
35	9	subid1d 11663	. . . . . 6 ⊢ (𝜑 → (1 − 0) = 1)
36	35, 9	eqeltrd 2860	. . . . 5 ⊢ (𝜑 → (1 − 0) ∈ ℂ)
37	34, 36	mulcld 11334	. . . 4 ⊢ (𝜑 → ((√‘𝑋) · (1 − 0)) ∈ ℂ)
38	37	addlidd 11516	. . 3 ⊢ (𝜑 → (0 + ((√‘𝑋) · (1 − 0))) = ((√‘𝑋) · (1 − 0)))
39	35	oveq2d 7438	. . 3 ⊢ (𝜑 → ((√‘𝑋) · (1 − 0)) = ((√‘𝑋) · 1))
40	34	mulridd 11331	. . 3 ⊢ (𝜑 → ((√‘𝑋) · 1) = (√‘𝑋))
41	38, 39, 40	3eqtrrd 2800	. 2 ⊢ (𝜑 → (√‘𝑋) = (0 + ((√‘𝑋) · (1 − 0))))
42		1red 11314	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
43	7, 42	readdcld 11343	. . . . 5 ⊢ (𝜑 → (𝑋 + 1) ∈ ℝ)
44	43	rehalfcld 12598	. . . 4 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ ℝ)
45		2rp 13130	. . . . . 6 ⊢ 2 ∈ ℝ⁺
46	45	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ⁺)
47		0red 11316	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
48	7	lep1d 12253	. . . . . 6 ⊢ (𝜑 → 𝑋 ≤ (𝑋 + 1))
49	47, 7, 43, 32, 48	letrd 11472	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝑋 + 1))
50	43, 46, 49	divge0d 13209	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝑋 + 1) / 2))
51	44, 50	absidd 15597	. . 3 ⊢ (𝜑 → (abs‘((𝑋 + 1) / 2)) = ((𝑋 + 1) / 2))
52	27	halfcld 12596	. . . . 5 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ ℂ)
53	52	subid1d 11663	. . . 4 ⊢ (𝜑 → (((𝑋 + 1) / 2) − 0) = ((𝑋 + 1) / 2))
54	53	fveq2d 6892	. . 3 ⊢ (𝜑 → (abs‘(((𝑋 + 1) / 2) − 0)) = (abs‘((𝑋 + 1) / 2)))
55		ax-icn 11264	. . . . . . . . 9 ⊢ i ∈ ℂ
56	55	a1i 11	. . . . . . . 8 ⊢ (𝜑 → i ∈ ℂ)
57	7, 42	resubcld 11749	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 − 1) ∈ ℝ)
58	57	rehalfcld 12598	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ ℝ)
59	58	recnd 11342	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ ℂ)
60	56, 59	mulneg2d 11775	. . . . . . 7 ⊢ (𝜑 → (i · -((𝑋 − 1) / 2)) = -(i · ((𝑋 − 1) / 2)))
61	60	oveq2d 7438	. . . . . 6 ⊢ (𝜑 → ((√‘𝑋) + (i · -((𝑋 − 1) / 2))) = ((√‘𝑋) + -(i · ((𝑋 − 1) / 2))))
62	26	constrcn 34440	. . . . . . 7 ⊢ (𝜑 → (i · ((𝑋 − 1) / 2)) ∈ ℂ)
63	34, 62	negsubd 11680	. . . . . 6 ⊢ (𝜑 → ((√‘𝑋) + -(i · ((𝑋 − 1) / 2))) = ((√‘𝑋) − (i · ((𝑋 − 1) / 2))))
64	61, 63	eqtr2d 2796	. . . . 5 ⊢ (𝜑 → ((√‘𝑋) − (i · ((𝑋 − 1) / 2))) = ((√‘𝑋) + (i · -((𝑋 − 1) / 2))))
65	64	fveq2d 6892	. . . 4 ⊢ (𝜑 → (abs‘((√‘𝑋) − (i · ((𝑋 − 1) / 2)))) = (abs‘((√‘𝑋) + (i · -((𝑋 − 1) / 2)))))
66	58	renegcld 11748	. . . . 5 ⊢ (𝜑 → -((𝑋 − 1) / 2) ∈ ℝ)
67		absreim 15467	. . . . 5 ⊢ (((√‘𝑋) ∈ ℝ ∧ -((𝑋 − 1) / 2) ∈ ℝ) → (abs‘((√‘𝑋) + (i · -((𝑋 − 1) / 2)))) = (√‘(((√‘𝑋)↑2) + ( -((𝑋 − 1) / 2)↑2))))
68	33, 66, 67	syl2anc 596	. . . 4 ⊢ (𝜑 → (abs‘((√‘𝑋) + (i · -((𝑋 − 1) / 2)))) = (√‘(((√‘𝑋)↑2) + ( -((𝑋 − 1) / 2)↑2))))
69		sq2 14345	. . . . . . . . . . 11 ⊢ (2↑2) = 4
70	69	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (2↑2) = 4)
71	70	oveq2d 7438	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) / (2↑2)) = ((4 · 𝑋) / 4))
72		4cn 12433	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
73	72	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 4 ∈ ℂ)
74	11, 13, 21	expne0d 14300	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑2) ≠ 0)
75	69, 74	eqnetrrid 3030	. . . . . . . . . 10 ⊢ (𝜑 → 4 ≠ 0)
76	8, 73, 75	divcan3d 12103	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) / 4) = 𝑋)
77	71, 76	eqtr2d 2796	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ((4 · 𝑋) / (2↑2)))
78	10, 11, 13	sqdivd 14307	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 − 1) / 2)↑2) = (((𝑋 − 1)↑2) / (2↑2)))
79	77, 78	oveq12d 7440	. . . . . . 7 ⊢ (𝜑 → (𝑋 + (((𝑋 − 1) / 2)↑2)) = (((4 · 𝑋) / (2↑2)) + (((𝑋 − 1)↑2) / (2↑2))))
80	8	sqsqrtd 15616	. . . . . . . 8 ⊢ (𝜑 → ((√‘𝑋)↑2) = 𝑋)
81	59	sqnegd 14264	. . . . . . . 8 ⊢ (𝜑 → ( -((𝑋 − 1) / 2)↑2) = (((𝑋 − 1) / 2)↑2))
82	80, 81	oveq12d 7440	. . . . . . 7 ⊢ (𝜑 → (((√‘𝑋)↑2) + ( -((𝑋 − 1) / 2)↑2)) = (𝑋 + (((𝑋 − 1) / 2)↑2)))
83	27, 11, 13	sqdivd 14307	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 + 1) / 2)↑2) = (((𝑋 + 1)↑2) / (2↑2)))
84	27	sqcld 14292	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 + 1)↑2) ∈ ℂ)
85	10	sqcld 14292	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 − 1)↑2) ∈ ℂ)
86	73, 8	mulcld 11334	. . . . . . . . . 10 ⊢ (𝜑 → (4 · 𝑋) ∈ ℂ)
87	8, 9	binom2subadd 33381	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑋 + 1)↑2) − ((𝑋 − 1)↑2)) = (4 · (𝑋 · 1)))
88	8	mulridd 11331	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 · 1) = 𝑋)
89	88	oveq2d 7438	. . . . . . . . . . 11 ⊢ (𝜑 → (4 · (𝑋 · 1)) = (4 · 𝑋))
90	87, 89	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑋 + 1)↑2) − ((𝑋 − 1)↑2)) = (4 · 𝑋))
91		subadd2 11566	. . . . . . . . . . 11 ⊢ ((((𝑋 + 1)↑2) ∈ ℂ ∧ ((𝑋 − 1)↑2) ∈ ℂ ∧ (4 · 𝑋) ∈ ℂ) → ((((𝑋 + 1)↑2) − ((𝑋 − 1)↑2)) = (4 · 𝑋) ↔ ((4 · 𝑋) + ((𝑋 − 1)↑2)) = ((𝑋 + 1)↑2)))
92	91	biimpa 482	. . . . . . . . . 10 ⊢ (((((𝑋 + 1)↑2) ∈ ℂ ∧ ((𝑋 − 1)↑2) ∈ ℂ ∧ (4 · 𝑋) ∈ ℂ) ∧ (((𝑋 + 1)↑2) − ((𝑋 − 1)↑2)) = (4 · 𝑋)) → ((4 · 𝑋) + ((𝑋 − 1)↑2)) = ((𝑋 + 1)↑2))
93	84, 85, 86, 90, 92	syl31anc 1400	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) + ((𝑋 − 1)↑2)) = ((𝑋 + 1)↑2))
94	93	oveq1d 7437	. . . . . . . 8 ⊢ (𝜑 → (((4 · 𝑋) + ((𝑋 − 1)↑2)) / (2↑2)) = (((𝑋 + 1)↑2) / (2↑2)))
95	11	sqcld 14292	. . . . . . . . 9 ⊢ (𝜑 → (2↑2) ∈ ℂ)
96	86, 85, 95, 74	divdird 12136	. . . . . . . 8 ⊢ (𝜑 → (((4 · 𝑋) + ((𝑋 − 1)↑2)) / (2↑2)) = (((4 · 𝑋) / (2↑2)) + (((𝑋 − 1)↑2) / (2↑2))))
97	83, 94, 96	3eqtr2d 2801	. . . . . . 7 ⊢ (𝜑 → (((𝑋 + 1) / 2)↑2) = (((4 · 𝑋) / (2↑2)) + (((𝑋 − 1)↑2) / (2↑2))))
98	79, 82, 97	3eqtr4d 2805	. . . . . 6 ⊢ (𝜑 → (((√‘𝑋)↑2) + ( -((𝑋 − 1) / 2)↑2)) = (((𝑋 + 1) / 2)↑2))
99	98	fveq2d 6892	. . . . 5 ⊢ (𝜑 → (√‘(((√‘𝑋)↑2) + ( -((𝑋 − 1) / 2)↑2))) = (√‘(((𝑋 + 1) / 2)↑2)))
100	44, 50	sqrtsqd 15594	. . . . 5 ⊢ (𝜑 → (√‘(((𝑋 + 1) / 2)↑2)) = ((𝑋 + 1) / 2))
101	99, 100	eqtrd 2795	. . . 4 ⊢ (𝜑 → (√‘(((√‘𝑋)↑2) + ( -((𝑋 − 1) / 2)↑2))) = ((𝑋 + 1) / 2))
102	65, 68, 101	3eqtrd 2799	. . 3 ⊢ (𝜑 → (abs‘((√‘𝑋) − (i · ((𝑋 − 1) / 2)))) = ((𝑋 + 1) / 2))
103	51, 54, 102	3eqtr4rd 2806	. 2 ⊢ (𝜑 → (abs‘((√‘𝑋) − (i · ((𝑋 − 1) / 2)))) = (abs‘(((𝑋 + 1) / 2) − 0)))
104	2, 4, 26, 31, 2, 33, 34, 41, 103	constrlccl 34437	1 ⊢ (𝜑 → (√‘𝑋) ∈ Constr)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ‘cfv 6541 (class class class)co 7422 ℂcc 11203 ℝcr 11204 0cc0 11205 1c1 11206 ici 11207 + caddc 11208 · cmul 11210 ≤ cle 11349 − cmin 11546 -cneg 11547 / cdiv 11978 2c2 12402 4c4 12404 ℤcz 12698 ℝ⁺crp 13125 ↑cexp 14209 √csqrt 15407 abscabs 15408 Constrcconstr 34409

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-inf2 9647 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-pre-sup 11283 ax-addf 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-of 7695 df-om 7880 df-1st 8003 df-2nd 8004 df-supp 8182 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-2o 8482 df-er 8722 df-map 8854 df-pm 8855 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-fsupp 9359 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9509 df-card 10025 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-div 11979 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-7 12415 df-8 12416 df-9 12417 df-n0 12612 df-z 12699 df-dec 12820 df-uz 12971 df-q 13081 df-rp 13126 df-xneg 13246 df-xadd 13247 df-xmul 13248 df-ioo 13485 df-ioc 13486 df-ico 13487 df-icc 13488 df-fz 13645 df-fzo 13794 df-fl 13937 df-mod 14015 df-seq 14150 df-exp 14210 df-fac 14423 df-bc 14452 df-hash 14480 df-shft 15227 df-cj 15273 df-re 15274 df-im 15275 df-sqrt 15409 df-abs 15410 df-limsup 15645 df-clim 15662 df-rlim 15663 df-sum 15861 df-ef 16240 df-sin 16242 df-cos 16243 df-pi 16245 df-struct 17332 df-sets 17349 df-slot 17367 df-ndx 17379 df-base 17395 df-ress 17416 df-plusg 17448 df-mulr 17449 df-starv 17450 df-sca 17451 df-vsca 17452 df-ip 17453 df-tset 17454 df-ple 17455 df-ds 17457 df-unif 17458 df-hom 17459 df-cco 17460 df-rest 17600 df-topn 17601 df-0g 17619 df-gsum 17620 df-topgen 17621 df-pt 17622 df-prds 17625 df-xrs 17681 df-qtop 17686 df-imas 17687 df-xps 17689 df-mre 17763 df-mrc 17764 df-acs 17766 df-mgm 18823 df-sgrp 18915 df-mnd 18931 df-submnd 18986 df-mulg 19285 df-cntz 19538 df-cmn 20003 df-psmet 21677 df-xmet 21678 df-met 21679 df-bl 21680 df-mopn 21681 df-fbas 21682 df-fg 21683 df-cnfld 21686 df-top 23219 df-topon 23236 df-topsp 23258 df-bases 23271 df-cld 23344 df-ntr 23345 df-cls 23346 df-nei 23423 df-lp 23461 df-perf 23462 df-cn 23552 df-cnp 23553 df-haus 23640 df-tx 23888 df-hmeo 24081 df-fil 24172 df-fm 24264 df-flim 24265 df-flf 24266 df-xms 24646 df-ms 24647 df-tms 24648 df-cncf 25206 df-limc 26193 df-dv 26194 df-log 26891 df-constr 34410

This theorem is used by: constrsqrtcl 34459

W3C validator