constrresqrtcl - Mathbox for Thierry Arnoux

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Theorem constrresqrtcl 34520

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 12745	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
2	1	zconstr 34507	. 2 ⊢ (𝜑 → 0 ∈ Constr)
3		1zzd 12767	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
4	3	zconstr 34507	. 2 ⊢ (𝜑 → 1 ∈ Constr)
5		iconstr 34509	. . . 4 ⊢ i ∈ Constr
6	5	a1i 11	. . 3 ⊢ (𝜑 → i ∈ Constr)
7		constrresqrtcl.2	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8	7	recnd 11376	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
9		1cnd 11341	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
10	8, 9	subcld 11708	. . . . 5 ⊢ (𝜑 → (𝑋 − 1) ∈ ℂ)
11		2cnd 12461	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
12		2ne0 12489	. . . . . 6 ⊢ 2 ≠ 0
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ≠ 0)
14	10, 11, 13	divrecd 12136	. . . 4 ⊢ (𝜑 → ((𝑋 − 1) / 2) = ((𝑋 − 1) · (1 / 2)))
15	8, 9	negsubd 11714	. . . . . 6 ⊢ (𝜑 → (𝑋 + -1) = (𝑋 − 1))
16		constrresqrtcl.1	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Constr)
17	4	constrnegcl 34506	. . . . . . 7 ⊢ (𝜑 → -1 ∈ Constr)
18	16, 17	constraddcl 34505	. . . . . 6 ⊢ (𝜑 → (𝑋 + -1) ∈ Constr)
19	15, 18	eqeltrrd 2865	. . . . 5 ⊢ (𝜑 → (𝑋 − 1) ∈ Constr)
20		2z 12768	. . . . . . . 8 ⊢ 2 ∈ ℤ
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ)
22	21	zconstr 34507	. . . . . 6 ⊢ (𝜑 → 2 ∈ Constr)
23	22, 13	constrinvcl 34516	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ Constr)
24	19, 23	constrmulcl 34514	. . . 4 ⊢ (𝜑 → ((𝑋 − 1) · (1 / 2)) ∈ Constr)
25	14, 24	eqeltrd 2864	. . 3 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ Constr)
26	6, 25	constrmulcl 34514	. 2 ⊢ (𝜑 → (i · ((𝑋 − 1) / 2)) ∈ Constr)
27	8, 9	addcld 11367	. . . 4 ⊢ (𝜑 → (𝑋 + 1) ∈ ℂ)
28	27, 11, 13	divrecd 12136	. . 3 ⊢ (𝜑 → ((𝑋 + 1) / 2) = ((𝑋 + 1) · (1 / 2)))
29	16, 4	constraddcl 34505	. . . 4 ⊢ (𝜑 → (𝑋 + 1) ∈ Constr)
30	29, 23	constrmulcl 34514	. . 3 ⊢ (𝜑 → ((𝑋 + 1) · (1 / 2)) ∈ Constr)
31	28, 30	eqeltrd 2864	. 2 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ Constr)
32		constrresqrtcl.3	. . 3 ⊢ (𝜑 → 0 ≤ 𝑋)
33	7, 32	resqrtcld 15629	. 2 ⊢ (𝜑 → (√‘𝑋) ∈ ℝ)
34	33	recnd 11376	. 2 ⊢ (𝜑 → (√‘𝑋) ∈ ℂ)
35	9	subid1d 11697	. . . . . 6 ⊢ (𝜑 → (1 − 0) = 1)
36	35, 9	eqeltrd 2864	. . . . 5 ⊢ (𝜑 → (1 − 0) ∈ ℂ)
37	34, 36	mulcld 11368	. . . 4 ⊢ (𝜑 → ((√‘𝑋) · (1 − 0)) ∈ ℂ)
38	37	addlidd 11550	. . 3 ⊢ (𝜑 → (0 + ((√‘𝑋) · (1 − 0))) = ((√‘𝑋) · (1 − 0)))
39	35	oveq2d 7467	. . 3 ⊢ (𝜑 → ((√‘𝑋) · (1 − 0)) = ((√‘𝑋) · 1))
40	34	mulridd 11365	. . 3 ⊢ (𝜑 → ((√‘𝑋) · 1) = (√‘𝑋))
41	38, 39, 40	3eqtrrd 2803	. 2 ⊢ (𝜑 → (√‘𝑋) = (0 + ((√‘𝑋) · (1 − 0))))
42		1red 11348	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
43	7, 42	readdcld 11377	. . . . 5 ⊢ (𝜑 → (𝑋 + 1) ∈ ℝ)
44	43	rehalfcld 12633	. . . 4 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ ℝ)
45		2rp 13165	. . . . . 6 ⊢ 2 ∈ ℝ⁺
46	45	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ⁺)
47		0red 11350	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
48	7	lep1d 12288	. . . . . 6 ⊢ (𝜑 → 𝑋 ≤ (𝑋 + 1))
49	47, 7, 43, 32, 48	letrd 11506	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝑋 + 1))
50	43, 46, 49	divge0d 13244	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝑋 + 1) / 2))
51	44, 50	absidd 15634	. . 3 ⊢ (𝜑 → (abs‘((𝑋 + 1) / 2)) = ((𝑋 + 1) / 2))
52	27	halfcld 12631	. . . . 5 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ ℂ)
53	52	subid1d 11697	. . . 4 ⊢ (𝜑 → (((𝑋 + 1) / 2) − 0) = ((𝑋 + 1) / 2))
54	53	fveq2d 6920	. . 3 ⊢ (𝜑 → (abs‘(((𝑋 + 1) / 2) − 0)) = (abs‘((𝑋 + 1) / 2)))
55		ax-icn 11298	. . . . . . . . 9 ⊢ i ∈ ℂ
56	55	a1i 11	. . . . . . . 8 ⊢ (𝜑 → i ∈ ℂ)
57	7, 42	resubcld 11783	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 − 1) ∈ ℝ)
58	57	rehalfcld 12633	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ ℝ)
59	58	recnd 11376	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ ℂ)
60	56, 59	mulneg2d 11809	. . . . . . 7 ⊢ (𝜑 → (i · -((𝑋 − 1) / 2)) = -(i · ((𝑋 − 1) / 2)))
61	60	oveq2d 7467	. . . . . 6 ⊢ (𝜑 → ((√‘𝑋) + (i · -((𝑋 − 1) / 2))) = ((√‘𝑋) + -(i · ((𝑋 − 1) / 2))))
62	26	constrcn 34503	. . . . . . 7 ⊢ (𝜑 → (i · ((𝑋 − 1) / 2)) ∈ ℂ)
63	34, 62	negsubd 11714	. . . . . 6 ⊢ (𝜑 → ((√‘𝑋) + -(i · ((𝑋 − 1) / 2))) = ((√‘𝑋) − (i · ((𝑋 − 1) / 2))))
64	61, 63	eqtr2d 2799	. . . . 5 ⊢ (𝜑 → ((√‘𝑋) − (i · ((𝑋 − 1) / 2))) = ((√‘𝑋) + (i · -((𝑋 − 1) / 2))))
65	64	fveq2d 6920	. . . 4 ⊢ (𝜑 → (abs‘((√‘𝑋) − (i · ((𝑋 − 1) / 2)))) = (abs‘((√‘𝑋) + (i · -((𝑋 − 1) / 2)))))
66	58	renegcld 11782	. . . . 5 ⊢ (𝜑 → -((𝑋 − 1) / 2) ∈ ℝ)
67		absreim 15503	. . . . 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 14380	. . . . . . . . . . 11 ⊢ (2↑2) = 4
70	69	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (2↑2) = 4)
71	70	oveq2d 7467	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) / (2↑2)) = ((4 · 𝑋) / 4))
72		4cn 12468	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
73	72	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 4 ∈ ℂ)
74	11, 13, 21	expne0d 14335	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑2) ≠ 0)
75	69, 74	eqnetrrid 3034	. . . . . . . . . 10 ⊢ (𝜑 → 4 ≠ 0)
76	8, 73, 75	divcan3d 12138	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) / 4) = 𝑋)
77	71, 76	eqtr2d 2799	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ((4 · 𝑋) / (2↑2)))
78	10, 11, 13	sqdivd 14342	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 − 1) / 2)↑2) = (((𝑋 − 1)↑2) / (2↑2)))
79	77, 78	oveq12d 7469	. . . . . . 7 ⊢ (𝜑 → (𝑋 + (((𝑋 − 1) / 2)↑2)) = (((4 · 𝑋) / (2↑2)) + (((𝑋 − 1)↑2) / (2↑2))))
80	8	sqsqrtd 15653	. . . . . . . 8 ⊢ (𝜑 → ((√‘𝑋)↑2) = 𝑋)
81	59	sqnegd 14299	. . . . . . . 8 ⊢ (𝜑 → ( -((𝑋 − 1) / 2)↑2) = (((𝑋 − 1) / 2)↑2))
82	80, 81	oveq12d 7469	. . . . . . 7 ⊢ (𝜑 → (((√‘𝑋)↑2) + ( -((𝑋 − 1) / 2)↑2)) = (𝑋 + (((𝑋 − 1) / 2)↑2)))
83	27, 11, 13	sqdivd 14342	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 + 1) / 2)↑2) = (((𝑋 + 1)↑2) / (2↑2)))
84	27	sqcld 14327	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 + 1)↑2) ∈ ℂ)
85	10	sqcld 14327	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 − 1)↑2) ∈ ℂ)
86	73, 8	mulcld 11368	. . . . . . . . . 10 ⊢ (𝜑 → (4 · 𝑋) ∈ ℂ)
87	8, 9	binom2subadd 33429	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑋 + 1)↑2) − ((𝑋 − 1)↑2)) = (4 · (𝑋 · 1)))
88	8	mulridd 11365	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 · 1) = 𝑋)
89	88	oveq2d 7467	. . . . . . . . . . 11 ⊢ (𝜑 → (4 · (𝑋 · 1)) = (4 · 𝑋))
90	87, 89	eqtrd 2798	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑋 + 1)↑2) − ((𝑋 − 1)↑2)) = (4 · 𝑋))
91		subadd2 11600	. . . . . . . . . . 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 1401	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) + ((𝑋 − 1)↑2)) = ((𝑋 + 1)↑2))
94	93	oveq1d 7466	. . . . . . . 8 ⊢ (𝜑 → (((4 · 𝑋) + ((𝑋 − 1)↑2)) / (2↑2)) = (((𝑋 + 1)↑2) / (2↑2)))
95	11	sqcld 14327	. . . . . . . . 9 ⊢ (𝜑 → (2↑2) ∈ ℂ)
96	86, 85, 95, 74	divdird 12171	. . . . . . . 8 ⊢ (𝜑 → (((4 · 𝑋) + ((𝑋 − 1)↑2)) / (2↑2)) = (((4 · 𝑋) / (2↑2)) + (((𝑋 − 1)↑2) / (2↑2))))
97	83, 94, 96	3eqtr2d 2804	. . . . . . 7 ⊢ (𝜑 → (((𝑋 + 1) / 2)↑2) = (((4 · 𝑋) / (2↑2)) + (((𝑋 − 1)↑2) / (2↑2))))
98	79, 82, 97	3eqtr4d 2808	. . . . . 6 ⊢ (𝜑 → (((√‘𝑋)↑2) + ( -((𝑋 − 1) / 2)↑2)) = (((𝑋 + 1) / 2)↑2))
99	98	fveq2d 6920	. . . . 5 ⊢ (𝜑 → (√‘(((√‘𝑋)↑2) + ( -((𝑋 − 1) / 2)↑2))) = (√‘(((𝑋 + 1) / 2)↑2)))
100	44, 50	sqrtsqd 15631	. . . . 5 ⊢ (𝜑 → (√‘(((𝑋 + 1) / 2)↑2)) = ((𝑋 + 1) / 2))
101	99, 100	eqtrd 2798	. . . 4 ⊢ (𝜑 → (√‘(((√‘𝑋)↑2) + ( -((𝑋 − 1) / 2)↑2))) = ((𝑋 + 1) / 2))
102	65, 68, 101	3eqtrd 2802	. . 3 ⊢ (𝜑 → (abs‘((√‘𝑋) − (i · ((𝑋 − 1) / 2)))) = ((𝑋 + 1) / 2))
103	51, 54, 102	3eqtr4rd 2809	. 2 ⊢ (𝜑 → (abs‘((√‘𝑋) − (i · ((𝑋 − 1) / 2)))) = (abs‘(((𝑋 + 1) / 2) − 0)))
104	2, 4, 26, 31, 2, 33, 34, 41, 103	constrlccl 34500	1 ⊢ (𝜑 → (√‘𝑋) ∈ Constr)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 ici 11241 + caddc 11242 · cmul 11244 ≤ cle 11383 − cmin 11580 -cneg 11581 / cdiv 12013 2c2 12437 4c4 12439 ℤcz 12733 ℝ⁺crp 13160 ↑cexp 14244 √csqrt 15443 abscabs 15444 Constrcconstr 34472

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-ef 16277 df-sin 16279 df-cos 16280 df-pi 16282 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-log 26941 df-constr 34473

This theorem is used by: constrsqrtcl 34522

W3C validator