constrresqrtcl - Mathbox for Thierry Arnoux

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Theorem constrresqrtcl 34287

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 12627	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
2	1	zconstr 34274	. 2 ⊢ (𝜑 → 0 ∈ Constr)
3		1zzd 12649	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
4	3	zconstr 34274	. 2 ⊢ (𝜑 → 1 ∈ Constr)
5		iconstr 34276	. . . 4 ⊢ i ∈ Constr
6	5	a1i 11	. . 3 ⊢ (𝜑 → i ∈ Constr)
7		constrresqrtcl.2	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8	7	recnd 11261	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
9		1cnd 11226	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
10	8, 9	subcld 11593	. . . . 5 ⊢ (𝜑 → (𝑋 − 1) ∈ ℂ)
11		2cnd 12343	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
12		2ne0 12371	. . . . . 6 ⊢ 2 ≠ 0
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ≠ 0)
14	10, 11, 13	divrecd 12018	. . . 4 ⊢ (𝜑 → ((𝑋 − 1) / 2) = ((𝑋 − 1) · (1 / 2)))
15	8, 9	negsubd 11599	. . . . . 6 ⊢ (𝜑 → (𝑋 + -1) = (𝑋 − 1))
16		constrresqrtcl.1	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Constr)
17	4	constrnegcl 34273	. . . . . . 7 ⊢ (𝜑 → -1 ∈ Constr)
18	16, 17	constraddcl 34272	. . . . . 6 ⊢ (𝜑 → (𝑋 + -1) ∈ Constr)
19	15, 18	eqeltrrd 2861	. . . . 5 ⊢ (𝜑 → (𝑋 − 1) ∈ Constr)
20		2z 12650	. . . . . . . 8 ⊢ 2 ∈ ℤ
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ)
22	21	zconstr 34274	. . . . . 6 ⊢ (𝜑 → 2 ∈ Constr)
23	22, 13	constrinvcl 34283	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ Constr)
24	19, 23	constrmulcl 34281	. . . 4 ⊢ (𝜑 → ((𝑋 − 1) · (1 / 2)) ∈ Constr)
25	14, 24	eqeltrd 2860	. . 3 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ Constr)
26	6, 25	constrmulcl 34281	. 2 ⊢ (𝜑 → (i · ((𝑋 − 1) / 2)) ∈ Constr)
27	8, 9	addcld 11252	. . . 4 ⊢ (𝜑 → (𝑋 + 1) ∈ ℂ)
28	27, 11, 13	divrecd 12018	. . 3 ⊢ (𝜑 → ((𝑋 + 1) / 2) = ((𝑋 + 1) · (1 / 2)))
29	16, 4	constraddcl 34272	. . . 4 ⊢ (𝜑 → (𝑋 + 1) ∈ Constr)
30	29, 23	constrmulcl 34281	. . 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 15505	. 2 ⊢ (𝜑 → (√‘𝑋) ∈ ℝ)
34	33	recnd 11261	. 2 ⊢ (𝜑 → (√‘𝑋) ∈ ℂ)
35	9	subid1d 11582	. . . . . 6 ⊢ (𝜑 → (1 − 0) = 1)
36	35, 9	eqeltrd 2860	. . . . 5 ⊢ (𝜑 → (1 − 0) ∈ ℂ)
37	34, 36	mulcld 11253	. . . 4 ⊢ (𝜑 → ((√‘𝑋) · (1 − 0)) ∈ ℂ)
38	37	addlidd 11435	. . 3 ⊢ (𝜑 → (0 + ((√‘𝑋) · (1 − 0))) = ((√‘𝑋) · (1 − 0)))
39	35	oveq2d 7429	. . 3 ⊢ (𝜑 → ((√‘𝑋) · (1 − 0)) = ((√‘𝑋) · 1))
40	34	mulridd 11250	. . 3 ⊢ (𝜑 → ((√‘𝑋) · 1) = (√‘𝑋))
41	38, 39, 40	3eqtrrd 2800	. 2 ⊢ (𝜑 → (√‘𝑋) = (0 + ((√‘𝑋) · (1 − 0))))
42		1red 11233	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
43	7, 42	readdcld 11262	. . . . 5 ⊢ (𝜑 → (𝑋 + 1) ∈ ℝ)
44	43	rehalfcld 12515	. . . 4 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ ℝ)
45		2rp 13047	. . . . . 6 ⊢ 2 ∈ ℝ⁺
46	45	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ⁺)
47		0red 11235	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
48	7	lep1d 12170	. . . . . 6 ⊢ (𝜑 → 𝑋 ≤ (𝑋 + 1))
49	47, 7, 43, 32, 48	letrd 11391	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝑋 + 1))
50	43, 46, 49	divge0d 13126	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝑋 + 1) / 2))
51	44, 50	absidd 15510	. . 3 ⊢ (𝜑 → (abs‘((𝑋 + 1) / 2)) = ((𝑋 + 1) / 2))
52	27	halfcld 12513	. . . . 5 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ ℂ)
53	52	subid1d 11582	. . . 4 ⊢ (𝜑 → (((𝑋 + 1) / 2) − 0) = ((𝑋 + 1) / 2))
54	53	fveq2d 6882	. . 3 ⊢ (𝜑 → (abs‘(((𝑋 + 1) / 2) − 0)) = (abs‘((𝑋 + 1) / 2)))
55		ax-icn 11183	. . . . . . . . 9 ⊢ i ∈ ℂ
56	55	a1i 11	. . . . . . . 8 ⊢ (𝜑 → i ∈ ℂ)
57	7, 42	resubcld 11666	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 − 1) ∈ ℝ)
58	57	rehalfcld 12515	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ ℝ)
59	58	recnd 11261	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ ℂ)
60	56, 59	mulneg2d 11692	. . . . . . 7 ⊢ (𝜑 → (i · -((𝑋 − 1) / 2)) = -(i · ((𝑋 − 1) / 2)))
61	60	oveq2d 7429	. . . . . 6 ⊢ (𝜑 → ((√‘𝑋) + (i · -((𝑋 − 1) / 2))) = ((√‘𝑋) + -(i · ((𝑋 − 1) / 2))))
62	26	constrcn 34270	. . . . . . 7 ⊢ (𝜑 → (i · ((𝑋 − 1) / 2)) ∈ ℂ)
63	34, 62	negsubd 11599	. . . . . 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 6882	. . . 4 ⊢ (𝜑 → (abs‘((√‘𝑋) − (i · ((𝑋 − 1) / 2)))) = (abs‘((√‘𝑋) + (i · -((𝑋 − 1) / 2)))))
66	58	renegcld 11665	. . . . 5 ⊢ (𝜑 → -((𝑋 − 1) / 2) ∈ ℝ)
67		absreim 15380	. . . . 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 14261	. . . . . . . . . . 11 ⊢ (2↑2) = 4
70	69	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (2↑2) = 4)
71	70	oveq2d 7429	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) / (2↑2)) = ((4 · 𝑋) / 4))
72		4cn 12350	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
73	72	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 4 ∈ ℂ)
74	11, 13, 21	expne0d 14216	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑2) ≠ 0)
75	69, 74	eqnetrrid 3030	. . . . . . . . . 10 ⊢ (𝜑 → 4 ≠ 0)
76	8, 73, 75	divcan3d 12020	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) / 4) = 𝑋)
77	71, 76	eqtr2d 2796	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ((4 · 𝑋) / (2↑2)))
78	10, 11, 13	sqdivd 14223	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 − 1) / 2)↑2) = (((𝑋 − 1)↑2) / (2↑2)))
79	77, 78	oveq12d 7431	. . . . . . 7 ⊢ (𝜑 → (𝑋 + (((𝑋 − 1) / 2)↑2)) = (((4 · 𝑋) / (2↑2)) + (((𝑋 − 1)↑2) / (2↑2))))
80	8	sqsqrtd 15529	. . . . . . . 8 ⊢ (𝜑 → ((√‘𝑋)↑2) = 𝑋)
81	59	sqnegd 14180	. . . . . . . 8 ⊢ (𝜑 → (-((𝑋 − 1) / 2)↑2) = (((𝑋 − 1) / 2)↑2))
82	80, 81	oveq12d 7431	. . . . . . 7 ⊢ (𝜑 → (((√‘𝑋)↑2) + (-((𝑋 − 1) / 2)↑2)) = (𝑋 + (((𝑋 − 1) / 2)↑2)))
83	27, 11, 13	sqdivd 14223	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 + 1) / 2)↑2) = (((𝑋 + 1)↑2) / (2↑2)))
84	27	sqcld 14208	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 + 1)↑2) ∈ ℂ)
85	10	sqcld 14208	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 − 1)↑2) ∈ ℂ)
86	73, 8	mulcld 11253	. . . . . . . . . 10 ⊢ (𝜑 → (4 · 𝑋) ∈ ℂ)
87	8, 9	binom2subadd 33212	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑋 + 1)↑2) − ((𝑋 − 1)↑2)) = (4 · (𝑋 · 1)))
88	8	mulridd 11250	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 · 1) = 𝑋)
89	88	oveq2d 7429	. . . . . . . . . . 11 ⊢ (𝜑 → (4 · (𝑋 · 1)) = (4 · 𝑋))
90	87, 89	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑋 + 1)↑2) − ((𝑋 − 1)↑2)) = (4 · 𝑋))
91		subadd2 11485	. . . . . . . . . . 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 7428	. . . . . . . 8 ⊢ (𝜑 → (((4 · 𝑋) + ((𝑋 − 1)↑2)) / (2↑2)) = (((𝑋 + 1)↑2) / (2↑2)))
95	11	sqcld 14208	. . . . . . . . 9 ⊢ (𝜑 → (2↑2) ∈ ℂ)
96	86, 85, 95, 74	divdird 12053	. . . . . . . 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 6882	. . . . 5 ⊢ (𝜑 → (√‘(((√‘𝑋)↑2) + (-((𝑋 − 1) / 2)↑2))) = (√‘(((𝑋 + 1) / 2)↑2)))
100	44, 50	sqrtsqd 15507	. . . . 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 34267	1 ⊢ (𝜑 → (√‘𝑋) ∈ Constr)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 class class class wbr 5103 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 ici 11126 + caddc 11127 · cmul 11129 ≤ cle 11268 − cmin 11465 -cneg 11466 / cdiv 11895 2c2 12319 4c4 12321 ℤcz 12615 ℝ⁺crp 13042 ↑cexp 14125 √csqrt 15320 abscabs 15321 Constrcconstr 34239

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

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 2100 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-fac 14338 df-bc 14367 df-hash 14395 df-shft 15140 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-ef 16153 df-sin 16155 df-cos 16156 df-pi 16158 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-limc 26093 df-dv 26094 df-log 26793 df-constr 34240

This theorem is used by: constrsqrtcl 34289

W3C validator