constrresqrtcl - Mathbox for Thierry Arnoux

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Theorem constrresqrtcl 33921

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 12536	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
2	1	zconstr 33908	. 2 ⊢ (𝜑 → 0 ∈ Constr)
3		1zzd 12558	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
4	3	zconstr 33908	. 2 ⊢ (𝜑 → 1 ∈ Constr)
5		iconstr 33910	. . . 4 ⊢ i ∈ Constr
6	5	a1i 11	. . 3 ⊢ (𝜑 → i ∈ Constr)
7		constrresqrtcl.2	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8	7	recnd 11173	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
9		1cnd 11139	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
10	8, 9	subcld 11505	. . . . 5 ⊢ (𝜑 → (𝑋 − 1) ∈ ℂ)
11		2cnd 12259	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
12		2ne0 12285	. . . . . 6 ⊢ 2 ≠ 0
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ≠ 0)
14	10, 11, 13	divrecd 11934	. . . 4 ⊢ (𝜑 → ((𝑋 − 1) / 2) = ((𝑋 − 1) · (1 / 2)))
15	8, 9	negsubd 11511	. . . . . 6 ⊢ (𝜑 → (𝑋 + -1) = (𝑋 − 1))
16		constrresqrtcl.1	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Constr)
17	4	constrnegcl 33907	. . . . . . 7 ⊢ (𝜑 → -1 ∈ Constr)
18	16, 17	constraddcl 33906	. . . . . 6 ⊢ (𝜑 → (𝑋 + -1) ∈ Constr)
19	15, 18	eqeltrrd 2837	. . . . 5 ⊢ (𝜑 → (𝑋 − 1) ∈ Constr)
20		2z 12559	. . . . . . . 8 ⊢ 2 ∈ ℤ
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ)
22	21	zconstr 33908	. . . . . 6 ⊢ (𝜑 → 2 ∈ Constr)
23	22, 13	constrinvcl 33917	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ Constr)
24	19, 23	constrmulcl 33915	. . . 4 ⊢ (𝜑 → ((𝑋 − 1) · (1 / 2)) ∈ Constr)
25	14, 24	eqeltrd 2836	. . 3 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ Constr)
26	6, 25	constrmulcl 33915	. 2 ⊢ (𝜑 → (i · ((𝑋 − 1) / 2)) ∈ Constr)
27	8, 9	addcld 11164	. . . 4 ⊢ (𝜑 → (𝑋 + 1) ∈ ℂ)
28	27, 11, 13	divrecd 11934	. . 3 ⊢ (𝜑 → ((𝑋 + 1) / 2) = ((𝑋 + 1) · (1 / 2)))
29	16, 4	constraddcl 33906	. . . 4 ⊢ (𝜑 → (𝑋 + 1) ∈ Constr)
30	29, 23	constrmulcl 33915	. . 3 ⊢ (𝜑 → ((𝑋 + 1) · (1 / 2)) ∈ Constr)
31	28, 30	eqeltrd 2836	. 2 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ Constr)
32		constrresqrtcl.3	. . 3 ⊢ (𝜑 → 0 ≤ 𝑋)
33	7, 32	resqrtcld 15380	. 2 ⊢ (𝜑 → (√‘𝑋) ∈ ℝ)
34	33	recnd 11173	. 2 ⊢ (𝜑 → (√‘𝑋) ∈ ℂ)
35	9	subid1d 11494	. . . . . 6 ⊢ (𝜑 → (1 − 0) = 1)
36	35, 9	eqeltrd 2836	. . . . 5 ⊢ (𝜑 → (1 − 0) ∈ ℂ)
37	34, 36	mulcld 11165	. . . 4 ⊢ (𝜑 → ((√‘𝑋) · (1 − 0)) ∈ ℂ)
38	37	addlidd 11347	. . 3 ⊢ (𝜑 → (0 + ((√‘𝑋) · (1 − 0))) = ((√‘𝑋) · (1 − 0)))
39	35	oveq2d 7383	. . 3 ⊢ (𝜑 → ((√‘𝑋) · (1 − 0)) = ((√‘𝑋) · 1))
40	34	mulridd 11162	. . 3 ⊢ (𝜑 → ((√‘𝑋) · 1) = (√‘𝑋))
41	38, 39, 40	3eqtrrd 2776	. 2 ⊢ (𝜑 → (√‘𝑋) = (0 + ((√‘𝑋) · (1 − 0))))
42		1red 11145	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
43	7, 42	readdcld 11174	. . . . 5 ⊢ (𝜑 → (𝑋 + 1) ∈ ℝ)
44	43	rehalfcld 12424	. . . 4 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ ℝ)
45		2rp 12947	. . . . . 6 ⊢ 2 ∈ ℝ⁺
46	45	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ⁺)
47		0red 11147	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
48	7	lep1d 12087	. . . . . 6 ⊢ (𝜑 → 𝑋 ≤ (𝑋 + 1))
49	47, 7, 43, 32, 48	letrd 11303	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝑋 + 1))
50	43, 46, 49	divge0d 13026	. . . 4 ⊢ (𝜑 → 0 ≤ ((𝑋 + 1) / 2))
51	44, 50	absidd 15385	. . 3 ⊢ (𝜑 → (abs‘((𝑋 + 1) / 2)) = ((𝑋 + 1) / 2))
52	27	halfcld 12422	. . . . 5 ⊢ (𝜑 → ((𝑋 + 1) / 2) ∈ ℂ)
53	52	subid1d 11494	. . . 4 ⊢ (𝜑 → (((𝑋 + 1) / 2) − 0) = ((𝑋 + 1) / 2))
54	53	fveq2d 6844	. . 3 ⊢ (𝜑 → (abs‘(((𝑋 + 1) / 2) − 0)) = (abs‘((𝑋 + 1) / 2)))
55		ax-icn 11097	. . . . . . . . 9 ⊢ i ∈ ℂ
56	55	a1i 11	. . . . . . . 8 ⊢ (𝜑 → i ∈ ℂ)
57	7, 42	resubcld 11578	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 − 1) ∈ ℝ)
58	57	rehalfcld 12424	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ ℝ)
59	58	recnd 11173	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 1) / 2) ∈ ℂ)
60	56, 59	mulneg2d 11604	. . . . . . 7 ⊢ (𝜑 → (i · -((𝑋 − 1) / 2)) = -(i · ((𝑋 − 1) / 2)))
61	60	oveq2d 7383	. . . . . 6 ⊢ (𝜑 → ((√‘𝑋) + (i · -((𝑋 − 1) / 2))) = ((√‘𝑋) + -(i · ((𝑋 − 1) / 2))))
62	26	constrcn 33904	. . . . . . 7 ⊢ (𝜑 → (i · ((𝑋 − 1) / 2)) ∈ ℂ)
63	34, 62	negsubd 11511	. . . . . 6 ⊢ (𝜑 → ((√‘𝑋) + -(i · ((𝑋 − 1) / 2))) = ((√‘𝑋) − (i · ((𝑋 − 1) / 2))))
64	61, 63	eqtr2d 2772	. . . . 5 ⊢ (𝜑 → ((√‘𝑋) − (i · ((𝑋 − 1) / 2))) = ((√‘𝑋) + (i · -((𝑋 − 1) / 2))))
65	64	fveq2d 6844	. . . 4 ⊢ (𝜑 → (abs‘((√‘𝑋) − (i · ((𝑋 − 1) / 2)))) = (abs‘((√‘𝑋) + (i · -((𝑋 − 1) / 2)))))
66	58	renegcld 11577	. . . . 5 ⊢ (𝜑 → -((𝑋 − 1) / 2) ∈ ℝ)
67		absreim 15255	. . . . 5 ⊢ (((√‘𝑋) ∈ ℝ ∧ -((𝑋 − 1) / 2) ∈ ℝ) → (abs‘((√‘𝑋) + (i · -((𝑋 − 1) / 2)))) = (√‘(((√‘𝑋)↑2) + (-((𝑋 − 1) / 2)↑2))))
68	33, 66, 67	syl2anc 585	. . . 4 ⊢ (𝜑 → (abs‘((√‘𝑋) + (i · -((𝑋 − 1) / 2)))) = (√‘(((√‘𝑋)↑2) + (-((𝑋 − 1) / 2)↑2))))
69		sq2 14159	. . . . . . . . . . 11 ⊢ (2↑2) = 4
70	69	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (2↑2) = 4)
71	70	oveq2d 7383	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) / (2↑2)) = ((4 · 𝑋) / 4))
72		4cn 12266	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
73	72	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 4 ∈ ℂ)
74	11, 13, 21	expne0d 14114	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑2) ≠ 0)
75	69, 74	eqnetrrid 3007	. . . . . . . . . 10 ⊢ (𝜑 → 4 ≠ 0)
76	8, 73, 75	divcan3d 11936	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) / 4) = 𝑋)
77	71, 76	eqtr2d 2772	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ((4 · 𝑋) / (2↑2)))
78	10, 11, 13	sqdivd 14121	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 − 1) / 2)↑2) = (((𝑋 − 1)↑2) / (2↑2)))
79	77, 78	oveq12d 7385	. . . . . . 7 ⊢ (𝜑 → (𝑋 + (((𝑋 − 1) / 2)↑2)) = (((4 · 𝑋) / (2↑2)) + (((𝑋 − 1)↑2) / (2↑2))))
80	8	sqsqrtd 15404	. . . . . . . 8 ⊢ (𝜑 → ((√‘𝑋)↑2) = 𝑋)
81	59	sqnegd 14078	. . . . . . . 8 ⊢ (𝜑 → (-((𝑋 − 1) / 2)↑2) = (((𝑋 − 1) / 2)↑2))
82	80, 81	oveq12d 7385	. . . . . . 7 ⊢ (𝜑 → (((√‘𝑋)↑2) + (-((𝑋 − 1) / 2)↑2)) = (𝑋 + (((𝑋 − 1) / 2)↑2)))
83	27, 11, 13	sqdivd 14121	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 + 1) / 2)↑2) = (((𝑋 + 1)↑2) / (2↑2)))
84	27	sqcld 14106	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 + 1)↑2) ∈ ℂ)
85	10	sqcld 14106	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 − 1)↑2) ∈ ℂ)
86	73, 8	mulcld 11165	. . . . . . . . . 10 ⊢ (𝜑 → (4 · 𝑋) ∈ ℂ)
87	8, 9	binom2subadd 32814	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑋 + 1)↑2) − ((𝑋 − 1)↑2)) = (4 · (𝑋 · 1)))
88	8	mulridd 11162	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 · 1) = 𝑋)
89	88	oveq2d 7383	. . . . . . . . . . 11 ⊢ (𝜑 → (4 · (𝑋 · 1)) = (4 · 𝑋))
90	87, 89	eqtrd 2771	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑋 + 1)↑2) − ((𝑋 − 1)↑2)) = (4 · 𝑋))
91		subadd2 11397	. . . . . . . . . . 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 476	. . . . . . . . . 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 1376	. . . . . . . . 9 ⊢ (𝜑 → ((4 · 𝑋) + ((𝑋 − 1)↑2)) = ((𝑋 + 1)↑2))
94	93	oveq1d 7382	. . . . . . . 8 ⊢ (𝜑 → (((4 · 𝑋) + ((𝑋 − 1)↑2)) / (2↑2)) = (((𝑋 + 1)↑2) / (2↑2)))
95	11	sqcld 14106	. . . . . . . . 9 ⊢ (𝜑 → (2↑2) ∈ ℂ)
96	86, 85, 95, 74	divdird 11969	. . . . . . . 8 ⊢ (𝜑 → (((4 · 𝑋) + ((𝑋 − 1)↑2)) / (2↑2)) = (((4 · 𝑋) / (2↑2)) + (((𝑋 − 1)↑2) / (2↑2))))
97	83, 94, 96	3eqtr2d 2777	. . . . . . 7 ⊢ (𝜑 → (((𝑋 + 1) / 2)↑2) = (((4 · 𝑋) / (2↑2)) + (((𝑋 − 1)↑2) / (2↑2))))
98	79, 82, 97	3eqtr4d 2781	. . . . . 6 ⊢ (𝜑 → (((√‘𝑋)↑2) + (-((𝑋 − 1) / 2)↑2)) = (((𝑋 + 1) / 2)↑2))
99	98	fveq2d 6844	. . . . 5 ⊢ (𝜑 → (√‘(((√‘𝑋)↑2) + (-((𝑋 − 1) / 2)↑2))) = (√‘(((𝑋 + 1) / 2)↑2)))
100	44, 50	sqrtsqd 15382	. . . . 5 ⊢ (𝜑 → (√‘(((𝑋 + 1) / 2)↑2)) = ((𝑋 + 1) / 2))
101	99, 100	eqtrd 2771	. . . 4 ⊢ (𝜑 → (√‘(((√‘𝑋)↑2) + (-((𝑋 − 1) / 2)↑2))) = ((𝑋 + 1) / 2))
102	65, 68, 101	3eqtrd 2775	. . 3 ⊢ (𝜑 → (abs‘((√‘𝑋) − (i · ((𝑋 − 1) / 2)))) = ((𝑋 + 1) / 2))
103	51, 54, 102	3eqtr4rd 2782	. 2 ⊢ (𝜑 → (abs‘((√‘𝑋) − (i · ((𝑋 − 1) / 2)))) = (abs‘(((𝑋 + 1) / 2) − 0)))
104	2, 4, 26, 31, 2, 33, 34, 41, 103	constrlccl 33901	1 ⊢ (𝜑 → (√‘𝑋) ∈ Constr)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 ici 11040 + caddc 11041 · cmul 11043 ≤ cle 11180 − cmin 11377 -cneg 11378 / cdiv 11807 2c2 12236 4c4 12238 ℤcz 12524 ℝ⁺crp 12942 ↑cexp 14023 √csqrt 15195 abscabs 15196 Constrcconstr 33873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520 df-constr 33874

This theorem is referenced by: constrsqrtcl 33923

W3C validator