sqrtnnaa - Mathbox for Ender Ting

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Theorem sqrtnnaa 47587

Description: Square root of a natural number is algebraic. (Contributed by Ender Ting, 21-Jul-2026.)

**Assertion**
Ref	Expression
sqrtnnaa	⊢ (𝐴 ∈ ℕ → (√‘𝐴) ∈ 𝔸)

Proof of Theorem sqrtnnaa Dummy variables 𝑥 𝑡 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nncn 12242	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
2		sqrtcl 15415	. . . 4 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
3	1, 2	syl 18	. . 3 ⊢ (𝐴 ∈ ℕ → (√‘𝐴) ∈ ℂ)
4		zsscn 12600	. . . . . . . 8 ⊢ ℤ ⊆ ℂ
5		1z 12625	. . . . . . . 8 ⊢ 1 ∈ ℤ
6		2nn0 12522	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		plypow 26343	. . . . . . . 8 ⊢ ((ℤ ⊆ ℂ ∧ 1 ∈ ℤ ∧ 2 ∈ ℕ₀) → (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈ (Poly‘ℤ))
8	4, 5, 6, 7	mp3an 1490	. . . . . . 7 ⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈ (Poly‘ℤ)
9	8	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈ (Poly‘ℤ))
10	4	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ℤ ⊆ ℂ)
11		nnz 12613	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
12		plyconst 26344	. . . . . . 7 ⊢ ((ℤ ⊆ ℂ ∧ 𝐴 ∈ ℤ) → (ℂ × {𝐴}) ∈ (Poly‘ℤ))
13	10, 11, 12	syl2anc 595	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (ℂ × {𝐴}) ∈ (Poly‘ℤ))
14		zaddcl 12635	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ) → (𝑥 + 𝑝) ∈ ℤ)
15	14	adantl 486	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ (𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ)) → (𝑥 + 𝑝) ∈ ℤ)
16		zmulcl 12644	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ) → (𝑥 · 𝑝) ∈ ℤ)
17	16	adantl 486	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ (𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ)) → (𝑥 · 𝑝) ∈ ℤ)
18		neg1z 12631	. . . . . . 7 ⊢ -1 ∈ ℤ
19	18	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ → -1 ∈ ℤ)
20	9, 13, 15, 17, 19	plysub 26357	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ (Poly‘ℤ))
21		0cnd 11200	. . . . . 6 ⊢ (𝐴 ∈ ℕ → 0 ∈ ℂ)
22		ovex 7445	. . . . . . . . . . . . 13 ⊢ (𝑡↑2) ∈ V
23	22	rgenw 3083	. . . . . . . . . . . 12 ⊢ ∀𝑡 ∈ ℂ (𝑡↑2) ∈ V
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ∀𝑡 ∈ ℂ (𝑡↑2) ∈ V)
25		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡ℂ
26	25	mptfnf 6672	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ ℂ (𝑡↑2) ∈ V ↔ (𝑡 ∈ ℂ ↦ (𝑡↑2)) Fn ℂ)
27	24, 26	sylib 221	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝑡 ∈ ℂ ↦ (𝑡↑2)) Fn ℂ)
28		fnconstg 6768	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (ℂ × {𝐴}) Fn ℂ)
29	27, 28	jca 520	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) Fn ℂ ∧ (ℂ × {𝐴}) Fn ℂ))
30		cnex 11182	. . . . . . . . . . 11 ⊢ ℂ ∈ V
31	30	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ℂ ∈ V)
32	31, 21	jca 520	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (ℂ ∈ V ∧ 0 ∈ ℂ))
33		fnfvof 7693	. . . . . . . . 9 ⊢ ((((𝑡 ∈ ℂ ↦ (𝑡↑2)) Fn ℂ ∧ (ℂ × {𝐴}) Fn ℂ) ∧ (ℂ ∈ V ∧ 0 ∈ ℂ)) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) = (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) − ((ℂ × {𝐴})‘0)))
34	29, 32, 33	syl2anc 595	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) = (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) − ((ℂ × {𝐴})‘0)))
35		0cn 11199	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
36		oveq1 7419	. . . . . . . . . . . . 13 ⊢ (𝑡 = 0 → (𝑡↑2) = (0↑2))
37		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) = (𝑡 ∈ ℂ ↦ (𝑡↑2))
38		ovex 7445	. . . . . . . . . . . . 13 ⊢ (0↑2) ∈ V
39	36, 37, 38	fvmpt 6991	. . . . . . . . . . . 12 ⊢ (0 ∈ ℂ → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = (0↑2))
40	35, 39	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = (0↑2)
41		sq0 14230	. . . . . . . . . . 11 ⊢ (0↑2) = 0
42	40, 41	eqtri 2786	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = 0
43	42	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = 0)
44		id 23	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ)
45		fvconst2g 7202	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 0 ∈ ℂ) → ((ℂ × {𝐴})‘0) = 𝐴)
46	44, 21, 45	syl2anc 595	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((ℂ × {𝐴})‘0) = 𝐴)
47	43, 46	oveq12d 7430	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) − ((ℂ × {𝐴})‘0)) = (0 − 𝐴))
48	34, 47	eqtrd 2798	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) = (0 − 𝐴))
49		nnne0 12271	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
50	49	necomd 3013	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 0 ≠ 𝐴)
51	21, 1, 50	subne0d 11579	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (0 − 𝐴) ≠ 0)
52	48, 51	eqnetrd 3025	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) ≠ 0)
53		ne0p 26345	. . . . . 6 ⊢ ((0 ∈ ℂ ∧ (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ≠ 0_𝑝)
54	21, 52, 53	syl2anc 595	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ≠ 0_𝑝)
55		eldifsn 4754	. . . . 5 ⊢ (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ↔ (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ (Poly‘ℤ) ∧ ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ≠ 0_𝑝))
56	20, 54, 55	sylanbrc 594	. . . 4 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ ((Poly‘ℤ) ∖ {0_𝑝}))
57	31, 3	jca 520	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (ℂ ∈ V ∧ (√‘𝐴) ∈ ℂ))
58		fnfvof 7693	. . . . . . 7 ⊢ ((((𝑡 ∈ ℂ ↦ (𝑡↑2)) Fn ℂ ∧ (ℂ × {𝐴}) Fn ℂ) ∧ (ℂ ∈ V ∧ (√‘𝐴) ∈ ℂ)) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) − ((ℂ × {𝐴})‘(√‘𝐴))))
59	29, 57, 58	syl2anc 595	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) − ((ℂ × {𝐴})‘(√‘𝐴))))
60		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑡 = (√‘𝐴) → (𝑡↑2) = ((√‘𝐴)↑2))
61		ovex 7445	. . . . . . . . . 10 ⊢ ((√‘𝐴)↑2) ∈ V
62	60, 37, 61	fvmpt 6991	. . . . . . . . 9 ⊢ ((√‘𝐴) ∈ ℂ → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) = ((√‘𝐴)↑2))
63	3, 62	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) = ((√‘𝐴)↑2))
64		sqrtth 15418	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴)
65	1, 64	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((√‘𝐴)↑2) = 𝐴)
66	63, 65	eqtrd 2798	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) = 𝐴)
67		fvconst2g 7202	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ (√‘𝐴) ∈ ℂ) → ((ℂ × {𝐴})‘(√‘𝐴)) = 𝐴)
68	44, 3, 67	syl2anc 595	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((ℂ × {𝐴})‘(√‘𝐴)) = 𝐴)
69	66, 68	oveq12d 7430	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) − ((ℂ × {𝐴})‘(√‘𝐴))) = (𝐴 − 𝐴))
70	59, 69	eqtrd 2798	. . . . 5 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = (𝐴 − 𝐴))
71		subid 11478	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0)
72	1, 71	syl 18	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐴 − 𝐴) = 0)
73	70, 72	eqtrd 2798	. . . 4 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0)
74		fveq1 6882	. . . . . 6 ⊢ (𝑥 = ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) → (𝑥‘(√‘𝐴)) = (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)))
75	74	eqeq1d 2765	. . . . 5 ⊢ (𝑥 = ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) → ((𝑥‘(√‘𝐴)) = 0 ↔ (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0))
76	75	rspcev 3582	. . . 4 ⊢ ((((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0) → ∃𝑥 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(𝑥‘(√‘𝐴)) = 0)
77	56, 73, 76	syl2anc 595	. . 3 ⊢ (𝐴 ∈ ℕ → ∃𝑥 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(𝑥‘(√‘𝐴)) = 0)
78	3, 77	jca 520	. 2 ⊢ (𝐴 ∈ ℕ → ((√‘𝐴) ∈ ℂ ∧ ∃𝑥 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(𝑥‘(√‘𝐴)) = 0))
79		elaa 26458	. 2 ⊢ ((√‘𝐴) ∈ 𝔸 ↔ ((√‘𝐴) ∈ ℂ ∧ ∃𝑥 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(𝑥‘(√‘𝐴)) = 0))
80	78, 79	sylibr 237	1 ⊢ (𝐴 ∈ ℕ → (√‘𝐴) ∈ 𝔸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ∖ cdif 3903 ⊆ wss 3906 {csn 4590 ↦ cmpt 5193 × cxp 5661 Fn wfn 6533 ‘cfv 6538 (class class class)co 7412 ∘_f cof 7674 ℂcc 11099 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 − cmin 11442 -cneg 11443 ℕcn 12234 2c2 12296 ℕ₀cn0 12505 ℤcz 12592 ↑cexp 14099 √csqrt 15286 0_𝑝c0p 25809 Polycply 26322 𝔸caa 26456

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-0p 25810 df-ply 26326 df-aa 26457

This theorem is referenced by: nthrucw 47590

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