sqrtnnaa - Mathbox for Ender Ting

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Theorem sqrtnnaa 47632

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 12245	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
2		sqrtcl 15418	. . . 4 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
3	1, 2	syl 18	. . 3 ⊢ (𝐴 ∈ ℕ → (√‘𝐴) ∈ ℂ)
4		zsscn 12603	. . . . . . . 8 ⊢ ℤ ⊆ ℂ
5		1z 12628	. . . . . . . 8 ⊢ 1 ∈ ℤ
6		2nn0 12525	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		plypow 26371	. . . . . . . 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 12616	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
12		plyconst 26372	. . . . . . 7 ⊢ ((ℤ ⊆ ℂ ∧ 𝐴 ∈ ℤ) → (ℂ × {𝐴}) ∈ (Poly‘ℤ))
13	10, 11, 12	syl2anc 595	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (ℂ × {𝐴}) ∈ (Poly‘ℤ))
14		zaddcl 12638	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ) → (𝑥 + 𝑝) ∈ ℤ)
15	14	adantl 486	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ (𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ)) → (𝑥 + 𝑝) ∈ ℤ)
16		zmulcl 12647	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ) → (𝑥 · 𝑝) ∈ ℤ)
17	16	adantl 486	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ (𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ)) → (𝑥 · 𝑝) ∈ ℤ)
18		neg1z 12634	. . . . . . 7 ⊢ -1 ∈ ℤ
19	18	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ → -1 ∈ ℤ)
20	9, 13, 15, 17, 19	plysub 26385	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ (Poly‘ℤ))
21		0cnd 11203	. . . . . 6 ⊢ (𝐴 ∈ ℕ → 0 ∈ ℂ)
22		ovex 7443	. . . . . . . . . . . . 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 6670	. . . . . . . . . . 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 6766	. . . . . . . . . 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 11185	. . . . . . . . . . 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 7691	. . . . . . . . 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 11202	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
36		oveq1 7417	. . . . . . . . . . . . 13 ⊢ (𝑡 = 0 → (𝑡↑2) = (0↑2))
37		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) = (𝑡 ∈ ℂ ↦ (𝑡↑2))
38		ovex 7443	. . . . . . . . . . . . 13 ⊢ (0↑2) ∈ V
39	36, 37, 38	fvmpt 6989	. . . . . . . . . . . 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 14233	. . . . . . . . . . 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 7200	. . . . . . . . . 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 7428	. . . . . . . 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 12274	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
50	49	necomd 3013	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 0 ≠ 𝐴)
51	21, 1, 50	subne0d 11582	. . . . . . 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 26373	. . . . . 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 4753	. . . . 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 7691	. . . . . . 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 7417	. . . . . . . . . 10 ⊢ (𝑡 = (√‘𝐴) → (𝑡↑2) = ((√‘𝐴)↑2))
61		ovex 7443	. . . . . . . . . 10 ⊢ ((√‘𝐴)↑2) ∈ V
62	60, 37, 61	fvmpt 6989	. . . . . . . . 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 15421	. . . . . . . . 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 7200	. . . . . . . 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 7428	. . . . . 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 11481	. . . . . 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 6880	. . . . . 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 3581	. . . 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 26486	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ∖ cdif 3902 ⊆ wss 3905 {csn 4589 ↦ cmpt 5192 × cxp 5659 Fn wfn 6531 ‘cfv 6536 (class class class)co 7410 ∘_f cof 7672 ℂcc 11102 0cc0 11104 1c1 11105 + caddc 11107 · cmul 11109 − cmin 11445 -cneg 11446 ℕcn 12237 2c2 12299 ℕ₀cn0 12508 ℤcz 12595 ↑cexp 14102 √csqrt 15289 0_𝑝c0p 25837 Polycply 26350 𝔸caa 26484

This proof depends on 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 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182

This proof 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 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-oi 9468 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-n0 12509 df-z 12596 df-uz 12867 df-rp 13021 df-fz 13540 df-fzo 13688 df-seq 14043 df-exp 14103 df-hash 14372 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-clim 15544 df-sum 15743 df-0p 25838 df-ply 26354 df-aa 26485

This theorem is used by: nthrucw 47635

W3C validator