sqrtnnaa - Mathbox for Ender Ting

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Theorem sqrtnnaa 47645

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 12259	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
2		sqrtcl 15439	. . . 4 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
3	1, 2	syl 18	. . 3 ⊢ (𝐴 ∈ ℕ → (√‘𝐴) ∈ ℂ)
4		zsscn 12617	. . . . . . . 8 ⊢ ℤ ⊆ ℂ
5		1z 12642	. . . . . . . 8 ⊢ 1 ∈ ℤ
6		2nn0 12539	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		plypow 26399	. . . . . . . 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 12630	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
12		plyconst 26400	. . . . . . 7 ⊢ ((ℤ ⊆ ℂ ∧ 𝐴 ∈ ℤ) → (ℂ × {𝐴}) ∈ (Poly‘ℤ))
13	10, 11, 12	syl2anc 596	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (ℂ × {𝐴}) ∈ (Poly‘ℤ))
14		zaddcl 12652	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ) → (𝑥 + 𝑝) ∈ ℤ)
15	14	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ (𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ)) → (𝑥 + 𝑝) ∈ ℤ)
16		zmulcl 12661	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ) → (𝑥 · 𝑝) ∈ ℤ)
17	16	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ (𝑥 ∈ ℤ ∧ 𝑝 ∈ ℤ)) → (𝑥 · 𝑝) ∈ ℤ)
18		neg1z 12648	. . . . . . 7 ⊢ -1 ∈ ℤ
19	18	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ → -1 ∈ ℤ)
20	9, 13, 15, 17, 19	plysub 26413	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ (Poly‘ℤ))
21		0cnd 11217	. . . . . 6 ⊢ (𝐴 ∈ ℕ → 0 ∈ ℂ)
22		ovex 7456	. . . . . . . . . . . . 13 ⊢ (𝑡↑2) ∈ V
23	22	rgenw 3086	. . . . . . . . . . . 12 ⊢ ∀𝑡 ∈ ℂ (𝑡↑2) ∈ V
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ∀𝑡 ∈ ℂ (𝑡↑2) ∈ V)
25		nfcv 2928	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡ℂ
26	25	mptfnf 6677	. . . . . . . . . . 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 6773	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (ℂ × {𝐴}) Fn ℂ)
29	27, 28	jca 521	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) Fn ℂ ∧ (ℂ × {𝐴}) Fn ℂ))
30		cnex 11199	. . . . . . . . . . 11 ⊢ ℂ ∈ V
31	30	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ℂ ∈ V)
32	31, 21	jca 521	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (ℂ ∈ V ∧ 0 ∈ ℂ))
33		fnfvof 7704	. . . . . . . . 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 596	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) = (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) − ((ℂ × {𝐴})‘0)))
35		0cn 11216	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
36		oveq1 7430	. . . . . . . . . . . . 13 ⊢ (𝑡 = 0 → (𝑡↑2) = (0↑2))
37		eqid 2766	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) = (𝑡 ∈ ℂ ↦ (𝑡↑2))
38		ovex 7456	. . . . . . . . . . . . 13 ⊢ (0↑2) ∈ V
39	36, 37, 38	fvmpt 6996	. . . . . . . . . . . 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 14248	. . . . . . . . . . 11 ⊢ (0↑2) = 0
42	40, 41	eqtri 2789	. . . . . . . . . 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 7207	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 0 ∈ ℂ) → ((ℂ × {𝐴})‘0) = 𝐴)
46	44, 21, 45	syl2anc 596	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((ℂ × {𝐴})‘0) = 𝐴)
47	43, 46	oveq12d 7441	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) − ((ℂ × {𝐴})‘0)) = (0 − 𝐴))
48	34, 47	eqtrd 2801	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) = (0 − 𝐴))
49		nnne0 12288	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
50	49	necomd 3016	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 0 ≠ 𝐴)
51	21, 1, 50	subne0d 11596	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (0 − 𝐴) ≠ 0)
52	48, 51	eqnetrd 3028	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) ≠ 0)
53		ne0p 26401	. . . . . 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 596	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ≠ 0_𝑝)
55		eldifsn 4758	. . . . 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 595	. . . 4 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ ((Poly‘ℤ) ∖ {0_𝑝}))
57	31, 3	jca 521	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (ℂ ∈ V ∧ (√‘𝐴) ∈ ℂ))
58		fnfvof 7704	. . . . . . 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 596	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) − ((ℂ × {𝐴})‘(√‘𝐴))))
60		oveq1 7430	. . . . . . . . . 10 ⊢ (𝑡 = (√‘𝐴) → (𝑡↑2) = ((√‘𝐴)↑2))
61		ovex 7456	. . . . . . . . . 10 ⊢ ((√‘𝐴)↑2) ∈ V
62	60, 37, 61	fvmpt 6996	. . . . . . . . 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 15442	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴)
65	1, 64	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((√‘𝐴)↑2) = 𝐴)
66	63, 65	eqtrd 2801	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) = 𝐴)
67		fvconst2g 7207	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ (√‘𝐴) ∈ ℂ) → ((ℂ × {𝐴})‘(√‘𝐴)) = 𝐴)
68	44, 3, 67	syl2anc 596	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((ℂ × {𝐴})‘(√‘𝐴)) = 𝐴)
69	66, 68	oveq12d 7441	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) − ((ℂ × {𝐴})‘(√‘𝐴))) = (𝐴 − 𝐴))
70	59, 69	eqtrd 2801	. . . . 5 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = (𝐴 − 𝐴))
71		subid 11495	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0)
72	1, 71	syl 18	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐴 − 𝐴) = 0)
73	70, 72	eqtrd 2801	. . . 4 ⊢ (𝐴 ∈ ℕ → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0)
74		fveq1 6887	. . . . . 6 ⊢ (𝑥 = ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) → (𝑥‘(√‘𝐴)) = (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)))
75	74	eqeq1d 2768	. . . . 5 ⊢ (𝑥 = ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) → ((𝑥‘(√‘𝐴)) = 0 ↔ (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0))
76	75	rspcev 3584	. . . 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 596	. . 3 ⊢ (𝐴 ∈ ℕ → ∃𝑥 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(𝑥‘(√‘𝐴)) = 0)
78	3, 77	jca 521	. 2 ⊢ (𝐴 ∈ ℕ → ((√‘𝐴) ∈ ℂ ∧ ∃𝑥 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(𝑥‘(√‘𝐴)) = 0))
79		elaa 26514	. 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 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 ∃wrex 3092 Vcvv 3458 ∖ cdif 3905 ⊆ wss 3908 {csn 4594 ↦ cmpt 5197 × cxp 5664 Fn wfn 6538 ‘cfv 6543 (class class class)co 7423 ∘_f cof 7685 ℂcc 11116 0cc0 11118 1c1 11119 + caddc 11121 · cmul 11123 − cmin 11459 -cneg 11460 ℕcn 12251 2c2 12313 ℕ₀cn0 12522 ℤcz 12609 ↑cexp 14117 √csqrt 15310 0_𝑝c0p 25865 Polycply 26378 𝔸caa 26512

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-z 12610 df-uz 12881 df-rp 13035 df-fz 13554 df-fzo 13702 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-sum 15764 df-0p 25866 df-ply 26382 df-aa 26513

This theorem is used by: nthrucw 47648

W3C validator