sqrtnzqaa - Mathbox for Ender Ting

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Theorem sqrtnzqaa 47740

Description: Square root of a nonzero rational is algebraic. (Contributed by Ender Ting, 22-Jul-2026.)

**Assertion**
Ref	Expression
sqrtnzqaa	⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (√‘𝐴) ∈ 𝔸)

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

Step	Hyp	Ref	Expression
1		simpl 488	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℚ)
2		qcn 13016	. . 3 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
3		sqrtcl 15453	. . 3 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
4	1, 2, 3	3syl 19	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (√‘𝐴) ∈ ℂ)
5		fveq1 6881	. . . 4 ⊢ (𝑥 = ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) → (𝑥‘(√‘𝐴)) = (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)))
6	5	eqeq1d 2764	. . 3 ⊢ (𝑥 = ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) → ((𝑥‘(√‘𝐴)) = 0 ↔ (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0))
7		qsscn 13013	. . . . . . 7 ⊢ ℚ ⊆ ℂ
8		1z 12652	. . . . . . . 8 ⊢ 1 ∈ ℤ
9		zq 13007	. . . . . . . 8 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ 1 ∈ ℚ
11		2nn0 12549	. . . . . . 7 ⊢ 2 ∈ ℕ₀
12		plypow 26437	. . . . . . 7 ⊢ ((ℚ ⊆ ℂ ∧ 1 ∈ ℚ ∧ 2 ∈ ℕ₀) → (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈ (Poly‘ℚ))
13	7, 10, 11, 12	mp3an 1490	. . . . . 6 ⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈ (Poly‘ℚ)
14	13	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈ (Poly‘ℚ))
15	7	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ℚ ⊆ ℂ)
16		plyconst 26438	. . . . . 6 ⊢ ((ℚ ⊆ ℂ ∧ 𝐴 ∈ ℚ) → (ℂ × {𝐴}) ∈ (Poly‘ℚ))
17	15, 1, 16	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (ℂ × {𝐴}) ∈ (Poly‘ℚ))
18		qaddcl 13019	. . . . . 6 ⊢ ((𝑥 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑥 + 𝑝) ∈ ℚ)
19	18	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℚ ∧ 𝑝 ∈ ℚ)) → (𝑥 + 𝑝) ∈ ℚ)
20		qmulcl 13021	. . . . . 6 ⊢ ((𝑥 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑥 · 𝑝) ∈ ℚ)
21	20	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℚ ∧ 𝑝 ∈ ℚ)) → (𝑥 · 𝑝) ∈ ℚ)
22		neg1z 12658	. . . . . . 7 ⊢ -1 ∈ ℤ
23		zq 13007	. . . . . . 7 ⊢ (-1 ∈ ℤ → -1 ∈ ℚ)
24	22, 23	ax-mp 5	. . . . . 6 ⊢ -1 ∈ ℚ
25	24	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → -1 ∈ ℚ)
26	14, 17, 19, 21, 25	plysub 26452	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ (Poly‘ℚ))
27		0cnd 11227	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → 0 ∈ ℂ)
28		fnconstg 6767	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → (ℂ × {𝐴}) Fn ℂ)
29	28	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (ℂ × {𝐴}) Fn ℂ)
30		ovex 7450	. . . . . . . . . . 11 ⊢ (𝑡↑2) ∈ V
31	30	rgenw 3082	. . . . . . . . . 10 ⊢ ∀𝑡 ∈ ℂ (𝑡↑2) ∈ V
32		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑡ℂ
33	32	mptfnf 6671	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ ℂ (𝑡↑2) ∈ V ↔ (𝑡 ∈ ℂ ↦ (𝑡↑2)) Fn ℂ)
34	31, 33	mpbi 233	. . . . . . . . 9 ⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) Fn ℂ
35	29, 34	jctil 529	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) Fn ℂ ∧ (ℂ × {𝐴}) Fn ℂ))
36		cnex 11209	. . . . . . . . . 10 ⊢ ℂ ∈ V
37		0cn 11226	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
38	36, 37	pm3.2i 476	. . . . . . . . 9 ⊢ (ℂ ∈ V ∧ 0 ∈ ℂ)
39	38	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (ℂ ∈ V ∧ 0 ∈ ℂ))
40		fnfvof 7699	. . . . . . . 8 ⊢ ((((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)))
41	35, 39, 40	syl2anc 596	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) = (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) − ((ℂ × {𝐴})‘0)))
42		oveq1 7424	. . . . . . . . . . . 12 ⊢ (𝑡 = 0 → (𝑡↑2) = (0↑2))
43		eqid 2762	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) = (𝑡 ∈ ℂ ↦ (𝑡↑2))
44		ovex 7450	. . . . . . . . . . . 12 ⊢ (0↑2) ∈ V
45	42, 43, 44	fvmpt 6990	. . . . . . . . . . 11 ⊢ (0 ∈ ℂ → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = (0↑2))
46	37, 45	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = (0↑2)
47		sq0 14260	. . . . . . . . . 10 ⊢ (0↑2) = 0
48	46, 47	eqtri 2785	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = 0
49	48	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = 0)
50		fvconst2g 7205	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℚ ∧ 0 ∈ ℂ) → ((ℂ × {𝐴})‘0) = 𝐴)
51	1, 27, 50	syl2anc 596	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((ℂ × {𝐴})‘0) = 𝐴)
52	49, 51	oveq12d 7435	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) − ((ℂ × {𝐴})‘0)) = (0 − 𝐴))
53	41, 52	eqtrd 2797	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) = (0 − 𝐴))
54	2	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
55		necom 3010	. . . . . . . 8 ⊢ (𝐴 ≠ 0 ↔ 0 ≠ 𝐴)
56	55	bilani 510	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → 0 ≠ 𝐴)
57	27, 54, 56	subne0d 11607	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (0 − 𝐴) ≠ 0)
58	53, 57	eqnetrd 3024	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) ≠ 0)
59		ne0p 26439	. . . . 5 ⊢ ((0 ∈ ℂ ∧ (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ≠ 0_𝑝)
60	27, 58, 59	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ≠ 0_𝑝)
61	26, 60	eldifsnd 4753	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ ((Poly‘ℚ) ∖ {0_𝑝}))
62	4, 36	jctil 529	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (ℂ ∈ V ∧ (√‘𝐴) ∈ ℂ))
63		fnfvof 7699	. . . . . 6 ⊢ ((((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))))
64	35, 62, 63	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) − ((ℂ × {𝐴})‘(√‘𝐴))))
65		oveq1 7424	. . . . . . . . 9 ⊢ (𝑡 = (√‘𝐴) → (𝑡↑2) = ((√‘𝐴)↑2))
66		ovex 7450	. . . . . . . . 9 ⊢ ((√‘𝐴)↑2) ∈ V
67	65, 43, 66	fvmpt 6990	. . . . . . . 8 ⊢ ((√‘𝐴) ∈ ℂ → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) = ((√‘𝐴)↑2))
68	54, 3, 67	3syl 19	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) = ((√‘𝐴)↑2))
69		sqrtth 15456	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴)
70	1, 2, 69	3syl 19	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((√‘𝐴)↑2) = 𝐴)
71	68, 70	eqtrd 2797	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) = 𝐴)
72		fvconst2g 7205	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ (√‘𝐴) ∈ ℂ) → ((ℂ × {𝐴})‘(√‘𝐴)) = 𝐴)
73	1, 4, 72	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((ℂ × {𝐴})‘(√‘𝐴)) = 𝐴)
74	71, 73	oveq12d 7435	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) − ((ℂ × {𝐴})‘(√‘𝐴))) = (𝐴 − 𝐴))
75	64, 74	eqtrd 2797	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = (𝐴 − 𝐴))
76		subid 11505	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0)
77	1, 2, 76	3syl 19	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (𝐴 − 𝐴) = 0)
78	75, 77	eqtrd 2797	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0)
79	6, 61, 78	rspcedvdw 3582	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0_𝑝})(𝑥‘(√‘𝐴)) = 0)
80		elqaa 26561	. 2 ⊢ ((√‘𝐴) ∈ 𝔸 ↔ ((√‘𝐴) ∈ ℂ ∧ ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0_𝑝})(𝑥‘(√‘𝐴)) = 0))
81	4, 79, 80	sylanbrc 595	1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (√‘𝐴) ∈ 𝔸)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 Vcvv 3453 ∖ cdif 3899 ⊆ wss 3902 {csn 4587 ↦ cmpt 5190 × cxp 5657 Fn wfn 6532 ‘cfv 6537 (class class class)co 7417 ∘_f cof 7680 ℂcc 11126 0cc0 11128 1c1 11129 + caddc 11131 · cmul 11133 − cmin 11469 -cneg 11470 2c2 12323 ℕ₀cn0 12532 ℤcz 12619 ℚcq 13001 ↑cexp 14129 √csqrt 15324 0_𝑝c0p 25903 Polycply 26416 𝔸caa 26553

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-pm 8833 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-sup 9416 df-inf 9417 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-n0 12533 df-z 12620 df-uz 12892 df-q 13002 df-rp 13047 df-fz 13566 df-fzo 13714 df-fl 13857 df-mod 13935 df-seq 14070 df-exp 14130 df-hash 14399 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579 df-rlim 15580 df-sum 15778 df-0p 25904 df-ply 26420 df-coe 26422 df-dgr 26423 df-aa 26554

This theorem is used by: sqrtqaa 47741

W3C validator