sqrtnzqaa - Mathbox for Ender Ting

	Mathbox for Ender Ting		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > sqrtnzqaa		Structured version Visualization version GIF version

Theorem sqrtnzqaa 48007

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 13093	. . 3 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
3		sqrtcl 15534	. . 3 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
4	1, 2, 3	3syl 19	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (√‘𝐴) ∈ ℂ)
5		fveq1 6884	. . . 4 ⊢ (𝑥 = ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) → (𝑥‘(√‘𝐴)) = (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)))
6	5	eqeq1d 2762	. . 3 ⊢ (𝑥 = ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) → ((𝑥‘(√‘𝐴)) = 0 ↔ (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0))
7		qsscn 13090	. . . . . . 7 ⊢ ℚ ⊆ ℂ
8		1z 12729	. . . . . . . 8 ⊢ 1 ∈ ℤ
9		zq 13084	. . . . . . . 8 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ 1 ∈ ℚ
11		2nn0 12626	. . . . . . 7 ⊢ 2 ∈ ℕ₀
12		plypow 26528	. . . . . . 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 26529	. . . . . 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 13096	. . . . . 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 13098	. . . . . 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 12735	. . . . . . 7 ⊢ -1 ∈ ℤ
23		zq 13084	. . . . . . 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 26543	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ (Poly‘ℚ))
27		0cnd 11302	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → 0 ∈ ℂ)
28		fnconstg 6770	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → (ℂ × {𝐴}) Fn ℂ)
29	28	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (ℂ × {𝐴}) Fn ℂ)
30		ovex 7453	. . . . . . . . . . 11 ⊢ (𝑡↑2) ∈ V
31	30	rgenw 3080	. . . . . . . . . 10 ⊢ ∀𝑡 ∈ ℂ (𝑡↑2) ∈ V
32		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑡ℂ
33	32	mptfnf 6674	. . . . . . . . . 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 11284	. . . . . . . . . 10 ⊢ ℂ ∈ V
37		0cn 11301	. . . . . . . . . 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 7710	. . . . . . . 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 7427	. . . . . . . . . . . 12 ⊢ (𝑡 = 0 → (𝑡↑2) = (0↑2))
43		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) = (𝑡 ∈ ℂ ↦ (𝑡↑2))
44		ovex 7453	. . . . . . . . . . . 12 ⊢ (0↑2) ∈ V
45	42, 43, 44	fvmpt 6993	. . . . . . . . . . 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 14338	. . . . . . . . . 10 ⊢ (0↑2) = 0
48	46, 47	eqtri 2783	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = 0
49	48	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = 0)
50		fvconst2g 7208	. . . . . . . . 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 7438	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) − ((ℂ × {𝐴})‘0)) = (0 − 𝐴))
53	41, 52	eqtrd 2795	. . . . . 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 3008	. . . . . . . 8 ⊢ (𝐴 ≠ 0 ↔ 0 ≠ 𝐴)
56	55	bilani 510	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → 0 ≠ 𝐴)
57	27, 54, 56	subne0d 11682	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (0 − 𝐴) ≠ 0)
58	53, 57	eqnetrd 3022	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) ≠ 0)
59		ne0p 26530	. . . . 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 4746	. . 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 7710	. . . . . 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 7427	. . . . . . . . 9 ⊢ (𝑡 = (√‘𝐴) → (𝑡↑2) = ((√‘𝐴)↑2))
66		ovex 7453	. . . . . . . . 9 ⊢ ((√‘𝐴)↑2) ∈ V
67	65, 43, 66	fvmpt 6993	. . . . . . . 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 15537	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴)
70	1, 2, 69	3syl 19	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((√‘𝐴)↑2) = 𝐴)
71	68, 70	eqtrd 2795	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) = 𝐴)
72		fvconst2g 7208	. . . . . . 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 7438	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) − ((ℂ × {𝐴})‘(√‘𝐴))) = (𝐴 − 𝐴))
75	64, 74	eqtrd 2795	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = (𝐴 − 𝐴))
76		subid 11580	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0)
77	1, 2, 76	3syl 19	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (𝐴 − 𝐴) = 0)
78	75, 77	eqtrd 2795	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0)
79	6, 61, 78	rspcedvdw 3579	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0_𝑝})(𝑥‘(√‘𝐴)) = 0)
80		elqaa 26650	. 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 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 {csn 4584 ↦ cmpt 5185 × cxp 5648 Fn wfn 6533 ‘cfv 6538 (class class class)co 7420 ∘_f cof 7691 ℂcc 11201 0cc0 11203 1c1 11204 + caddc 11206 · cmul 11208 − cmin 11544 -cneg 11545 2c2 12400 ℕ₀cn0 12609 ℤcz 12696 ℚcq 13078 ↑cexp 14207 √csqrt 15405 0_𝑝c0p 25995 Polycply 26507 𝔸caa 26642

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 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 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-er 8720 df-map 8852 df-pm 8853 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9437 df-inf 9438 df-oi 9507 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-n0 12610 df-z 12697 df-uz 12969 df-q 13079 df-rp 13124 df-fz 13643 df-fzo 13792 df-fl 13935 df-mod 14013 df-seq 14148 df-exp 14208 df-hash 14478 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-clim 15660 df-rlim 15661 df-sum 15859 df-0p 25996 df-ply 26511 df-coe 26513 df-dgr 26514 df-aa 26643

This theorem is used by: sqrtqaa 48008

W3C validator