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 48129

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 13120	. . 3 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
3		sqrtcl 15561	. . 3 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
4	1, 2, 3	3syl 19	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (√‘𝐴) ∈ ℂ)
5		fveq1 6910	. . . 4 ⊢ (𝑥 = ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) → (𝑥‘(√‘𝐴)) = (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)))
6	5	eqeq1d 2765	. . 3 ⊢ (𝑥 = ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) → ((𝑥‘(√‘𝐴)) = 0 ↔ (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0))
7		qsscn 13117	. . . . . . 7 ⊢ ℚ ⊆ ℂ
8		1z 12756	. . . . . . . 8 ⊢ 1 ∈ ℤ
9		zq 13111	. . . . . . . 8 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ 1 ∈ ℚ
11		2nn0 12653	. . . . . . 7 ⊢ 2 ∈ ℕ₀
12		plypow 26556	. . . . . . 7 ⊢ ((ℚ ⊆ ℂ ∧ 1 ∈ ℚ ∧ 2 ∈ ℕ₀) → (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈ (Poly‘ℚ))
13	7, 10, 11, 12	mp3an 1491	. . . . . 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 26557	. . . . . 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 13123	. . . . . 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 13125	. . . . . 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 12762	. . . . . . 7 ⊢ -1 ∈ ℤ
23		zq 13111	. . . . . . 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 26571	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴})) ∈ (Poly‘ℚ))
27		0cnd 11329	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → 0 ∈ ℂ)
28		fnconstg 6795	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → (ℂ × {𝐴}) Fn ℂ)
29	28	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (ℂ × {𝐴}) Fn ℂ)
30		ovex 7479	. . . . . . . . . . 11 ⊢ (𝑡↑2) ∈ V
31	30	rgenw 3084	. . . . . . . . . 10 ⊢ ∀𝑡 ∈ ℂ (𝑡↑2) ∈ V
32		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑡ℂ
33	32	mptfnf 6699	. . . . . . . . . 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 11311	. . . . . . . . . 10 ⊢ ℂ ∈ V
37		0cn 11328	. . . . . . . . . 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 7737	. . . . . . . 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 7453	. . . . . . . . . . . 12 ⊢ (𝑡 = 0 → (𝑡↑2) = (0↑2))
43		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) = (𝑡 ∈ ℂ ↦ (𝑡↑2))
44		ovex 7479	. . . . . . . . . . . 12 ⊢ (0↑2) ∈ V
45	42, 43, 44	fvmpt 7019	. . . . . . . . . . 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 14365	. . . . . . . . . 10 ⊢ (0↑2) = 0
48	46, 47	eqtri 2786	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = 0
49	48	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) = 0)
50		fvconst2g 7234	. . . . . . . . 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 7464	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘0) − ((ℂ × {𝐴})‘0)) = (0 − 𝐴))
53	41, 52	eqtrd 2798	. . . . . 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 3012	. . . . . . . 8 ⊢ (𝐴 ≠ 0 ↔ 0 ≠ 𝐴)
56	55	bilani 510	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → 0 ≠ 𝐴)
57	27, 54, 56	subne0d 11709	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (0 − 𝐴) ≠ 0)
58	53, 57	eqnetrd 3026	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘0) ≠ 0)
59		ne0p 26558	. . . . 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 4750	. . 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 7737	. . . . . 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 7453	. . . . . . . . 9 ⊢ (𝑡 = (√‘𝐴) → (𝑡↑2) = ((√‘𝐴)↑2))
66		ovex 7479	. . . . . . . . 9 ⊢ ((√‘𝐴)↑2) ∈ V
67	65, 43, 66	fvmpt 7019	. . . . . . . 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 15564	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴)
70	1, 2, 69	3syl 19	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((√‘𝐴)↑2) = 𝐴)
71	68, 70	eqtrd 2798	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) = 𝐴)
72		fvconst2g 7234	. . . . . . 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 7464	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2))‘(√‘𝐴)) − ((ℂ × {𝐴})‘(√‘𝐴))) = (𝐴 − 𝐴))
75	64, 74	eqtrd 2798	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = (𝐴 − 𝐴))
76		subid 11607	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0)
77	1, 2, 76	3syl 19	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (𝐴 − 𝐴) = 0)
78	75, 77	eqtrd 2798	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (((𝑡 ∈ ℂ ↦ (𝑡↑2)) ∘_f − (ℂ × {𝐴}))‘(√‘𝐴)) = 0)
79	6, 61, 78	rspcedvdw 3583	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0_𝑝})(𝑥‘(√‘𝐴)) = 0)
80		elqaa 26678	. 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 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 {csn 4588 ↦ cmpt 5190 × cxp 5654 Fn wfn 6556 ‘cfv 6561 (class class class)co 7446 ∘_f cof 7718 ℂcc 11228 0cc0 11230 1c1 11231 + caddc 11233 · cmul 11235 − cmin 11571 -cneg 11572 2c2 12427 ℕ₀cn0 12636 ℤcz 12723 ℚcq 13105 ↑cexp 14234 √csqrt 15432 0_𝑝c0p 26023 Polycply 26535 𝔸caa 26670

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-inf2 9672 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-of 7720 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-er 8748 df-map 8880 df-pm 8881 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-sup 9464 df-inf 9465 df-oi 9534 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-n0 12637 df-z 12724 df-uz 12996 df-q 13106 df-rp 13151 df-fz 13670 df-fzo 13819 df-fl 13962 df-mod 14040 df-seq 14175 df-exp 14235 df-hash 14505 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-clim 15687 df-rlim 15688 df-sum 15886 df-0p 26024 df-ply 26539 df-coe 26541 df-dgr 26542 df-aa 26671

This theorem is used by: sqrtqaa 48130

W3C validator