Theorem reusq0 14660

Description: A complex number is the square of exactly one complex number iff the given complex number is zero. (Contributed by AV, 21-Jun-2023.)

Assertion

Ref	Expression
reusq0	⊢ (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ 𝑋 = 0))

Distinct variable group:   𝑥,𝑋

Proof of Theorem reusq0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2a1 28	. . 3 ⊢ (𝑋 = 0 → (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 → 𝑋 = 0)))
2		sqrtcl 14559	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (√‘𝑋) ∈ ℂ)
3	2	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ∈ ℂ)
4	2	negcld 10838	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → -(√‘𝑋) ∈ ℂ)
5	4	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → -(√‘𝑋) ∈ ℂ)
6	2	eqnegd 11215	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) ↔ (√‘𝑋) = 0))
7		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 ∈ ℂ)
8		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → (√‘𝑋) = 0)
9	7, 8	sqr00d 14639	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 = 0)
10	9	ex 413	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋) = 0 → 𝑋 = 0))
11	6, 10	sylbid 241	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) → 𝑋 = 0))
12	11	necon3bd 3000	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (¬ 𝑋 = 0 → (√‘𝑋) ≠ -(√‘𝑋)))
13	12	imp 407	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ≠ -(√‘𝑋))
14	3, 5, 13	3jca 1121	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)))
15		sqrtth 14562	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋)↑2) = 𝑋)
16		sqneg 13336	. . . . . . . . . 10 ⊢ ((√‘𝑋) ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
17	2, 16	syl 17	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
18	17, 15	eqtrd 2833	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (-(√‘𝑋)↑2) = 𝑋)
19	15, 18	jca 512	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋))
20	19	adantr 481	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋))
21		oveq1 7030	. . . . . . . 8 ⊢ (𝑥 = (√‘𝑋) → (𝑥↑2) = ((√‘𝑋)↑2))
22	21	eqeq1d 2799	. . . . . . 7 ⊢ (𝑥 = (√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ ((√‘𝑋)↑2) = 𝑋))
23		oveq1 7030	. . . . . . . 8 ⊢ (𝑥 = -(√‘𝑋) → (𝑥↑2) = (-(√‘𝑋)↑2))
24	23	eqeq1d 2799	. . . . . . 7 ⊢ (𝑥 = -(√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ (-(√‘𝑋)↑2) = 𝑋))
25	22, 24	2nreu 4313	. . . . . 6 ⊢ (((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)) → ((((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋) → ¬ ∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋))
26	14, 20, 25	sylc 65	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ¬ ∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋)
27	26	pm2.21d 121	. . . 4 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 → 𝑋 = 0))
28	27	expcom 414	. . 3 ⊢ (¬ 𝑋 = 0 → (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 → 𝑋 = 0)))
29	1, 28	pm2.61i 183	. 2 ⊢ (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 → 𝑋 = 0))
30		2nn 11564	. . . . . 6 ⊢ 2 ∈ ℕ
31		0cnd 10487	. . . . . . 7 ⊢ (2 ∈ ℕ → 0 ∈ ℂ)
32		oveq1 7030	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥↑2) = (0↑2))
33	32	eqeq1d 2799	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥↑2) = 0 ↔ (0↑2) = 0))
34		eqeq1 2801	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
35	34	imbi2d 342	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 0 = 𝑦)))
36	35	ralbidv 3166	. . . . . . . . 9 ⊢ (𝑥 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
37	33, 36	anbi12d 630	. . . . . . . 8 ⊢ (𝑥 = 0 → (((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)) ↔ ((0↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦))))
38	37	adantl 482	. . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 𝑥 = 0) → (((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)) ↔ ((0↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦))))
39		0exp 13318	. . . . . . . 8 ⊢ (2 ∈ ℕ → (0↑2) = 0)
40		sqeq0 13340	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
41	40	biimpd 230	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 𝑦 = 0))
42		eqcom 2804	. . . . . . . . . . 11 ⊢ (0 = 𝑦 ↔ 𝑦 = 0)
43	41, 42	syl6ibr 253	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 0 = 𝑦))
44	43	adantl 482	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑦 ∈ ℂ) → ((𝑦↑2) = 0 → 0 = 𝑦))
45	44	ralrimiva 3151	. . . . . . . 8 ⊢ (2 ∈ ℕ → ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦))
46	39, 45	jca 512	. . . . . . 7 ⊢ (2 ∈ ℕ → ((0↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
47	31, 38, 46	rspcedvd 3568	. . . . . 6 ⊢ (2 ∈ ℕ → ∃𝑥 ∈ ℂ ((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
48	30, 47	mp1i 13	. . . . 5 ⊢ (𝑋 = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
49		eqeq2 2808	. . . . . . 7 ⊢ (𝑋 = 0 → ((𝑥↑2) = 𝑋 ↔ (𝑥↑2) = 0))
50		eqeq2 2808	. . . . . . . . 9 ⊢ (𝑋 = 0 → ((𝑦↑2) = 𝑋 ↔ (𝑦↑2) = 0))
51	50	imbi1d 343	. . . . . . . 8 ⊢ (𝑋 = 0 → (((𝑦↑2) = 𝑋 → 𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
52	51	ralbidv 3166	. . . . . . 7 ⊢ (𝑋 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
53	49, 52	anbi12d 630	. . . . . 6 ⊢ (𝑋 = 0 → (((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦)) ↔ ((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦))))
54	53	rexbidv 3262	. . . . 5 ⊢ (𝑋 = 0 → (∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦)) ↔ ∃𝑥 ∈ ℂ ((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦))))
55	48, 54	mpbird 258	. . . 4 ⊢ (𝑋 = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦)))
56	55	a1i 11	. . 3 ⊢ (𝑋 ∈ ℂ → (𝑋 = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦))))
57		oveq1 7030	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
58	57	eqeq1d 2799	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥↑2) = 𝑋 ↔ (𝑦↑2) = 𝑋))
59	58	reu8 3663	. . 3 ⊢ (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦)))
60	56, 59	syl6ibr 253	. 2 ⊢ (𝑋 ∈ ℂ → (𝑋 = 0 → ∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋))
61	29, 60	impbid 213	1 ⊢ (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ 𝑋 = 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083   ≠ wne 2986  ∀wral 3107  ∃wrex 3108  ∃!wreu 3109  ‘cfv 6232  (class class class)co 7023  ℂcc 10388  0cc0 10390  -cneg 10724  ℕcn 11492  2c2 11546  ↑cexp 13283  √csqrt 14430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-sup 8759  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-n0 11752  df-z 11836  df-uz 12098  df-rp 12244  df-seq 13224  df-exp 13284  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433

This theorem is referenced by:  addsq2reu  25702