Theorem reusq0 15174

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 15073	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (√‘𝑋) ∈ ℂ)
3	2	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ∈ ℂ)
4	2	negcld 11319	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → -(√‘𝑋) ∈ ℂ)
5	4	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → -(√‘𝑋) ∈ ℂ)
6	2	eqnegd 11696	. . . . . . . . . 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 15153	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 = 0)
10	9	ex 413	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋) = 0 → 𝑋 = 0))
11	6, 10	sylbid 239	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) → 𝑋 = 0))
12	11	necon3bd 2957	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (¬ 𝑋 = 0 → (√‘𝑋) ≠ -(√‘𝑋)))
13	12	imp 407	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ≠ -(√‘𝑋))
14	3, 5, 13	3jca 1127	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)))
15		sqrtth 15076	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋)↑2) = 𝑋)
16		sqneg 13836	. . . . . . . . . 10 ⊢ ((√‘𝑋) ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
17	2, 16	syl 17	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
18	17, 15	eqtrd 2778	. . . . . . . 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 7282	. . . . . . . 8 ⊢ (𝑥 = (√‘𝑋) → (𝑥↑2) = ((√‘𝑋)↑2))
22	21	eqeq1d 2740	. . . . . . 7 ⊢ (𝑥 = (√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ ((√‘𝑋)↑2) = 𝑋))
23		oveq1 7282	. . . . . . . 8 ⊢ (𝑥 = -(√‘𝑋) → (𝑥↑2) = (-(√‘𝑋)↑2))
24	23	eqeq1d 2740	. . . . . . 7 ⊢ (𝑥 = -(√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ (-(√‘𝑋)↑2) = 𝑋))
25	22, 24	2nreu 4375	. . . . . 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 182	. 2 ⊢ (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 → 𝑋 = 0))
30		2nn 12046	. . . . . 6 ⊢ 2 ∈ ℕ
31		0cnd 10968	. . . . . . 7 ⊢ (2 ∈ ℕ → 0 ∈ ℂ)
32		oveq1 7282	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥↑2) = (0↑2))
33	32	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥↑2) = 0 ↔ (0↑2) = 0))
34		eqeq1 2742	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
35	34	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 0 = 𝑦)))
36	35	ralbidv 3112	. . . . . . . . 9 ⊢ (𝑥 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
37	33, 36	anbi12d 631	. . . . . . . 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 13818	. . . . . . . 8 ⊢ (2 ∈ ℕ → (0↑2) = 0)
40		sqeq0 13840	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
41	40	biimpd 228	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 𝑦 = 0))
42		eqcom 2745	. . . . . . . . . . 11 ⊢ (0 = 𝑦 ↔ 𝑦 = 0)
43	41, 42	syl6ibr 251	. . . . . . . . . 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 3103	. . . . . . . 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 3563	. . . . . 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 2750	. . . . . . 7 ⊢ (𝑋 = 0 → ((𝑥↑2) = 𝑋 ↔ (𝑥↑2) = 0))
50		eqeq2 2750	. . . . . . . . 9 ⊢ (𝑋 = 0 → ((𝑦↑2) = 𝑋 ↔ (𝑦↑2) = 0))
51	50	imbi1d 342	. . . . . . . 8 ⊢ (𝑋 = 0 → (((𝑦↑2) = 𝑋 → 𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
52	51	ralbidv 3112	. . . . . . 7 ⊢ (𝑋 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
53	49, 52	anbi12d 631	. . . . . 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 3226	. . . . 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 256	. . . 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 7282	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
58	57	eqeq1d 2740	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥↑2) = 𝑋 ↔ (𝑦↑2) = 𝑋))
59	58	reu8 3668	. . 3 ⊢ (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦)))
60	56, 59	syl6ibr 251	. 2 ⊢ (𝑋 ∈ ℂ → (𝑋 = 0 → ∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋))
61	29, 60	impbid 211	1 ⊢ (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ 𝑋 = 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  ∃!wreu 3066  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  0cc0 10871  -cneg 11206  ℕcn 11973  2c2 12028  ↑cexp 13782  √csqrt 14944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947

This theorem is referenced by:  addsq2reu  26588