Theorem reusq0 15102

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 15001	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (√‘𝑋) ∈ ℂ)
3	2	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ∈ ℂ)
4	2	negcld 11249	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → -(√‘𝑋) ∈ ℂ)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → -(√‘𝑋) ∈ ℂ)
6	2	eqnegd 11626	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) ↔ (√‘𝑋) = 0))
7		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 ∈ ℂ)
8		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → (√‘𝑋) = 0)
9	7, 8	sqr00d 15081	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 = 0)
10	9	ex 412	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋) = 0 → 𝑋 = 0))
11	6, 10	sylbid 239	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) → 𝑋 = 0))
12	11	necon3bd 2956	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (¬ 𝑋 = 0 → (√‘𝑋) ≠ -(√‘𝑋)))
13	12	imp 406	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ≠ -(√‘𝑋))
14	3, 5, 13	3jca 1126	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)))
15		sqrtth 15004	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋)↑2) = 𝑋)
16		sqneg 13764	. . . . . . . . . 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 511	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋))
20	19	adantr 480	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋))
21		oveq1 7262	. . . . . . . 8 ⊢ (𝑥 = (√‘𝑋) → (𝑥↑2) = ((√‘𝑋)↑2))
22	21	eqeq1d 2740	. . . . . . 7 ⊢ (𝑥 = (√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ ((√‘𝑋)↑2) = 𝑋))
23		oveq1 7262	. . . . . . . 8 ⊢ (𝑥 = -(√‘𝑋) → (𝑥↑2) = (-(√‘𝑋)↑2))
24	23	eqeq1d 2740	. . . . . . 7 ⊢ (𝑥 = -(√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ (-(√‘𝑋)↑2) = 𝑋))
25	22, 24	2nreu 4372	. . . . . 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 413	. . 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 11976	. . . . . 6 ⊢ 2 ∈ ℕ
31		0cnd 10899	. . . . . . 7 ⊢ (2 ∈ ℕ → 0 ∈ ℂ)
32		oveq1 7262	. . . . . . . . . 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 340	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 0 = 𝑦)))
36	35	ralbidv 3120	. . . . . . . . 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 481	. . . . . . 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 13746	. . . . . . . 8 ⊢ (2 ∈ ℕ → (0↑2) = 0)
40		sqeq0 13768	. . . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑦 ∈ ℂ) → ((𝑦↑2) = 0 → 0 = 𝑦))
45	44	ralrimiva 3107	. . . . . . . 8 ⊢ (2 ∈ ℕ → ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦))
46	39, 45	jca 511	. . . . . . 7 ⊢ (2 ∈ ℕ → ((0↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
47	31, 38, 46	rspcedvd 3555	. . . . . 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 341	. . . . . . . 8 ⊢ (𝑋 = 0 → (((𝑦↑2) = 𝑋 → 𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
52	51	ralbidv 3120	. . . . . . 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 3225	. . . . 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 7262	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
58	57	eqeq1d 2740	. . . 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 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 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  ∃!wreu 3065  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  0cc0 10802  -cneg 11136  ℕcn 11903  2c2 11958  ↑cexp 13710  √csqrt 14872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875

This theorem is referenced by:  addsq2reu  26493