Theorem reusq0 15502

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 15401	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (√‘𝑋) ∈ ℂ)
3	2	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ∈ ℂ)
4	2	negcld 11608	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → -(√‘𝑋) ∈ ℂ)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → -(√‘𝑋) ∈ ℂ)
6	2	eqnegd 11989	. . . . . . . . . 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 15481	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 = 0)
10	9	ex 412	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋) = 0 → 𝑋 = 0))
11	6, 10	sylbid 240	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) → 𝑋 = 0))
12	11	necon3bd 2953	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (¬ 𝑋 = 0 → (√‘𝑋) ≠ -(√‘𝑋)))
13	12	imp 406	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ≠ -(√‘𝑋))
14	3, 5, 13	3jca 1128	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)))
15		sqrtth 15404	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋)↑2) = 𝑋)
16		sqneg 14157	. . . . . . . . . 10 ⊢ ((√‘𝑋) ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
17	2, 16	syl 17	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
18	17, 15	eqtrd 2776	. . . . . . . 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 7439	. . . . . . . 8 ⊢ (𝑥 = (√‘𝑋) → (𝑥↑2) = ((√‘𝑋)↑2))
22	21	eqeq1d 2738	. . . . . . 7 ⊢ (𝑥 = (√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ ((√‘𝑋)↑2) = 𝑋))
23		oveq1 7439	. . . . . . . 8 ⊢ (𝑥 = -(√‘𝑋) → (𝑥↑2) = (-(√‘𝑋)↑2))
24	23	eqeq1d 2738	. . . . . . 7 ⊢ (𝑥 = -(√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ (-(√‘𝑋)↑2) = 𝑋))
25	22, 24	2nreu 4443	. . . . . 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 12340	. . . . . 6 ⊢ 2 ∈ ℕ
31		0cnd 11255	. . . . . . 7 ⊢ (2 ∈ ℕ → 0 ∈ ℂ)
32		oveq1 7439	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥↑2) = (0↑2))
33	32	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥↑2) = 0 ↔ (0↑2) = 0))
34		eqeq1 2740	. . . . . . . . . . 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 3177	. . . . . . . . 9 ⊢ (𝑥 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
37	33, 36	anbi12d 632	. . . . . . . 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 14139	. . . . . . . 8 ⊢ (2 ∈ ℕ → (0↑2) = 0)
40		sqeq0 14161	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
41	40	biimpd 229	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 𝑦 = 0))
42		eqcom 2743	. . . . . . . . . . 11 ⊢ (0 = 𝑦 ↔ 𝑦 = 0)
43	41, 42	imbitrrdi 252	. . . . . . . . . 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 3145	. . . . . . . 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 3623	. . . . . 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 2748	. . . . . . 7 ⊢ (𝑋 = 0 → ((𝑥↑2) = 𝑋 ↔ (𝑥↑2) = 0))
50		eqeq2 2748	. . . . . . . . 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 3177	. . . . . . 7 ⊢ (𝑋 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
53	49, 52	anbi12d 632	. . . . . 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 3178	. . . . 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 257	. . . 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 7439	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
58	57	eqeq1d 2738	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥↑2) = 𝑋 ↔ (𝑦↑2) = 𝑋))
59	58	reu8 3738	. . 3 ⊢ (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦)))
60	56, 59	imbitrrdi 252	. 2 ⊢ (𝑋 ∈ ℂ → (𝑋 = 0 → ∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋))
61	29, 60	impbid 212	1 ⊢ (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ 𝑋 = 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  ∃!wreu 3377  ‘cfv 6560  (class class class)co 7432  ℂcc 11154  0cc0 11156  -cneg 11494  ℕcn 12267  2c2 12322  ↑cexp 14103  √csqrt 15273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-sup 9483  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-seq 14044  df-exp 14104  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276

This theorem is referenced by:  addsq2reu  27485