Theorem reusq0 15418

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 15315	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (√‘𝑋) ∈ ℂ)
3	2	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ∈ ℂ)
4	2	negcld 11483	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → -(√‘𝑋) ∈ ℂ)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → -(√‘𝑋) ∈ ℂ)
6	2	eqnegd 11867	. . . . . . . . . 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 15397	. . . . . . . . . . 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 2947	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (¬ 𝑋 = 0 → (√‘𝑋) ≠ -(√‘𝑋)))
13	12	imp 406	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ≠ -(√‘𝑋))
14	3, 5, 13	3jca 1129	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)))
15		sqrtth 15318	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → ((√‘𝑋)↑2) = 𝑋)
16		sqneg 14068	. . . . . . . . . 10 ⊢ ((√‘𝑋) ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
17	2, 16	syl 17	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
18	17, 15	eqtrd 2772	. . . . . . . 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 7367	. . . . . . . 8 ⊢ (𝑥 = (√‘𝑋) → (𝑥↑2) = ((√‘𝑋)↑2))
22	21	eqeq1d 2739	. . . . . . 7 ⊢ (𝑥 = (√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ ((√‘𝑋)↑2) = 𝑋))
23		oveq1 7367	. . . . . . . 8 ⊢ (𝑥 = -(√‘𝑋) → (𝑥↑2) = (-(√‘𝑋)↑2))
24	23	eqeq1d 2739	. . . . . . 7 ⊢ (𝑥 = -(√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ (-(√‘𝑋)↑2) = 𝑋))
25	22, 24	2nreu 4385	. . . . . 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 12245	. . . . . 6 ⊢ 2 ∈ ℕ
31		0cnd 11128	. . . . . . 7 ⊢ (2 ∈ ℕ → 0 ∈ ℂ)
32		oveq1 7367	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥↑2) = (0↑2))
33	32	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥↑2) = 0 ↔ (0↑2) = 0))
34		eqeq1 2741	. . . . . . . . . . 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 3161	. . . . . . . . 9 ⊢ (𝑥 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
37	33, 36	anbi12d 633	. . . . . . . 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 14050	. . . . . . . 8 ⊢ (2 ∈ ℕ → (0↑2) = 0)
40		sqeq0 14073	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
41	40	biimpd 229	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 𝑦 = 0))
42		eqcom 2744	. . . . . . . . . . 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 3130	. . . . . . . 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 3567	. . . . . 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 2749	. . . . . . 7 ⊢ (𝑋 = 0 → ((𝑥↑2) = 𝑋 ↔ (𝑥↑2) = 0))
50		eqeq2 2749	. . . . . . . . 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 3161	. . . . . . 7 ⊢ (𝑋 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
53	49, 52	anbi12d 633	. . . . . 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 3162	. . . . 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 7367	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
58	57	eqeq1d 2739	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥↑2) = 𝑋 ↔ (𝑦↑2) = 𝑋))
59	58	reu8 3680	. . 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 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  ∃!wreu 3341  ‘cfv 6492  (class class class)co 7360  ℂcc 11027  0cc0 11029  -cneg 11369  ℕcn 12165  2c2 12227  ↑cexp 14014  √csqrt 15186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189

This theorem is referenced by:  addsq2reu  27417