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Theorem reusq0 14816
 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) = 𝑋𝑋 = 0))
Distinct variable group:   𝑥,𝑋

Proof of Theorem reusq0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 2a1 28 . . 3 (𝑋 = 0 → (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0)))
2 sqrtcl 14715 . . . . . . . 8 (𝑋 ∈ ℂ → (√‘𝑋) ∈ ℂ)
32adantr 484 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ∈ ℂ)
42negcld 10975 . . . . . . . 8 (𝑋 ∈ ℂ → -(√‘𝑋) ∈ ℂ)
54adantr 484 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → -(√‘𝑋) ∈ ℂ)
62eqnegd 11352 . . . . . . . . . 10 (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) ↔ (√‘𝑋) = 0))
7 simpl 486 . . . . . . . . . . . 12 ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 ∈ ℂ)
8 simpr 488 . . . . . . . . . . . 12 ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → (√‘𝑋) = 0)
97, 8sqr00d 14795 . . . . . . . . . . 11 ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 = 0)
109ex 416 . . . . . . . . . 10 (𝑋 ∈ ℂ → ((√‘𝑋) = 0 → 𝑋 = 0))
116, 10sylbid 243 . . . . . . . . 9 (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) → 𝑋 = 0))
1211necon3bd 3001 . . . . . . . 8 (𝑋 ∈ ℂ → (¬ 𝑋 = 0 → (√‘𝑋) ≠ -(√‘𝑋)))
1312imp 410 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ≠ -(√‘𝑋))
143, 5, 133jca 1125 . . . . . 6 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)))
15 sqrtth 14718 . . . . . . . 8 (𝑋 ∈ ℂ → ((√‘𝑋)↑2) = 𝑋)
16 sqneg 13480 . . . . . . . . . 10 ((√‘𝑋) ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
172, 16syl 17 . . . . . . . . 9 (𝑋 ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
1817, 15eqtrd 2833 . . . . . . . 8 (𝑋 ∈ ℂ → (-(√‘𝑋)↑2) = 𝑋)
1915, 18jca 515 . . . . . . 7 (𝑋 ∈ ℂ → (((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋))
2019adantr 484 . . . . . 6 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋))
21 oveq1 7142 . . . . . . . 8 (𝑥 = (√‘𝑋) → (𝑥↑2) = ((√‘𝑋)↑2))
2221eqeq1d 2800 . . . . . . 7 (𝑥 = (√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ ((√‘𝑋)↑2) = 𝑋))
23 oveq1 7142 . . . . . . . 8 (𝑥 = -(√‘𝑋) → (𝑥↑2) = (-(√‘𝑋)↑2))
2423eqeq1d 2800 . . . . . . 7 (𝑥 = -(√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ (-(√‘𝑋)↑2) = 𝑋))
2522, 242nreu 4349 . . . . . 6 (((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)) → ((((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋) → ¬ ∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋))
2614, 20, 25sylc 65 . . . . 5 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ¬ ∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋)
2726pm2.21d 121 . . . 4 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0))
2827expcom 417 . . 3 𝑋 = 0 → (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0)))
291, 28pm2.61i 185 . 2 (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0))
30 2nn 11700 . . . . . 6 2 ∈ ℕ
31 0cnd 10625 . . . . . . 7 (2 ∈ ℕ → 0 ∈ ℂ)
32 oveq1 7142 . . . . . . . . . 10 (𝑥 = 0 → (𝑥↑2) = (0↑2))
3332eqeq1d 2800 . . . . . . . . 9 (𝑥 = 0 → ((𝑥↑2) = 0 ↔ (0↑2) = 0))
34 eqeq1 2802 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
3534imbi2d 344 . . . . . . . . . 10 (𝑥 = 0 → (((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 0 = 𝑦)))
3635ralbidv 3162 . . . . . . . . 9 (𝑥 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
3733, 36anbi12d 633 . . . . . . . 8 (𝑥 = 0 → (((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)) ↔ ((0↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦))))
3837adantl 485 . . . . . . 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 13462 . . . . . . . 8 (2 ∈ ℕ → (0↑2) = 0)
40 sqeq0 13484 . . . . . . . . . . . 12 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
4140biimpd 232 . . . . . . . . . . 11 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 𝑦 = 0))
42 eqcom 2805 . . . . . . . . . . 11 (0 = 𝑦𝑦 = 0)
4341, 42syl6ibr 255 . . . . . . . . . 10 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 0 = 𝑦))
4443adantl 485 . . . . . . . . 9 ((2 ∈ ℕ ∧ 𝑦 ∈ ℂ) → ((𝑦↑2) = 0 → 0 = 𝑦))
4544ralrimiva 3149 . . . . . . . 8 (2 ∈ ℕ → ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦))
4639, 45jca 515 . . . . . . 7 (2 ∈ ℕ → ((0↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
4731, 38, 46rspcedvd 3574 . . . . . 6 (2 ∈ ℕ → ∃𝑥 ∈ ℂ ((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
4830, 47mp1i 13 . . . . 5 (𝑋 = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
49 eqeq2 2810 . . . . . . 7 (𝑋 = 0 → ((𝑥↑2) = 𝑋 ↔ (𝑥↑2) = 0))
50 eqeq2 2810 . . . . . . . . 9 (𝑋 = 0 → ((𝑦↑2) = 𝑋 ↔ (𝑦↑2) = 0))
5150imbi1d 345 . . . . . . . 8 (𝑋 = 0 → (((𝑦↑2) = 𝑋𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
5251ralbidv 3162 . . . . . . 7 (𝑋 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
5349, 52anbi12d 633 . . . . . 6 (𝑋 = 0 → (((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦)) ↔ ((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦))))
5453rexbidv 3256 . . . . 5 (𝑋 = 0 → (∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦)) ↔ ∃𝑥 ∈ ℂ ((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦))))
5548, 54mpbird 260 . . . 4 (𝑋 = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦)))
5655a1i 11 . . 3 (𝑋 ∈ ℂ → (𝑋 = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦))))
57 oveq1 7142 . . . . 5 (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
5857eqeq1d 2800 . . . 4 (𝑥 = 𝑦 → ((𝑥↑2) = 𝑋 ↔ (𝑦↑2) = 𝑋))
5958reu8 3672 . . 3 (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦)))
6056, 59syl6ibr 255 . 2 (𝑋 ∈ ℂ → (𝑋 = 0 → ∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋))
6129, 60impbid 215 1 (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  ‘cfv 6324  (class class class)co 7135  ℂcc 10526  0cc0 10528  -cneg 10862  ℕcn 11627  2c2 11682  ↑cexp 13427  √csqrt 14586 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605  ax-pre-sup 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-sup 8892  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-2 11690  df-3 11691  df-n0 11888  df-z 11972  df-uz 12234  df-rp 12380  df-seq 13367  df-exp 13428  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589 This theorem is referenced by:  addsq2reu  26031
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