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Theorem reusq0 14660
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 14559 . . . . . . . 8 (𝑋 ∈ ℂ → (√‘𝑋) ∈ ℂ)
32adantr 481 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ∈ ℂ)
42negcld 10838 . . . . . . . 8 (𝑋 ∈ ℂ → -(√‘𝑋) ∈ ℂ)
54adantr 481 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → -(√‘𝑋) ∈ ℂ)
62eqnegd 11215 . . . . . . . . . 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)
97, 8sqr00d 14639 . . . . . . . . . . 11 ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 = 0)
109ex 413 . . . . . . . . . 10 (𝑋 ∈ ℂ → ((√‘𝑋) = 0 → 𝑋 = 0))
116, 10sylbid 241 . . . . . . . . 9 (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) → 𝑋 = 0))
1211necon3bd 3000 . . . . . . . 8 (𝑋 ∈ ℂ → (¬ 𝑋 = 0 → (√‘𝑋) ≠ -(√‘𝑋)))
1312imp 407 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ≠ -(√‘𝑋))
143, 5, 133jca 1121 . . . . . 6 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)))
15 sqrtth 14562 . . . . . . . 8 (𝑋 ∈ ℂ → ((√‘𝑋)↑2) = 𝑋)
16 sqneg 13336 . . . . . . . . . 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 512 . . . . . . 7 (𝑋 ∈ ℂ → (((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋))
2019adantr 481 . . . . . 6 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋))
21 oveq1 7030 . . . . . . . 8 (𝑥 = (√‘𝑋) → (𝑥↑2) = ((√‘𝑋)↑2))
2221eqeq1d 2799 . . . . . . 7 (𝑥 = (√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ ((√‘𝑋)↑2) = 𝑋))
23 oveq1 7030 . . . . . . . 8 (𝑥 = -(√‘𝑋) → (𝑥↑2) = (-(√‘𝑋)↑2))
2423eqeq1d 2799 . . . . . . 7 (𝑥 = -(√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ (-(√‘𝑋)↑2) = 𝑋))
2522, 242nreu 4313 . . . . . 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 414 . . 3 𝑋 = 0 → (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0)))
291, 28pm2.61i 183 . 2 (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0))
30 2nn 11564 . . . . . 6 2 ∈ ℕ
31 0cnd 10487 . . . . . . 7 (2 ∈ ℕ → 0 ∈ ℂ)
32 oveq1 7030 . . . . . . . . . 10 (𝑥 = 0 → (𝑥↑2) = (0↑2))
3332eqeq1d 2799 . . . . . . . . 9 (𝑥 = 0 → ((𝑥↑2) = 0 ↔ (0↑2) = 0))
34 eqeq1 2801 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
3534imbi2d 342 . . . . . . . . . 10 (𝑥 = 0 → (((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 0 = 𝑦)))
3635ralbidv 3166 . . . . . . . . 9 (𝑥 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
3733, 36anbi12d 630 . . . . . . . 8 (𝑥 = 0 → (((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)) ↔ ((0↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦))))
3837adantl 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 13318 . . . . . . . 8 (2 ∈ ℕ → (0↑2) = 0)
40 sqeq0 13340 . . . . . . . . . . . 12 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
4140biimpd 230 . . . . . . . . . . 11 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 𝑦 = 0))
42 eqcom 2804 . . . . . . . . . . 11 (0 = 𝑦𝑦 = 0)
4341, 42syl6ibr 253 . . . . . . . . . 10 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 0 = 𝑦))
4443adantl 482 . . . . . . . . 9 ((2 ∈ ℕ ∧ 𝑦 ∈ ℂ) → ((𝑦↑2) = 0 → 0 = 𝑦))
4544ralrimiva 3151 . . . . . . . 8 (2 ∈ ℕ → ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦))
4639, 45jca 512 . . . . . . 7 (2 ∈ ℕ → ((0↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
4731, 38, 46rspcedvd 3568 . . . . . 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 2808 . . . . . . 7 (𝑋 = 0 → ((𝑥↑2) = 𝑋 ↔ (𝑥↑2) = 0))
50 eqeq2 2808 . . . . . . . . 9 (𝑋 = 0 → ((𝑦↑2) = 𝑋 ↔ (𝑦↑2) = 0))
5150imbi1d 343 . . . . . . . 8 (𝑋 = 0 → (((𝑦↑2) = 𝑋𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
5251ralbidv 3166 . . . . . . 7 (𝑋 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
5349, 52anbi12d 630 . . . . . 6 (𝑋 = 0 → (((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦)) ↔ ((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦))))
5453rexbidv 3262 . . . . 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 258 . . . 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 7030 . . . . 5 (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
5857eqeq1d 2799 . . . 4 (𝑥 = 𝑦 → ((𝑥↑2) = 𝑋 ↔ (𝑦↑2) = 𝑋))
5958reu8 3663 . . 3 (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦)))
6056, 59syl6ibr 253 . 2 (𝑋 ∈ ℂ → (𝑋 = 0 → ∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋))
6129, 60impbid 213 1 (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  wne 2986  wral 3107  wrex 3108  ∃!wreu 3109  cfv 6232  (class class class)co 7023  cc 10388  0cc0 10390  -cneg 10724  cn 11492  2c2 11546  cexp 13283  csqrt 14430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-sup 8759  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-n0 11752  df-z 11836  df-uz 12098  df-rp 12244  df-seq 13224  df-exp 13284  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433
This theorem is referenced by:  addsq2reu  25702
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