MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reusq0 Structured version   Visualization version   GIF version

Theorem reusq0 15445
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 15344 . . . . . . . 8 (𝑋 ∈ ℂ → (√‘𝑋) ∈ ℂ)
32adantr 479 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ∈ ℂ)
42negcld 11590 . . . . . . . 8 (𝑋 ∈ ℂ → -(√‘𝑋) ∈ ℂ)
54adantr 479 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → -(√‘𝑋) ∈ ℂ)
62eqnegd 11968 . . . . . . . . . 10 (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) ↔ (√‘𝑋) = 0))
7 simpl 481 . . . . . . . . . . . 12 ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 ∈ ℂ)
8 simpr 483 . . . . . . . . . . . 12 ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → (√‘𝑋) = 0)
97, 8sqr00d 15424 . . . . . . . . . . 11 ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 = 0)
109ex 411 . . . . . . . . . 10 (𝑋 ∈ ℂ → ((√‘𝑋) = 0 → 𝑋 = 0))
116, 10sylbid 239 . . . . . . . . 9 (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) → 𝑋 = 0))
1211necon3bd 2943 . . . . . . . 8 (𝑋 ∈ ℂ → (¬ 𝑋 = 0 → (√‘𝑋) ≠ -(√‘𝑋)))
1312imp 405 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ≠ -(√‘𝑋))
143, 5, 133jca 1125 . . . . . 6 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)))
15 sqrtth 15347 . . . . . . . 8 (𝑋 ∈ ℂ → ((√‘𝑋)↑2) = 𝑋)
16 sqneg 14116 . . . . . . . . . 10 ((√‘𝑋) ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
172, 16syl 17 . . . . . . . . 9 (𝑋 ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
1817, 15eqtrd 2765 . . . . . . . 8 (𝑋 ∈ ℂ → (-(√‘𝑋)↑2) = 𝑋)
1915, 18jca 510 . . . . . . 7 (𝑋 ∈ ℂ → (((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋))
2019adantr 479 . . . . . 6 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (((√‘𝑋)↑2) = 𝑋 ∧ (-(√‘𝑋)↑2) = 𝑋))
21 oveq1 7426 . . . . . . . 8 (𝑥 = (√‘𝑋) → (𝑥↑2) = ((√‘𝑋)↑2))
2221eqeq1d 2727 . . . . . . 7 (𝑥 = (√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ ((√‘𝑋)↑2) = 𝑋))
23 oveq1 7426 . . . . . . . 8 (𝑥 = -(√‘𝑋) → (𝑥↑2) = (-(√‘𝑋)↑2))
2423eqeq1d 2727 . . . . . . 7 (𝑥 = -(√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ (-(√‘𝑋)↑2) = 𝑋))
2522, 242nreu 4443 . . . . . 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 412 . . 3 𝑋 = 0 → (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0)))
291, 28pm2.61i 182 . 2 (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0))
30 2nn 12318 . . . . . 6 2 ∈ ℕ
31 0cnd 11239 . . . . . . 7 (2 ∈ ℕ → 0 ∈ ℂ)
32 oveq1 7426 . . . . . . . . . 10 (𝑥 = 0 → (𝑥↑2) = (0↑2))
3332eqeq1d 2727 . . . . . . . . 9 (𝑥 = 0 → ((𝑥↑2) = 0 ↔ (0↑2) = 0))
34 eqeq1 2729 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
3534imbi2d 339 . . . . . . . . . 10 (𝑥 = 0 → (((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 0 = 𝑦)))
3635ralbidv 3167 . . . . . . . . 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 480 . . . . . . 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 14098 . . . . . . . 8 (2 ∈ ℕ → (0↑2) = 0)
40 sqeq0 14120 . . . . . . . . . . . 12 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
4140biimpd 228 . . . . . . . . . . 11 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 𝑦 = 0))
42 eqcom 2732 . . . . . . . . . . 11 (0 = 𝑦𝑦 = 0)
4341, 42imbitrrdi 251 . . . . . . . . . 10 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 0 = 𝑦))
4443adantl 480 . . . . . . . . 9 ((2 ∈ ℕ ∧ 𝑦 ∈ ℂ) → ((𝑦↑2) = 0 → 0 = 𝑦))
4544ralrimiva 3135 . . . . . . . 8 (2 ∈ ℕ → ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦))
4639, 45jca 510 . . . . . . 7 (2 ∈ ℕ → ((0↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
4731, 38, 46rspcedvd 3608 . . . . . 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 2737 . . . . . . 7 (𝑋 = 0 → ((𝑥↑2) = 𝑋 ↔ (𝑥↑2) = 0))
50 eqeq2 2737 . . . . . . . . 9 (𝑋 = 0 → ((𝑦↑2) = 𝑋 ↔ (𝑦↑2) = 0))
5150imbi1d 340 . . . . . . . 8 (𝑋 = 0 → (((𝑦↑2) = 𝑋𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
5251ralbidv 3167 . . . . . . 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 3168 . . . . 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 256 . . . 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 7426 . . . . 5 (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
5857eqeq1d 2727 . . . 4 (𝑥 = 𝑦 → ((𝑥↑2) = 𝑋 ↔ (𝑦↑2) = 𝑋))
5958reu8 3725 . . 3 (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦)))
6056, 59imbitrrdi 251 . 2 (𝑋 ∈ ℂ → (𝑋 = 0 → ∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋))
6129, 60impbid 211 1 (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  ∃!wreu 3361  cfv 6549  (class class class)co 7419  cc 11138  0cc0 11140  -cneg 11477  cn 12245  2c2 12300  cexp 14062  csqrt 15216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9467  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-seq 14003  df-exp 14063  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219
This theorem is referenced by:  addsq2reu  27418
  Copyright terms: Public domain W3C validator