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Theorem reusq0 15347
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 15246 . . . . . . . 8 (𝑋 ∈ ℂ → (√‘𝑋) ∈ ℂ)
32adantr 481 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ∈ ℂ)
42negcld 11499 . . . . . . . 8 (𝑋 ∈ ℂ → -(√‘𝑋) ∈ ℂ)
54adantr 481 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → -(√‘𝑋) ∈ ℂ)
62eqnegd 11876 . . . . . . . . . 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 15326 . . . . . . . . . . 11 ((𝑋 ∈ ℂ ∧ (√‘𝑋) = 0) → 𝑋 = 0)
109ex 413 . . . . . . . . . 10 (𝑋 ∈ ℂ → ((√‘𝑋) = 0 → 𝑋 = 0))
116, 10sylbid 239 . . . . . . . . 9 (𝑋 ∈ ℂ → ((√‘𝑋) = -(√‘𝑋) → 𝑋 = 0))
1211necon3bd 2957 . . . . . . . 8 (𝑋 ∈ ℂ → (¬ 𝑋 = 0 → (√‘𝑋) ≠ -(√‘𝑋)))
1312imp 407 . . . . . . 7 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → (√‘𝑋) ≠ -(√‘𝑋))
143, 5, 133jca 1128 . . . . . 6 ((𝑋 ∈ ℂ ∧ ¬ 𝑋 = 0) → ((√‘𝑋) ∈ ℂ ∧ -(√‘𝑋) ∈ ℂ ∧ (√‘𝑋) ≠ -(√‘𝑋)))
15 sqrtth 15249 . . . . . . . 8 (𝑋 ∈ ℂ → ((√‘𝑋)↑2) = 𝑋)
16 sqneg 14021 . . . . . . . . . 10 ((√‘𝑋) ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
172, 16syl 17 . . . . . . . . 9 (𝑋 ∈ ℂ → (-(√‘𝑋)↑2) = ((√‘𝑋)↑2))
1817, 15eqtrd 2776 . . . . . . . 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 7364 . . . . . . . 8 (𝑥 = (√‘𝑋) → (𝑥↑2) = ((√‘𝑋)↑2))
2221eqeq1d 2738 . . . . . . 7 (𝑥 = (√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ ((√‘𝑋)↑2) = 𝑋))
23 oveq1 7364 . . . . . . . 8 (𝑥 = -(√‘𝑋) → (𝑥↑2) = (-(√‘𝑋)↑2))
2423eqeq1d 2738 . . . . . . 7 (𝑥 = -(√‘𝑋) → ((𝑥↑2) = 𝑋 ↔ (-(√‘𝑋)↑2) = 𝑋))
2522, 242nreu 4401 . . . . . 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 182 . 2 (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋𝑋 = 0))
30 2nn 12226 . . . . . 6 2 ∈ ℕ
31 0cnd 11148 . . . . . . 7 (2 ∈ ℕ → 0 ∈ ℂ)
32 oveq1 7364 . . . . . . . . . 10 (𝑥 = 0 → (𝑥↑2) = (0↑2))
3332eqeq1d 2738 . . . . . . . . 9 (𝑥 = 0 → ((𝑥↑2) = 0 ↔ (0↑2) = 0))
34 eqeq1 2740 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
3534imbi2d 340 . . . . . . . . . 10 (𝑥 = 0 → (((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 0 = 𝑦)))
3635ralbidv 3174 . . . . . . . . 9 (𝑥 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 0 = 𝑦)))
3733, 36anbi12d 631 . . . . . . . 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 14003 . . . . . . . 8 (2 ∈ ℕ → (0↑2) = 0)
40 sqeq0 14025 . . . . . . . . . . . 12 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 ↔ 𝑦 = 0))
4140biimpd 228 . . . . . . . . . . 11 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 𝑦 = 0))
42 eqcom 2743 . . . . . . . . . . 11 (0 = 𝑦𝑦 = 0)
4341, 42syl6ibr 251 . . . . . . . . . 10 (𝑦 ∈ ℂ → ((𝑦↑2) = 0 → 0 = 𝑦))
4443adantl 482 . . . . . . . . 9 ((2 ∈ ℕ ∧ 𝑦 ∈ ℂ) → ((𝑦↑2) = 0 → 0 = 𝑦))
4544ralrimiva 3143 . . . . . . . 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 3583 . . . . . 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 2748 . . . . . . 7 (𝑋 = 0 → ((𝑥↑2) = 𝑋 ↔ (𝑥↑2) = 0))
50 eqeq2 2748 . . . . . . . . 9 (𝑋 = 0 → ((𝑦↑2) = 𝑋 ↔ (𝑦↑2) = 0))
5150imbi1d 341 . . . . . . . 8 (𝑋 = 0 → (((𝑦↑2) = 𝑋𝑥 = 𝑦) ↔ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
5251ralbidv 3174 . . . . . . 7 (𝑋 = 0 → (∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦) ↔ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦)))
5349, 52anbi12d 631 . . . . . 6 (𝑋 = 0 → (((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦)) ↔ ((𝑥↑2) = 0 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 0 → 𝑥 = 𝑦))))
5453rexbidv 3175 . . . . 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 7364 . . . . 5 (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2))
5857eqeq1d 2738 . . . 4 (𝑥 = 𝑦 → ((𝑥↑2) = 𝑋 ↔ (𝑦↑2) = 𝑋))
5958reu8 3691 . . 3 (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝑋 ∧ ∀𝑦 ∈ ℂ ((𝑦↑2) = 𝑋𝑥 = 𝑦)))
6056, 59syl6ibr 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 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  ∃!wreu 3351  cfv 6496  (class class class)co 7357  cc 11049  0cc0 11051  -cneg 11386  cn 12153  2c2 12208  cexp 13967  csqrt 15118
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121
This theorem is referenced by:  addsq2reu  26788
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