Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  addsqn2reu Structured version   Visualization version   GIF version

Theorem addsqn2reu 26069
 Description: For each complex number 𝐶, there does not exist a unique complex number 𝑏, squared and added to a unique another complex number 𝑎 resulting in the given complex number 𝐶. Actually, for each complex number 𝑏, 𝑎 = (𝐶 − (𝑏↑2)) is unique. Remark: This, together with addsq2reu 26068, shows that commutation of two unique quantifications need not be equivalent, and provides an evident justification of the fact that considering the pair of variables is necessary to obtain what we intuitively understand as "double unique existence". (Proposed by GL, 23-Jun-2023.). (Contributed by AV, 23-Jun-2023.)
Assertion
Ref Expression
addsqn2reu (𝐶 ∈ ℂ → ¬ ∃!𝑏 ∈ ℂ ∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)
Distinct variable group:   𝐶,𝑎,𝑏

Proof of Theorem addsqn2reu
StepHypRef Expression
1 ax-1cn 10602 . . 3 1 ∈ ℂ
2 neg1cn 11757 . . 3 -1 ∈ ℂ
3 1nn 11654 . . . 4 1 ∈ ℕ
4 nnneneg 11678 . . . 4 (1 ∈ ℕ → 1 ≠ -1)
53, 4ax-mp 5 . . 3 1 ≠ -1
61, 2, 53pm3.2i 1336 . 2 (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1)
7 1cnd 10643 . . . . 5 (𝐶 ∈ ℂ → 1 ∈ ℂ)
8 negeu 10883 . . . . 5 ((1 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)
97, 8mpancom 687 . . . 4 (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)
10 sq1 13574 . . . . . . . . 9 (1↑2) = 1
1110a1i 11 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (1↑2) = 1)
1211oveq2d 7161 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (𝑎 + 1))
13 id 22 . . . . . . . . 9 (𝑎 ∈ ℂ → 𝑎 ∈ ℂ)
14 1cnd 10643 . . . . . . . . 9 (𝑎 ∈ ℂ → 1 ∈ ℂ)
1513, 14addcomd 10849 . . . . . . . 8 (𝑎 ∈ ℂ → (𝑎 + 1) = (1 + 𝑎))
1615adantl 485 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + 1) = (1 + 𝑎))
1712, 16eqtrd 2833 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (1 + 𝑎))
1817eqeq1d 2800 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶))
1918reubidva 3342 . . . 4 (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶))
209, 19mpbird 260 . . 3 (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶)
21 neg1sqe1 13575 . . . . . . . . 9 (-1↑2) = 1
2221a1i 11 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (-1↑2) = 1)
2322oveq2d 7161 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (𝑎 + 1))
2423, 16eqtrd 2833 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (1 + 𝑎))
2524eqeq1d 2800 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (-1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶))
2625reubidva 3342 . . . 4 (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶))
279, 26mpbird 260 . . 3 (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶)
2820, 27jca 515 . 2 (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ∧ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶))
29 oveq1 7152 . . . . . 6 (𝑏 = 1 → (𝑏↑2) = (1↑2))
3029oveq2d 7161 . . . . 5 (𝑏 = 1 → (𝑎 + (𝑏↑2)) = (𝑎 + (1↑2)))
3130eqeq1d 2800 . . . 4 (𝑏 = 1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (1↑2)) = 𝐶))
3231reubidv 3343 . . 3 (𝑏 = 1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶))
33 oveq1 7152 . . . . . 6 (𝑏 = -1 → (𝑏↑2) = (-1↑2))
3433oveq2d 7161 . . . . 5 (𝑏 = -1 → (𝑎 + (𝑏↑2)) = (𝑎 + (-1↑2)))
3534eqeq1d 2800 . . . 4 (𝑏 = -1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (-1↑2)) = 𝐶))
3635reubidv 3343 . . 3 (𝑏 = -1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶))
3732, 362nreu 4352 . 2 ((1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1) → ((∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ∧ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶) → ¬ ∃!𝑏 ∈ ℂ ∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶))
386, 28, 37mpsyl 68 1 (𝐶 ∈ ℂ → ¬ ∃!𝑏 ∈ ℂ ∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃!wreu 3108  (class class class)co 7145  ℂcc 10542  1c1 10545   + caddc 10547  -cneg 10878  ℕcn 11643  2c2 11698  ↑cexp 13445 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 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 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 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-div 11305  df-nn 11644  df-2 11706  df-n0 11904  df-z 11990  df-uz 12252  df-seq 13385  df-exp 13446 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator