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Theorem addsqn2reu 26322
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 26321, 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 10787 . . 3 1 ∈ ℂ
2 neg1cn 11944 . . 3 -1 ∈ ℂ
3 1nn 11841 . . . 4 1 ∈ ℕ
4 nnneneg 11865 . . . 4 (1 ∈ ℕ → 1 ≠ -1)
53, 4ax-mp 5 . . 3 1 ≠ -1
61, 2, 53pm3.2i 1341 . 2 (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1)
7 1cnd 10828 . . . . 5 (𝐶 ∈ ℂ → 1 ∈ ℂ)
8 negeu 11068 . . . . 5 ((1 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)
97, 8mpancom 688 . . . 4 (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)
10 sq1 13764 . . . . . . . . 9 (1↑2) = 1
1110a1i 11 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (1↑2) = 1)
1211oveq2d 7229 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (𝑎 + 1))
13 id 22 . . . . . . . . 9 (𝑎 ∈ ℂ → 𝑎 ∈ ℂ)
14 1cnd 10828 . . . . . . . . 9 (𝑎 ∈ ℂ → 1 ∈ ℂ)
1513, 14addcomd 11034 . . . . . . . 8 (𝑎 ∈ ℂ → (𝑎 + 1) = (1 + 𝑎))
1615adantl 485 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + 1) = (1 + 𝑎))
1712, 16eqtrd 2777 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (1 + 𝑎))
1817eqeq1d 2739 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶))
1918reubidva 3300 . . . 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 13765 . . . . . . . . 9 (-1↑2) = 1
2221a1i 11 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (-1↑2) = 1)
2322oveq2d 7229 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (𝑎 + 1))
2423, 16eqtrd 2777 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (1 + 𝑎))
2524eqeq1d 2739 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (-1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶))
2625reubidva 3300 . . . 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 7220 . . . . . 6 (𝑏 = 1 → (𝑏↑2) = (1↑2))
3029oveq2d 7229 . . . . 5 (𝑏 = 1 → (𝑎 + (𝑏↑2)) = (𝑎 + (1↑2)))
3130eqeq1d 2739 . . . 4 (𝑏 = 1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (1↑2)) = 𝐶))
3231reubidv 3301 . . 3 (𝑏 = 1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶))
33 oveq1 7220 . . . . . 6 (𝑏 = -1 → (𝑏↑2) = (-1↑2))
3433oveq2d 7229 . . . . 5 (𝑏 = -1 → (𝑎 + (𝑏↑2)) = (𝑎 + (-1↑2)))
3534eqeq1d 2739 . . . 4 (𝑏 = -1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (-1↑2)) = 𝐶))
3635reubidv 3301 . . 3 (𝑏 = -1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶))
3732, 362nreu 4356 . 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 1089   = wceq 1543  wcel 2110  wne 2940  ∃!wreu 3063  (class class class)co 7213  cc 10727  1c1 10730   + caddc 10732  -cneg 11063  cn 11830  2c2 11885  cexp 13635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-n0 12091  df-z 12177  df-uz 12439  df-seq 13575  df-exp 13636
This theorem is referenced by: (None)
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