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Theorem addsq2reu 27484
Description: For each complex number 𝐶, there exists a unique complex number 𝑎 added to the square of a unique another complex number 𝑏 resulting in the given complex number 𝐶. The unique complex number 𝑎 is 𝐶, and the unique another complex number 𝑏 is 0.

Remark: This, together with addsqnreup 27487, is an example showing that the pattern ∃!𝑎𝐴∃!𝑏𝐵𝜑 does not necessarily mean "There are unique sets 𝑎 and 𝑏 fulfilling 𝜑). See also comments for df-eu 2569 and 2eu4 2655. For more details see comment for addsqnreup 27487. (Contributed by AV, 21-Jun-2023.)

Assertion
Ref Expression
addsq2reu (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)
Distinct variable group:   𝐶,𝑎,𝑏

Proof of Theorem addsq2reu
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝐶 ∈ ℂ → 𝐶 ∈ ℂ)
2 oveq1 7438 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 + (𝑏↑2)) = (𝐶 + (𝑏↑2)))
32eqeq1d 2739 . . . . . 6 (𝑎 = 𝐶 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝐶 + (𝑏↑2)) = 𝐶))
43reubidv 3398 . . . . 5 (𝑎 = 𝐶 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶))
5 eqeq1 2741 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 = 𝑐𝐶 = 𝑐))
65imbi2d 340 . . . . . 6 (𝑎 = 𝐶 → ((∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐) ↔ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐)))
76ralbidv 3178 . . . . 5 (𝑎 = 𝐶 → (∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐) ↔ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐)))
84, 7anbi12d 632 . . . 4 (𝑎 = 𝐶 → ((∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐)) ↔ (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐))))
98adantl 481 . . 3 ((𝐶 ∈ ℂ ∧ 𝑎 = 𝐶) → ((∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐)) ↔ (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐))))
10 0cnd 11254 . . . . . 6 (𝐶 ∈ ℂ → 0 ∈ ℂ)
11 reueq 3743 . . . . . 6 (0 ∈ ℂ ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0)
1210, 11sylib 218 . . . . 5 (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ 𝑏 = 0)
13 subid 11528 . . . . . . . . 9 (𝐶 ∈ ℂ → (𝐶𝐶) = 0)
1413adantr 480 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶𝐶) = 0)
1514eqeq1d 2739 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶𝐶) = (𝑏↑2) ↔ 0 = (𝑏↑2)))
16 simpl 482 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
17 simpr 484 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝑏 ∈ ℂ)
1817sqcld 14184 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
1916, 16, 18subaddd 11638 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶𝐶) = (𝑏↑2) ↔ (𝐶 + (𝑏↑2)) = 𝐶))
20 eqcom 2744 . . . . . . . . 9 (0 = (𝑏↑2) ↔ (𝑏↑2) = 0)
21 sqeq0 14160 . . . . . . . . 9 (𝑏 ∈ ℂ → ((𝑏↑2) = 0 ↔ 𝑏 = 0))
2220, 21bitrid 283 . . . . . . . 8 (𝑏 ∈ ℂ → (0 = (𝑏↑2) ↔ 𝑏 = 0))
2322adantl 481 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (0 = (𝑏↑2) ↔ 𝑏 = 0))
2415, 19, 233bitr3d 309 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 + (𝑏↑2)) = 𝐶𝑏 = 0))
2524reubidva 3396 . . . . 5 (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0))
2612, 25mpbird 257 . . . 4 (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶)
27 simpr 484 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝑐 ∈ ℂ)
2827adantr 480 . . . . . . . 8 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝑐 ∈ ℂ)
29 sqcl 14158 . . . . . . . . 9 (𝑏 ∈ ℂ → (𝑏↑2) ∈ ℂ)
3029adantl 481 . . . . . . . 8 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
31 simpl 482 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝐶 ∈ ℂ)
3231adantr 480 . . . . . . . 8 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
3328, 30, 32addrsub 11680 . . . . . . 7 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → ((𝑐 + (𝑏↑2)) = 𝐶 ↔ (𝑏↑2) = (𝐶𝑐)))
3433reubidva 3396 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶𝑐)))
35 subcl 11507 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝐶𝑐) ∈ ℂ)
36 reusq0 15501 . . . . . . . 8 ((𝐶𝑐) ∈ ℂ → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶𝑐) ↔ (𝐶𝑐) = 0))
3735, 36syl 17 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶𝑐) ↔ (𝐶𝑐) = 0))
38 subeq0 11535 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶𝑐) = 0 ↔ 𝐶 = 𝑐))
3938biimpd 229 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶𝑐) = 0 → 𝐶 = 𝑐))
4037, 39sylbid 240 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶𝑐) → 𝐶 = 𝑐))
4134, 40sylbid 240 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐))
4241ralrimiva 3146 . . . 4 (𝐶 ∈ ℂ → ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐))
4326, 42jca 511 . . 3 (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐)))
441, 9, 43rspcedvd 3624 . 2 (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐)))
45 oveq1 7438 . . . . 5 (𝑎 = 𝑐 → (𝑎 + (𝑏↑2)) = (𝑐 + (𝑏↑2)))
4645eqeq1d 2739 . . . 4 (𝑎 = 𝑐 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑐 + (𝑏↑2)) = 𝐶))
4746reubidv 3398 . . 3 (𝑎 = 𝑐 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶))
4847reu8 3739 . 2 (∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐)))
4944, 48sylibr 234 1 (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  ∃!wreu 3378  (class class class)co 7431  cc 11153  0cc0 11155   + caddc 11158  cmin 11492  2c2 12321  cexp 14102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275
This theorem is referenced by: (None)
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