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Theorem addsq2reu 27180
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 27183, 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 2562 and 2eu4 2649. For more details see comment for addsqnreup 27183. (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 7419 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 + (𝑏↑2)) = (𝐶 + (𝑏↑2)))
32eqeq1d 2733 . . . . . 6 (𝑎 = 𝐶 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝐶 + (𝑏↑2)) = 𝐶))
43reubidv 3393 . . . . 5 (𝑎 = 𝐶 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶))
5 eqeq1 2735 . . . . . . 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 3176 . . . . 5 (𝑎 = 𝐶 → (∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐) ↔ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐)))
84, 7anbi12d 630 . . . 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 11212 . . . . . 6 (𝐶 ∈ ℂ → 0 ∈ ℂ)
11 reueq 3733 . . . . . 6 (0 ∈ ℂ ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0)
1210, 11sylib 217 . . . . 5 (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ 𝑏 = 0)
13 subid 11484 . . . . . . . . 9 (𝐶 ∈ ℂ → (𝐶𝐶) = 0)
1413adantr 480 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶𝐶) = 0)
1514eqeq1d 2733 . . . . . . 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 14114 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
1916, 16, 18subaddd 11594 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶𝐶) = (𝑏↑2) ↔ (𝐶 + (𝑏↑2)) = 𝐶))
20 eqcom 2738 . . . . . . . . 9 (0 = (𝑏↑2) ↔ (𝑏↑2) = 0)
21 sqeq0 14090 . . . . . . . . 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 3391 . . . . 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 14088 . . . . . . . . 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 11636 . . . . . . 7 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → ((𝑐 + (𝑏↑2)) = 𝐶 ↔ (𝑏↑2) = (𝐶𝑐)))
3433reubidva 3391 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶𝑐)))
35 subcl 11464 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝐶𝑐) ∈ ℂ)
36 reusq0 15414 . . . . . . . 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 11491 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶𝑐) = 0 ↔ 𝐶 = 𝑐))
3938biimpd 228 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶𝑐) = 0 → 𝐶 = 𝑐))
4037, 39sylbid 239 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶𝑐) → 𝐶 = 𝑐))
4134, 40sylbid 239 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐))
4241ralrimiva 3145 . . . 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 3614 . 2 (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐)))
45 oveq1 7419 . . . . 5 (𝑎 = 𝑐 → (𝑎 + (𝑏↑2)) = (𝑐 + (𝑏↑2)))
4645eqeq1d 2733 . . . 4 (𝑎 = 𝑐 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑐 + (𝑏↑2)) = 𝐶))
4746reubidv 3393 . . 3 (𝑎 = 𝑐 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶))
4847reu8 3729 . 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 233 1 (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wral 3060  wrex 3069  ∃!wreu 3373  (class class class)co 7412  cc 11112  0cc0 11114   + caddc 11117  cmin 11449  2c2 12272  cexp 14032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-sup 9441  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-n0 12478  df-z 12564  df-uz 12828  df-rp 12980  df-seq 13972  df-exp 14033  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188
This theorem is referenced by: (None)
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