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Theorem addsq2reu 26804
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 26807, 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 2568 and 2eu4 2655. For more details see comment for addsqnreup 26807. (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 7369 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 + (𝑏↑2)) = (𝐶 + (𝑏↑2)))
32eqeq1d 2739 . . . . . 6 (𝑎 = 𝐶 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝐶 + (𝑏↑2)) = 𝐶))
43reubidv 3374 . . . . 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 341 . . . . . 6 (𝑎 = 𝐶 → ((∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐) ↔ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐)))
76ralbidv 3175 . . . . 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 483 . . 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 11155 . . . . . 6 (𝐶 ∈ ℂ → 0 ∈ ℂ)
11 reueq 3700 . . . . . 6 (0 ∈ ℂ ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0)
1210, 11sylib 217 . . . . 5 (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ 𝑏 = 0)
13 subid 11427 . . . . . . . . 9 (𝐶 ∈ ℂ → (𝐶𝐶) = 0)
1413adantr 482 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶𝐶) = 0)
1514eqeq1d 2739 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶𝐶) = (𝑏↑2) ↔ 0 = (𝑏↑2)))
16 simpl 484 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
17 simpr 486 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝑏 ∈ ℂ)
1817sqcld 14056 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
1916, 16, 18subaddd 11537 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶𝐶) = (𝑏↑2) ↔ (𝐶 + (𝑏↑2)) = 𝐶))
20 eqcom 2744 . . . . . . . . 9 (0 = (𝑏↑2) ↔ (𝑏↑2) = 0)
21 sqeq0 14032 . . . . . . . . 9 (𝑏 ∈ ℂ → ((𝑏↑2) = 0 ↔ 𝑏 = 0))
2220, 21bitrid 283 . . . . . . . 8 (𝑏 ∈ ℂ → (0 = (𝑏↑2) ↔ 𝑏 = 0))
2322adantl 483 . . . . . . 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 3372 . . . . 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 486 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝑐 ∈ ℂ)
2827adantr 482 . . . . . . . 8 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝑐 ∈ ℂ)
29 sqcl 14030 . . . . . . . . 9 (𝑏 ∈ ℂ → (𝑏↑2) ∈ ℂ)
3029adantl 483 . . . . . . . 8 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
31 simpl 484 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝐶 ∈ ℂ)
3231adantr 482 . . . . . . . 8 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
3328, 30, 32addrsub 11579 . . . . . . 7 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → ((𝑐 + (𝑏↑2)) = 𝐶 ↔ (𝑏↑2) = (𝐶𝑐)))
3433reubidva 3372 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶𝑐)))
35 subcl 11407 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝐶𝑐) ∈ ℂ)
36 reusq0 15354 . . . . . . . 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 11434 . . . . . . . 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 3144 . . . 4 (𝐶 ∈ ℂ → ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐))
4326, 42jca 513 . . 3 (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐)))
441, 9, 43rspcedvd 3586 . 2 (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐)))
45 oveq1 7369 . . . . 5 (𝑎 = 𝑐 → (𝑎 + (𝑏↑2)) = (𝑐 + (𝑏↑2)))
4645eqeq1d 2739 . . . 4 (𝑎 = 𝑐 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑐 + (𝑏↑2)) = 𝐶))
4746reubidv 3374 . . 3 (𝑎 = 𝑐 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶))
4847reu8 3696 . 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 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  ∃!wreu 3354  (class class class)co 7362  cc 11056  0cc0 11058   + caddc 11061  cmin 11392  2c2 12215  cexp 13974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9385  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-z 12507  df-uz 12771  df-rp 12923  df-seq 13914  df-exp 13975  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128
This theorem is referenced by: (None)
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