Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  addsq2reu Structured version   Visualization version   GIF version

 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 26139, is an example showing that the pattern ∃!𝑎 ∈ 𝐴∃!𝑏 ∈ 𝐵𝜑 does not necessarily mean "There are unique sets 𝑎 and 𝑏 fulfilling 𝜑). See also comments for df-eu 2588 and 2eu4 2675. For more details see comment for addsqnreup 26139. (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

Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝐶 ∈ ℂ → 𝐶 ∈ ℂ)
2 oveq1 7163 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 + (𝑏↑2)) = (𝐶 + (𝑏↑2)))
32eqeq1d 2760 . . . . . 6 (𝑎 = 𝐶 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝐶 + (𝑏↑2)) = 𝐶))
43reubidv 3307 . . . . 5 (𝑎 = 𝐶 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶))
5 eqeq1 2762 . . . . . . 7 (𝑎 = 𝐶 → (𝑎 = 𝑐𝐶 = 𝑐))
65imbi2d 344 . . . . . 6 (𝑎 = 𝐶 → ((∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐) ↔ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐)))
76ralbidv 3126 . . . . 5 (𝑎 = 𝐶 → (∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐) ↔ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐)))
84, 7anbi12d 633 . . . 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 485 . . 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 10685 . . . . . 6 (𝐶 ∈ ℂ → 0 ∈ ℂ)
11 reueq 3653 . . . . . 6 (0 ∈ ℂ ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0)
1210, 11sylib 221 . . . . 5 (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ 𝑏 = 0)
13 subid 10956 . . . . . . . . 9 (𝐶 ∈ ℂ → (𝐶𝐶) = 0)
1413adantr 484 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶𝐶) = 0)
1514eqeq1d 2760 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶𝐶) = (𝑏↑2) ↔ 0 = (𝑏↑2)))
16 simpl 486 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
17 simpr 488 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝑏 ∈ ℂ)
1817sqcld 13571 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
1916, 16, 18subaddd 11066 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶𝐶) = (𝑏↑2) ↔ (𝐶 + (𝑏↑2)) = 𝐶))
20 eqcom 2765 . . . . . . . . 9 (0 = (𝑏↑2) ↔ (𝑏↑2) = 0)
21 sqeq0 13549 . . . . . . . . 9 (𝑏 ∈ ℂ → ((𝑏↑2) = 0 ↔ 𝑏 = 0))
2220, 21syl5bb 286 . . . . . . . 8 (𝑏 ∈ ℂ → (0 = (𝑏↑2) ↔ 𝑏 = 0))
2322adantl 485 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (0 = (𝑏↑2) ↔ 𝑏 = 0))
2415, 19, 233bitr3d 312 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 + (𝑏↑2)) = 𝐶𝑏 = 0))
2524reubidva 3306 . . . . 5 (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0))
2612, 25mpbird 260 . . . 4 (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶)
27 simpr 488 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝑐 ∈ ℂ)
2827adantr 484 . . . . . . . 8 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝑐 ∈ ℂ)
29 sqcl 13547 . . . . . . . . 9 (𝑏 ∈ ℂ → (𝑏↑2) ∈ ℂ)
3029adantl 485 . . . . . . . 8 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
31 simpl 486 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝐶 ∈ ℂ)
3231adantr 484 . . . . . . . 8 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
3328, 30, 32addrsub 11108 . . . . . . 7 (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → ((𝑐 + (𝑏↑2)) = 𝐶 ↔ (𝑏↑2) = (𝐶𝑐)))
3433reubidva 3306 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶𝑐)))
35 subcl 10936 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝐶𝑐) ∈ ℂ)
36 reusq0 14883 . . . . . . . 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 10963 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶𝑐) = 0 ↔ 𝐶 = 𝑐))
3938biimpd 232 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶𝑐) = 0 → 𝐶 = 𝑐))
4037, 39sylbid 243 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶𝑐) → 𝐶 = 𝑐))
4134, 40sylbid 243 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐))
4241ralrimiva 3113 . . . 4 (𝐶 ∈ ℂ → ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐))
4326, 42jca 515 . . 3 (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝐶 = 𝑐)))
441, 9, 43rspcedvd 3546 . 2 (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶𝑎 = 𝑐)))
45 oveq1 7163 . . . . 5 (𝑎 = 𝑐 → (𝑎 + (𝑏↑2)) = (𝑐 + (𝑏↑2)))
4645eqeq1d 2760 . . . 4 (𝑎 = 𝑐 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑐 + (𝑏↑2)) = 𝐶))
4746reubidv 3307 . . 3 (𝑎 = 𝑐 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶))
4847reu8 3649 . 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 237 1 (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  ∃!wreu 3072  (class class class)co 7156  ℂcc 10586  0cc0 10588   + caddc 10591   − cmin 10921  2c2 11742  ↑cexp 13492 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 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665  ax-pre-sup 10666 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 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-sup 8952  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-div 11349  df-nn 11688  df-2 11750  df-3 11751  df-n0 11948  df-z 12034  df-uz 12296  df-rp 12444  df-seq 13432  df-exp 13493  df-cj 14519  df-re 14520  df-im 14521  df-sqrt 14655  df-abs 14656 This theorem is referenced by: (None)
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