Theorem addsq2reu 27401

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 27404, 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 2568 and 2eu4 2654. For more details see comment for addsqnreup 27404. (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.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝐶 ∈ ℂ → 𝐶 ∈ ℂ)
2		oveq1 7410	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 + (𝑏↑2)) = (𝐶 + (𝑏↑2)))
3	2	eqeq1d 2737	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝐶 + (𝑏↑2)) = 𝐶))
4	3	reubidv 3377	. . . . 5 ⊢ (𝑎 = 𝐶 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶))
5		eqeq1 2739	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 = 𝑐 ↔ 𝐶 = 𝑐))
6	5	imbi2d 340	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐) ↔ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)))
7	6	ralbidv 3163	. . . . 5 ⊢ (𝑎 = 𝐶 → (∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)))
8	4, 7	anbi12d 632	. . . 4 ⊢ (𝑎 = 𝐶 → ((∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)) ↔ (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))))
9	8	adantl 481	. . 3 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 = 𝐶) → ((∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)) ↔ (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))))
10		0cnd 11226	. . . . . 6 ⊢ (𝐶 ∈ ℂ → 0 ∈ ℂ)
11		reueq 3720	. . . . . 6 ⊢ (0 ∈ ℂ ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0)
12	10, 11	sylib 218	. . . . 5 ⊢ (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ 𝑏 = 0)
13		subid 11500	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (𝐶 − 𝐶) = 0)
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − 𝐶) = 0)
15	14	eqeq1d 2737	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − 𝐶) = (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)
18	17	sqcld 14160	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
19	16, 16, 18	subaddd 11610	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − 𝐶) = (𝑏↑2) ↔ (𝐶 + (𝑏↑2)) = 𝐶))
20		eqcom 2742	. . . . . . . . 9 ⊢ (0 = (𝑏↑2) ↔ (𝑏↑2) = 0)
21		sqeq0 14136	. . . . . . . . 9 ⊢ (𝑏 ∈ ℂ → ((𝑏↑2) = 0 ↔ 𝑏 = 0))
22	20, 21	bitrid 283	. . . . . . . 8 ⊢ (𝑏 ∈ ℂ → (0 = (𝑏↑2) ↔ 𝑏 = 0))
23	22	adantl 481	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (0 = (𝑏↑2) ↔ 𝑏 = 0))
24	15, 19, 23	3bitr3d 309	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 + (𝑏↑2)) = 𝐶 ↔ 𝑏 = 0))
25	24	reubidva 3375	. . . . 5 ⊢ (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0))
26	12, 25	mpbird 257	. . . 4 ⊢ (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶)
27		simpr 484	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝑐 ∈ ℂ)
28	27	adantr 480	. . . . . . . 8 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝑐 ∈ ℂ)
29		sqcl 14134	. . . . . . . . 9 ⊢ (𝑏 ∈ ℂ → (𝑏↑2) ∈ ℂ)
30	29	adantl 481	. . . . . . . 8 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
31		simpl 482	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝐶 ∈ ℂ)
32	31	adantr 480	. . . . . . . 8 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
33	28, 30, 32	addrsub 11652	. . . . . . 7 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → ((𝑐 + (𝑏↑2)) = 𝐶 ↔ (𝑏↑2) = (𝐶 − 𝑐)))
34	33	reubidva 3375	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐)))
35		subcl 11479	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝐶 − 𝑐) ∈ ℂ)
36		reusq0 15479	. . . . . . . 8 ⊢ ((𝐶 − 𝑐) ∈ ℂ → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐) ↔ (𝐶 − 𝑐) = 0))
37	35, 36	syl 17	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐) ↔ (𝐶 − 𝑐) = 0))
38		subeq0 11507	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶 − 𝑐) = 0 ↔ 𝐶 = 𝑐))
39	38	biimpd 229	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶 − 𝑐) = 0 → 𝐶 = 𝑐))
40	37, 39	sylbid 240	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐) → 𝐶 = 𝑐))
41	34, 40	sylbid 240	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))
42	41	ralrimiva 3132	. . . 4 ⊢ (𝐶 ∈ ℂ → ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))
43	26, 42	jca 511	. . 3 ⊢ (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)))
44	1, 9, 43	rspcedvd 3603	. 2 ⊢ (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)))
45		oveq1 7410	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎 + (𝑏↑2)) = (𝑐 + (𝑏↑2)))
46	45	eqeq1d 2737	. . . 4 ⊢ (𝑎 = 𝑐 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑐 + (𝑏↑2)) = 𝐶))
47	46	reubidv 3377	. . 3 ⊢ (𝑎 = 𝑐 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶))
48	47	reu8 3716	. 2 ⊢ (∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)))
49	44, 48	sylibr 234	1 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  ∃!wreu 3357  (class class class)co 7403  ℂcc 11125  0cc0 11127   + caddc 11130   − cmin 11464  2c2 12293  ↑cexp 14077

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9452  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-3 12302  df-n0 12500  df-z 12587  df-uz 12851  df-rp 13007  df-seq 14018  df-exp 14078  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253

This theorem is referenced by: (None)