Theorem addsq2reu 27428

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 27431, 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 2573 and 2eu4 2659. For more details see comment for addsqnreup 27431. (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 7370	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 + (𝑏↑2)) = (𝐶 + (𝑏↑2)))
3	2	eqeq1d 2742	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝐶 + (𝑏↑2)) = 𝐶))
4	3	reubidv 3361	. . . . 5 ⊢ (𝑎 = 𝐶 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶))
5		eqeq1 2744	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 = 𝑐 ↔ 𝐶 = 𝑐))
6	5	imbi2d 341	. . . . . 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 638	. . . 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 482	. . 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 11135	. . . . . 6 ⊢ (𝐶 ∈ ℂ → 0 ∈ ℂ)
11		reueq 3685	. . . . . 6 ⊢ (0 ∈ ℂ ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0)
12	10, 11	sylib 219	. . . . 5 ⊢ (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ 𝑏 = 0)
13		subid 11411	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (𝐶 − 𝐶) = 0)
14	13	adantr 481	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − 𝐶) = 0)
15	14	eqeq1d 2742	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − 𝐶) = (𝑏↑2) ↔ 0 = (𝑏↑2)))
16		simpl 483	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
17		simpr 485	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝑏 ∈ ℂ)
18	17	sqcld 14104	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
19	16, 16, 18	subaddd 11521	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − 𝐶) = (𝑏↑2) ↔ (𝐶 + (𝑏↑2)) = 𝐶))
20		eqcom 2747	. . . . . . . . 9 ⊢ (0 = (𝑏↑2) ↔ (𝑏↑2) = 0)
21		sqeq0 14080	. . . . . . . . 9 ⊢ (𝑏 ∈ ℂ → ((𝑏↑2) = 0 ↔ 𝑏 = 0))
22	20, 21	bitrid 284	. . . . . . . 8 ⊢ (𝑏 ∈ ℂ → (0 = (𝑏↑2) ↔ 𝑏 = 0))
23	22	adantl 482	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (0 = (𝑏↑2) ↔ 𝑏 = 0))
24	15, 19, 23	3bitr3d 310	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 + (𝑏↑2)) = 𝐶 ↔ 𝑏 = 0))
25	24	reubidva 3359	. . . . 5 ⊢ (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0))
26	12, 25	mpbird 258	. . . 4 ⊢ (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶)
27		simpr 485	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝑐 ∈ ℂ)
28	27	adantr 481	. . . . . . . 8 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝑐 ∈ ℂ)
29		sqcl 14078	. . . . . . . . 9 ⊢ (𝑏 ∈ ℂ → (𝑏↑2) ∈ ℂ)
30	29	adantl 482	. . . . . . . 8 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
31		simpl 483	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝐶 ∈ ℂ)
32	31	adantr 481	. . . . . . . 8 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
33	28, 30, 32	addrsub 11565	. . . . . . 7 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → ((𝑐 + (𝑏↑2)) = 𝐶 ↔ (𝑏↑2) = (𝐶 − 𝑐)))
34	33	reubidva 3359	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐)))
35		subcl 11390	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝐶 − 𝑐) ∈ ℂ)
36		reusq0 15425	. . . . . . . 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 11418	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶 − 𝑐) = 0 ↔ 𝐶 = 𝑐))
39	38	biimpd 230	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶 − 𝑐) = 0 → 𝐶 = 𝑐))
40	37, 39	sylbid 241	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐) → 𝐶 = 𝑐))
41	34, 40	sylbid 241	. . . . 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 516	. . 3 ⊢ (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)))
44	1, 9, 43	rspcedvd 3569	. 2 ⊢ (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)))
45		oveq1 7370	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎 + (𝑏↑2)) = (𝑐 + (𝑏↑2)))
46	45	eqeq1d 2742	. . . 4 ⊢ (𝑎 = 𝑐 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑐 + (𝑏↑2)) = 𝐶))
47	46	reubidv 3361	. . 3 ⊢ (𝑎 = 𝑐 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶))
48	47	reu8 3681	. 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 235	1 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  ∃!wreu 3343  (class class class)co 7363  ℂcc 11034  0cc0 11036   + caddc 11039   − cmin 11375  2c2 12234  ↑cexp 14021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9352  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-n0 12436  df-z 12523  df-uz 12787  df-rp 12941  df-seq 13962  df-exp 14022  df-cj 15059  df-re 15060  df-im 15061  df-sqrt 15195  df-abs 15196

This theorem is referenced by: (None)