Theorem addsq2reu 27407

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 27410, 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 2569 and 2eu4 2655. For more details see comment for addsqnreup 27410. (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 7365	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 + (𝑏↑2)) = (𝐶 + (𝑏↑2)))
3	2	eqeq1d 2738	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝐶 + (𝑏↑2)) = 𝐶))
4	3	reubidv 3366	. . . . 5 ⊢ (𝑎 = 𝐶 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶))
5		eqeq1 2740	. . . . . . 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 3159	. . . . 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 11125	. . . . . 6 ⊢ (𝐶 ∈ ℂ → 0 ∈ ℂ)
11		reueq 3695	. . . . . 6 ⊢ (0 ∈ ℂ ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0)
12	10, 11	sylib 218	. . . . 5 ⊢ (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ 𝑏 = 0)
13		subid 11400	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (𝐶 − 𝐶) = 0)
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − 𝐶) = 0)
15	14	eqeq1d 2738	. . . . . . 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 14067	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
19	16, 16, 18	subaddd 11510	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − 𝐶) = (𝑏↑2) ↔ (𝐶 + (𝑏↑2)) = 𝐶))
20		eqcom 2743	. . . . . . . . 9 ⊢ (0 = (𝑏↑2) ↔ (𝑏↑2) = 0)
21		sqeq0 14043	. . . . . . . . 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 3364	. . . . 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 14041	. . . . . . . . 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 11554	. . . . . . 7 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → ((𝑐 + (𝑏↑2)) = 𝐶 ↔ (𝑏↑2) = (𝐶 − 𝑐)))
34	33	reubidva 3364	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐)))
35		subcl 11379	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝐶 − 𝑐) ∈ ℂ)
36		reusq0 15388	. . . . . . . 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 11407	. . . . . . . 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 3128	. . . 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 3578	. 2 ⊢ (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)))
45		oveq1 7365	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎 + (𝑏↑2)) = (𝑐 + (𝑏↑2)))
46	45	eqeq1d 2738	. . . 4 ⊢ (𝑎 = 𝑐 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑐 + (𝑏↑2)) = 𝐶))
47	46	reubidv 3366	. . 3 ⊢ (𝑎 = 𝑐 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶))
48	47	reu8 3691	. 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 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  ∃!wreu 3348  (class class class)co 7358  ℂcc 11024  0cc0 11026   + caddc 11029   − cmin 11364  2c2 12200  ↑cexp 13984

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-rp 12906  df-seq 13925  df-exp 13985  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159

This theorem is referenced by: (None)