Theorem addsqnreup 27501

Description: There is no unique decomposition of a complex number as a sum of a complex number and a square of a complex number.

Remark: This theorem, together with addsq2reu 27498, is a real life example (about a numerical property) 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 2566 and 2eu4 2652.

In the case of decompositions of complex numbers as a sum of a complex number and a square of a complex number, the only/unique complex number to which the square of a unique complex number is added yields in the given complex number is the given number itself, and the unique complex number to be squared is 0 (see comment for addsq2reu 27498). There are, however, complex numbers to which the square of more than one other complex numbers can be added to yield the given complex number (see addsqrexnreu 27500). For example, ⟨1, (√‘(𝐶 − 1))⟩ and ⟨1, -(√‘(𝐶 − 1))⟩ are two different decompositions of 𝐶 (if 𝐶 ≠ 1). Therefore, there is no unique decomposition of any complex number as a sum of a complex number and a square of a complex number, as generally proved by this theorem.

As a consequence, a theorem must claim the existence of a unique pair of sets to express "There are unique 𝑎 and 𝑏 so that .." (more formally ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 with 𝑝 = ⟨𝑎, 𝑏⟩), or by showing (∃!𝑥 ∈ 𝐴∃𝑦 ∈ 𝐵𝜑 ∧ ∃!𝑦 ∈ 𝐵∃𝑥 ∈ 𝐴𝜑) (see 2reu4 4528 resp. 2eu4 2652). These two representations are equivalent (see opreu2reurex 6315). An analogon of this theorem using the latter variant is given in addsqn2reurex2 27503. In some cases, however, the variant with (ordered!) pairs may be possible only for ordered sets (like ℝ or ℙ) and claiming that the first component is less than or equal to the second component (see, for example, 2sqreunnltb 27519 and 2sqreuopb 27526). Alternatively, (proper) unordered pairs can be used: ∃!𝑝𝑒𝒫 𝐴((♯‘𝑝) = 2 ∧ 𝜑), or, using the definition of proper pairs: ∃!𝑝 ∈ (Pairs_proper‘𝐴)𝜑 (see, for example, inlinecirc02preu 48637). (Contributed by AV, 21-Jun-2023.)

Assertion

Ref	Expression
addsqnreup	⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶)

Distinct variable group:   𝐶,𝑝

Proof of Theorem addsqnreup

Step	Hyp	Ref	Expression
1		ax-1cn 11210	. . . . . . 7 ⊢ 1 ∈ ℂ
2		0cn 11250	. . . . . . 7 ⊢ 0 ∈ ℂ
3		opelxpi 5725	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨1, 0⟩ ∈ (ℂ × ℂ))
4	1, 2, 3	mp2an 692	. . . . . 6 ⊢ ⟨1, 0⟩ ∈ (ℂ × ℂ)
5		3cn 12344	. . . . . . . 8 ⊢ 3 ∈ ℂ
6	5	negcli 11574	. . . . . . 7 ⊢ -3 ∈ ℂ
7		2cn 12338	. . . . . . 7 ⊢ 2 ∈ ℂ
8		opelxpi 5725	. . . . . . 7 ⊢ ((-3 ∈ ℂ ∧ 2 ∈ ℂ) → ⟨-3, 2⟩ ∈ (ℂ × ℂ))
9	6, 7, 8	mp2an 692	. . . . . 6 ⊢ ⟨-3, 2⟩ ∈ (ℂ × ℂ)
10		0ne2 12470	. . . . . . . 8 ⊢ 0 ≠ 2
11	10	olci 866	. . . . . . 7 ⊢ (1 ≠ -3 ∨ 0 ≠ 2)
12		1ex 11254	. . . . . . . 8 ⊢ 1 ∈ V
13		c0ex 11252	. . . . . . . 8 ⊢ 0 ∈ V
14	12, 13	opthne 5492	. . . . . . 7 ⊢ (⟨1, 0⟩ ≠ ⟨-3, 2⟩ ↔ (1 ≠ -3 ∨ 0 ≠ 2))
15	11, 14	mpbir 231	. . . . . 6 ⊢ ⟨1, 0⟩ ≠ ⟨-3, 2⟩
16	4, 9, 15	3pm3.2i 1338	. . . . 5 ⊢ (⟨1, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨-3, 2⟩ ∈ (ℂ × ℂ) ∧ ⟨1, 0⟩ ≠ ⟨-3, 2⟩)
17	12, 13	op1st 8020	. . . . . . . 8 ⊢ (1st ‘⟨1, 0⟩) = 1
18	12, 13	op2nd 8021	. . . . . . . . . 10 ⊢ (2nd ‘⟨1, 0⟩) = 0
19	18	oveq1i 7440	. . . . . . . . 9 ⊢ ((2nd ‘⟨1, 0⟩)↑2) = (0↑2)
20		sq0 14227	. . . . . . . . 9 ⊢ (0↑2) = 0
21	19, 20	eqtri 2762	. . . . . . . 8 ⊢ ((2nd ‘⟨1, 0⟩)↑2) = 0
22	17, 21	oveq12i 7442	. . . . . . 7 ⊢ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = (1 + 0)
23		1p0e1 12387	. . . . . . 7 ⊢ (1 + 0) = 1
24	22, 23	eqtri 2762	. . . . . 6 ⊢ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1
25		negex 11503	. . . . . . . . 9 ⊢ -3 ∈ V
26		2ex 12340	. . . . . . . . 9 ⊢ 2 ∈ V
27	25, 26	op1st 8020	. . . . . . . 8 ⊢ (1st ‘⟨-3, 2⟩) = -3
28	25, 26	op2nd 8021	. . . . . . . . . 10 ⊢ (2nd ‘⟨-3, 2⟩) = 2
29	28	oveq1i 7440	. . . . . . . . 9 ⊢ ((2nd ‘⟨-3, 2⟩)↑2) = (2↑2)
30		sq2 14232	. . . . . . . . 9 ⊢ (2↑2) = 4
31	29, 30	eqtri 2762	. . . . . . . 8 ⊢ ((2nd ‘⟨-3, 2⟩)↑2) = 4
32	27, 31	oveq12i 7442	. . . . . . 7 ⊢ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = (-3 + 4)
33		4cn 12348	. . . . . . . 8 ⊢ 4 ∈ ℂ
34	33, 5	negsubi 11584	. . . . . . . . 9 ⊢ (4 + -3) = (4 − 3)
35		3p1e4 12408	. . . . . . . . . 10 ⊢ (3 + 1) = 4
36	33, 5, 1, 35	subaddrii 11595	. . . . . . . . 9 ⊢ (4 − 3) = 1
37	34, 36	eqtri 2762	. . . . . . . 8 ⊢ (4 + -3) = 1
38	33, 6, 37	addcomli 11450	. . . . . . 7 ⊢ (-3 + 4) = 1
39	32, 38	eqtri 2762	. . . . . 6 ⊢ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1
40	24, 39	pm3.2i 470	. . . . 5 ⊢ (((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1 ∧ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1)
41		fveq2 6906	. . . . . . . 8 ⊢ (𝑝 = ⟨1, 0⟩ → (1st ‘𝑝) = (1st ‘⟨1, 0⟩))
42		fveq2 6906	. . . . . . . . 9 ⊢ (𝑝 = ⟨1, 0⟩ → (2nd ‘𝑝) = (2nd ‘⟨1, 0⟩))
43	42	oveq1d 7445	. . . . . . . 8 ⊢ (𝑝 = ⟨1, 0⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨1, 0⟩)↑2))
44	41, 43	oveq12d 7448	. . . . . . 7 ⊢ (𝑝 = ⟨1, 0⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)))
45	44	eqeq1d 2736	. . . . . 6 ⊢ (𝑝 = ⟨1, 0⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1 ↔ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1))
46		fveq2 6906	. . . . . . . 8 ⊢ (𝑝 = ⟨-3, 2⟩ → (1st ‘𝑝) = (1st ‘⟨-3, 2⟩))
47		fveq2 6906	. . . . . . . . 9 ⊢ (𝑝 = ⟨-3, 2⟩ → (2nd ‘𝑝) = (2nd ‘⟨-3, 2⟩))
48	47	oveq1d 7445	. . . . . . . 8 ⊢ (𝑝 = ⟨-3, 2⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨-3, 2⟩)↑2))
49	46, 48	oveq12d 7448	. . . . . . 7 ⊢ (𝑝 = ⟨-3, 2⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)))
50	49	eqeq1d 2736	. . . . . 6 ⊢ (𝑝 = ⟨-3, 2⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1 ↔ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1))
51	45, 50	2nreu 4449	. . . . 5 ⊢ ((⟨1, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨-3, 2⟩ ∈ (ℂ × ℂ) ∧ ⟨1, 0⟩ ≠ ⟨-3, 2⟩) → ((((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1 ∧ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1) → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1))
52	16, 40, 51	mp2 9	. . . 4 ⊢ ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1
53		eqeq2 2746	. . . . 5 ⊢ (𝐶 = 1 → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1))
54	53	reubidv 3395	. . . 4 ⊢ (𝐶 = 1 → (∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1))
55	52, 54	mtbiri 327	. . 3 ⊢ (𝐶 = 1 → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶)
56	55	a1d 25	. 2 ⊢ (𝐶 = 1 → (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶))
57		id 22	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 𝐶 ∈ ℂ)
58		0cnd 11251	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 0 ∈ ℂ)
59	57, 58	opelxpd 5727	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
60	59	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
61		1cnd 11253	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ)
62		peano2cnm 11572	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ)
63	62	sqrtcld 15472	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (√‘(𝐶 − 1)) ∈ ℂ)
64	61, 63	opelxpd 5727	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
65	64	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
66		animorrl 982	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1))))
67		0cnd 11251	. . . . . . 7 ⊢ (𝐶 ≠ 1 → 0 ∈ ℂ)
68		opthneg 5491	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
69	67, 68	sylan2 593	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
70	66, 69	mpbird 257	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩)
71	60, 65, 70	3jca 1127	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ) ∧ ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩))
72		op1stg 8024	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (1st ‘⟨𝐶, 0⟩) = 𝐶)
73	58, 72	mpdan 687	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (1st ‘⟨𝐶, 0⟩) = 𝐶)
74		op2ndg 8025	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (2nd ‘⟨𝐶, 0⟩) = 0)
75	58, 74	mpdan 687	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (2nd ‘⟨𝐶, 0⟩) = 0)
76	75	sq0id 14229	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨𝐶, 0⟩)↑2) = 0)
77	73, 76	oveq12d 7448	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = (𝐶 + 0))
78		addrid 11438	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (𝐶 + 0) = 𝐶)
79	77, 78	eqtrd 2774	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶)
80		op1stg 8024	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
81	61, 63, 80	syl2anc 584	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
82		op2ndg 8025	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
83	61, 63, 82	syl2anc 584	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℂ → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
84	83	oveq1d 7445	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = ((√‘(𝐶 − 1))↑2))
85	62	sqsqrtd 15474	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → ((√‘(𝐶 − 1))↑2) = (𝐶 − 1))
86	84, 85	eqtrd 2774	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = (𝐶 − 1))
87	81, 86	oveq12d 7448	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = (1 + (𝐶 − 1)))
88	61, 57	pncan3d 11620	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (1 + (𝐶 − 1)) = 𝐶)
89	87, 88	eqtrd 2774	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶)
90	79, 89	jca 511	. . . . 5 ⊢ (𝐶 ∈ ℂ → (((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶 ∧ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
91	90	adantr 480	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶 ∧ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
92		fveq2 6906	. . . . . . 7 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (1st ‘𝑝) = (1st ‘⟨𝐶, 0⟩))
93		fveq2 6906	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝐶, 0⟩))
94	93	oveq1d 7445	. . . . . . 7 ⊢ (𝑝 = ⟨𝐶, 0⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨𝐶, 0⟩)↑2))
95	92, 94	oveq12d 7448	. . . . . 6 ⊢ (𝑝 = ⟨𝐶, 0⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)))
96	95	eqeq1d 2736	. . . . 5 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶))
97		fveq2 6906	. . . . . . 7 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (1st ‘𝑝) = (1st ‘⟨1, (√‘(𝐶 − 1))⟩))
98		fveq2 6906	. . . . . . . 8 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (2nd ‘𝑝) = (2nd ‘⟨1, (√‘(𝐶 − 1))⟩))
99	98	oveq1d 7445	. . . . . . 7 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2))
100	97, 99	oveq12d 7448	. . . . . 6 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)))
101	100	eqeq1d 2736	. . . . 5 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
102	96, 101	2nreu 4449	. . . 4 ⊢ ((⟨𝐶, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ) ∧ ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩) → ((((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶 ∧ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶) → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶))
103	71, 91, 102	sylc 65	. . 3 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶)
104	103	expcom 413	. 2 ⊢ (𝐶 ≠ 1 → (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶))
105	56, 104	pm2.61ine 3022	1 ⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∃!wreu 3375  ⟨cop 4636   × cxp 5686  ‘cfv 6562  (class class class)co 7430  1^st c1st 8010  2^nd c2nd 8011  ℂcc 11150  0cc0 11152  1c1 11153   + caddc 11155   − cmin 11489  -cneg 11490  2c2 12318  3c3 12319  4c4 12320  ↑cexp 14098  √csqrt 15268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271

This theorem is referenced by: (None)