Theorem addsqnreup 27352

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 27349, 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 2562 and 2eu4 2648.

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 27349). 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 27351). 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 4474 resp. 2eu4 2648). These two representations are equivalent (see opreu2reurex 6242). An analogon of this theorem using the latter variant is given in addsqn2reurex2 27354. 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 27370 and 2sqreuopb 27377). 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 48777). (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 11067	. . . . . . 7 ⊢ 1 ∈ ℂ
2		0cn 11107	. . . . . . 7 ⊢ 0 ∈ ℂ
3		opelxpi 5656	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨1, 0⟩ ∈ (ℂ × ℂ))
4	1, 2, 3	mp2an 692	. . . . . 6 ⊢ ⟨1, 0⟩ ∈ (ℂ × ℂ)
5		3cn 12209	. . . . . . . 8 ⊢ 3 ∈ ℂ
6	5	negcli 11432	. . . . . . 7 ⊢ -3 ∈ ℂ
7		2cn 12203	. . . . . . 7 ⊢ 2 ∈ ℂ
8		opelxpi 5656	. . . . . . 7 ⊢ ((-3 ∈ ℂ ∧ 2 ∈ ℂ) → ⟨-3, 2⟩ ∈ (ℂ × ℂ))
9	6, 7, 8	mp2an 692	. . . . . 6 ⊢ ⟨-3, 2⟩ ∈ (ℂ × ℂ)
10		0ne2 12330	. . . . . . . 8 ⊢ 0 ≠ 2
11	10	olci 866	. . . . . . 7 ⊢ (1 ≠ -3 ∨ 0 ≠ 2)
12		1ex 11111	. . . . . . . 8 ⊢ 1 ∈ V
13		c0ex 11109	. . . . . . . 8 ⊢ 0 ∈ V
14	12, 13	opthne 5425	. . . . . . 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 1340	. . . . 5 ⊢ (⟨1, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨-3, 2⟩ ∈ (ℂ × ℂ) ∧ ⟨1, 0⟩ ≠ ⟨-3, 2⟩)
17	12, 13	op1st 7932	. . . . . . . 8 ⊢ (1st ‘⟨1, 0⟩) = 1
18	12, 13	op2nd 7933	. . . . . . . . . 10 ⊢ (2nd ‘⟨1, 0⟩) = 0
19	18	oveq1i 7359	. . . . . . . . 9 ⊢ ((2nd ‘⟨1, 0⟩)↑2) = (0↑2)
20		sq0 14099	. . . . . . . . 9 ⊢ (0↑2) = 0
21	19, 20	eqtri 2752	. . . . . . . 8 ⊢ ((2nd ‘⟨1, 0⟩)↑2) = 0
22	17, 21	oveq12i 7361	. . . . . . 7 ⊢ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = (1 + 0)
23		1p0e1 12247	. . . . . . 7 ⊢ (1 + 0) = 1
24	22, 23	eqtri 2752	. . . . . 6 ⊢ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1
25		negex 11361	. . . . . . . . 9 ⊢ -3 ∈ V
26		2ex 12205	. . . . . . . . 9 ⊢ 2 ∈ V
27	25, 26	op1st 7932	. . . . . . . 8 ⊢ (1st ‘⟨-3, 2⟩) = -3
28	25, 26	op2nd 7933	. . . . . . . . . 10 ⊢ (2nd ‘⟨-3, 2⟩) = 2
29	28	oveq1i 7359	. . . . . . . . 9 ⊢ ((2nd ‘⟨-3, 2⟩)↑2) = (2↑2)
30		sq2 14104	. . . . . . . . 9 ⊢ (2↑2) = 4
31	29, 30	eqtri 2752	. . . . . . . 8 ⊢ ((2nd ‘⟨-3, 2⟩)↑2) = 4
32	27, 31	oveq12i 7361	. . . . . . 7 ⊢ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = (-3 + 4)
33		4cn 12213	. . . . . . . 8 ⊢ 4 ∈ ℂ
34	33, 5	negsubi 11442	. . . . . . . . 9 ⊢ (4 + -3) = (4 − 3)
35		3p1e4 12268	. . . . . . . . . 10 ⊢ (3 + 1) = 4
36	33, 5, 1, 35	subaddrii 11453	. . . . . . . . 9 ⊢ (4 − 3) = 1
37	34, 36	eqtri 2752	. . . . . . . 8 ⊢ (4 + -3) = 1
38	33, 6, 37	addcomli 11308	. . . . . . 7 ⊢ (-3 + 4) = 1
39	32, 38	eqtri 2752	. . . . . 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 6822	. . . . . . . 8 ⊢ (𝑝 = ⟨1, 0⟩ → (1st ‘𝑝) = (1st ‘⟨1, 0⟩))
42		fveq2 6822	. . . . . . . . 9 ⊢ (𝑝 = ⟨1, 0⟩ → (2nd ‘𝑝) = (2nd ‘⟨1, 0⟩))
43	42	oveq1d 7364	. . . . . . . 8 ⊢ (𝑝 = ⟨1, 0⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨1, 0⟩)↑2))
44	41, 43	oveq12d 7367	. . . . . . 7 ⊢ (𝑝 = ⟨1, 0⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)))
45	44	eqeq1d 2731	. . . . . 6 ⊢ (𝑝 = ⟨1, 0⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1 ↔ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1))
46		fveq2 6822	. . . . . . . 8 ⊢ (𝑝 = ⟨-3, 2⟩ → (1st ‘𝑝) = (1st ‘⟨-3, 2⟩))
47		fveq2 6822	. . . . . . . . 9 ⊢ (𝑝 = ⟨-3, 2⟩ → (2nd ‘𝑝) = (2nd ‘⟨-3, 2⟩))
48	47	oveq1d 7364	. . . . . . . 8 ⊢ (𝑝 = ⟨-3, 2⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨-3, 2⟩)↑2))
49	46, 48	oveq12d 7367	. . . . . . 7 ⊢ (𝑝 = ⟨-3, 2⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)))
50	49	eqeq1d 2731	. . . . . 6 ⊢ (𝑝 = ⟨-3, 2⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1 ↔ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1))
51	45, 50	2nreu 4395	. . . . 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 2741	. . . . 5 ⊢ (𝐶 = 1 → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1))
54	53	reubidv 3361	. . . 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 11108	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 0 ∈ ℂ)
59	57, 58	opelxpd 5658	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
60	59	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
61		1cnd 11110	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ)
62		peano2cnm 11430	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ)
63	62	sqrtcld 15347	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (√‘(𝐶 − 1)) ∈ ℂ)
64	61, 63	opelxpd 5658	. . . . . 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 11108	. . . . . . 7 ⊢ (𝐶 ≠ 1 → 0 ∈ ℂ)
68		opthneg 5424	. . . . . . 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 1128	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ) ∧ ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩))
72		op1stg 7936	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (1st ‘⟨𝐶, 0⟩) = 𝐶)
73	58, 72	mpdan 687	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (1st ‘⟨𝐶, 0⟩) = 𝐶)
74		op2ndg 7937	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (2nd ‘⟨𝐶, 0⟩) = 0)
75	58, 74	mpdan 687	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (2nd ‘⟨𝐶, 0⟩) = 0)
76	75	sq0id 14101	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨𝐶, 0⟩)↑2) = 0)
77	73, 76	oveq12d 7367	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = (𝐶 + 0))
78		addrid 11296	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (𝐶 + 0) = 𝐶)
79	77, 78	eqtrd 2764	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶)
80		op1stg 7936	. . . . . . . . 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 7937	. . . . . . . . . . 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 7364	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = ((√‘(𝐶 − 1))↑2))
85	62	sqsqrtd 15349	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → ((√‘(𝐶 − 1))↑2) = (𝐶 − 1))
86	84, 85	eqtrd 2764	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = (𝐶 − 1))
87	81, 86	oveq12d 7367	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = (1 + (𝐶 − 1)))
88	61, 57	pncan3d 11478	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (1 + (𝐶 − 1)) = 𝐶)
89	87, 88	eqtrd 2764	. . . . . 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 6822	. . . . . . 7 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (1st ‘𝑝) = (1st ‘⟨𝐶, 0⟩))
93		fveq2 6822	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝐶, 0⟩))
94	93	oveq1d 7364	. . . . . . 7 ⊢ (𝑝 = ⟨𝐶, 0⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨𝐶, 0⟩)↑2))
95	92, 94	oveq12d 7367	. . . . . 6 ⊢ (𝑝 = ⟨𝐶, 0⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)))
96	95	eqeq1d 2731	. . . . 5 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶))
97		fveq2 6822	. . . . . . 7 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (1st ‘𝑝) = (1st ‘⟨1, (√‘(𝐶 − 1))⟩))
98		fveq2 6822	. . . . . . . 8 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (2nd ‘𝑝) = (2nd ‘⟨1, (√‘(𝐶 − 1))⟩))
99	98	oveq1d 7364	. . . . . . 7 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2))
100	97, 99	oveq12d 7367	. . . . . 6 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)))
101	100	eqeq1d 2731	. . . . 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 4395	. . . 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 3008	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 1540   ∈ wcel 2109   ≠ wne 2925  ∃!wreu 3341  ⟨cop 4583   × cxp 5617  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  ℂcc 11007  0cc0 11009  1c1 11010   + caddc 11012   − cmin 11347  -cneg 11348  2c2 12183  3c3 12184  4c4 12185  ↑cexp 13968  √csqrt 15140

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-sup 9332  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143

This theorem is referenced by: (None)