Theorem addsqnreup 25706

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 25703, 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 2612 and 2eu4 2711.

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 25703). 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 25705). 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 4384 resp. 2eu4 2711). These two representations are equivalent (see opreu2reurex 6025). An analogon of this theorem using the latter variant is given in addsqn2reurex2 25708. 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 25724 and 2sqreuopb 25731). 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 44283). (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 10446	. . . . . . 7 ⊢ 1 ∈ ℂ
2		0cn 10484	. . . . . . 7 ⊢ 0 ∈ ℂ
3		opelxpi 5485	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨1, 0⟩ ∈ (ℂ × ℂ))
4	1, 2, 3	mp2an 688	. . . . . 6 ⊢ ⟨1, 0⟩ ∈ (ℂ × ℂ)
5		3cn 11571	. . . . . . . 8 ⊢ 3 ∈ ℂ
6	5	negcli 10807	. . . . . . 7 ⊢ -3 ∈ ℂ
7		2cn 11565	. . . . . . 7 ⊢ 2 ∈ ℂ
8		opelxpi 5485	. . . . . . 7 ⊢ ((-3 ∈ ℂ ∧ 2 ∈ ℂ) → ⟨-3, 2⟩ ∈ (ℂ × ℂ))
9	6, 7, 8	mp2an 688	. . . . . 6 ⊢ ⟨-3, 2⟩ ∈ (ℂ × ℂ)
10		0ne2 11697	. . . . . . . 8 ⊢ 0 ≠ 2
11	10	olci 861	. . . . . . 7 ⊢ (1 ≠ -3 ∨ 0 ≠ 2)
12		1ex 10488	. . . . . . . 8 ⊢ 1 ∈ V
13		c0ex 10486	. . . . . . . 8 ⊢ 0 ∈ V
14	12, 13	opthne 5271	. . . . . . 7 ⊢ (⟨1, 0⟩ ≠ ⟨-3, 2⟩ ↔ (1 ≠ -3 ∨ 0 ≠ 2))
15	11, 14	mpbir 232	. . . . . 6 ⊢ ⟨1, 0⟩ ≠ ⟨-3, 2⟩
16	4, 9, 15	3pm3.2i 1332	. . . . 5 ⊢ (⟨1, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨-3, 2⟩ ∈ (ℂ × ℂ) ∧ ⟨1, 0⟩ ≠ ⟨-3, 2⟩)
17	12, 13	op1st 7558	. . . . . . . 8 ⊢ (1st ‘⟨1, 0⟩) = 1
18	12, 13	op2nd 7559	. . . . . . . . . 10 ⊢ (2nd ‘⟨1, 0⟩) = 0
19	18	oveq1i 7031	. . . . . . . . 9 ⊢ ((2nd ‘⟨1, 0⟩)↑2) = (0↑2)
20		sq0 13410	. . . . . . . . 9 ⊢ (0↑2) = 0
21	19, 20	eqtri 2819	. . . . . . . 8 ⊢ ((2nd ‘⟨1, 0⟩)↑2) = 0
22	17, 21	oveq12i 7033	. . . . . . 7 ⊢ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = (1 + 0)
23		1p0e1 11614	. . . . . . 7 ⊢ (1 + 0) = 1
24	22, 23	eqtri 2819	. . . . . 6 ⊢ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1
25		negex 10736	. . . . . . . . 9 ⊢ -3 ∈ V
26		2ex 11567	. . . . . . . . 9 ⊢ 2 ∈ V
27	25, 26	op1st 7558	. . . . . . . 8 ⊢ (1st ‘⟨-3, 2⟩) = -3
28	25, 26	op2nd 7559	. . . . . . . . . 10 ⊢ (2nd ‘⟨-3, 2⟩) = 2
29	28	oveq1i 7031	. . . . . . . . 9 ⊢ ((2nd ‘⟨-3, 2⟩)↑2) = (2↑2)
30		sq2 13415	. . . . . . . . 9 ⊢ (2↑2) = 4
31	29, 30	eqtri 2819	. . . . . . . 8 ⊢ ((2nd ‘⟨-3, 2⟩)↑2) = 4
32	27, 31	oveq12i 7033	. . . . . . 7 ⊢ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = (-3 + 4)
33		4cn 11575	. . . . . . . 8 ⊢ 4 ∈ ℂ
34	33, 5	negsubi 10817	. . . . . . . . 9 ⊢ (4 + -3) = (4 − 3)
35		3p1e4 11635	. . . . . . . . . 10 ⊢ (3 + 1) = 4
36	33, 5, 1, 35	subaddrii 10828	. . . . . . . . 9 ⊢ (4 − 3) = 1
37	34, 36	eqtri 2819	. . . . . . . 8 ⊢ (4 + -3) = 1
38	33, 6, 37	addcomli 10684	. . . . . . 7 ⊢ (-3 + 4) = 1
39	32, 38	eqtri 2819	. . . . . 6 ⊢ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1
40	24, 39	pm3.2i 471	. . . . 5 ⊢ (((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1 ∧ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1)
41		fveq2 6543	. . . . . . . 8 ⊢ (𝑝 = ⟨1, 0⟩ → (1st ‘𝑝) = (1st ‘⟨1, 0⟩))
42		fveq2 6543	. . . . . . . . 9 ⊢ (𝑝 = ⟨1, 0⟩ → (2nd ‘𝑝) = (2nd ‘⟨1, 0⟩))
43	42	oveq1d 7036	. . . . . . . 8 ⊢ (𝑝 = ⟨1, 0⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨1, 0⟩)↑2))
44	41, 43	oveq12d 7039	. . . . . . 7 ⊢ (𝑝 = ⟨1, 0⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)))
45	44	eqeq1d 2797	. . . . . 6 ⊢ (𝑝 = ⟨1, 0⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1 ↔ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1))
46		fveq2 6543	. . . . . . . 8 ⊢ (𝑝 = ⟨-3, 2⟩ → (1st ‘𝑝) = (1st ‘⟨-3, 2⟩))
47		fveq2 6543	. . . . . . . . 9 ⊢ (𝑝 = ⟨-3, 2⟩ → (2nd ‘𝑝) = (2nd ‘⟨-3, 2⟩))
48	47	oveq1d 7036	. . . . . . . 8 ⊢ (𝑝 = ⟨-3, 2⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨-3, 2⟩)↑2))
49	46, 48	oveq12d 7039	. . . . . . 7 ⊢ (𝑝 = ⟨-3, 2⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)))
50	49	eqeq1d 2797	. . . . . 6 ⊢ (𝑝 = ⟨-3, 2⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1 ↔ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1))
51	45, 50	2nreu 4311	. . . . 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 2806	. . . . 5 ⊢ (𝐶 = 1 → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1))
54	53	reubidv 3349	. . . 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 328	. . 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 10485	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 0 ∈ ℂ)
59	57, 58	opelxpd 5486	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
60	59	adantr 481	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
61		1cnd 10487	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ)
62		peano2cnm 10805	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ)
63	62	sqrtcld 14636	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (√‘(𝐶 − 1)) ∈ ℂ)
64	61, 63	opelxpd 5486	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
65	64	adantr 481	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
66		animorrl 975	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1))))
67		0cnd 10485	. . . . . . 7 ⊢ (𝐶 ≠ 1 → 0 ∈ ℂ)
68		opthneg 5270	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
69	67, 68	sylan2 592	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
70	66, 69	mpbird 258	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩)
71	60, 65, 70	3jca 1121	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ) ∧ ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩))
72		op1stg 7562	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (1st ‘⟨𝐶, 0⟩) = 𝐶)
73	58, 72	mpdan 683	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (1st ‘⟨𝐶, 0⟩) = 𝐶)
74		op2ndg 7563	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (2nd ‘⟨𝐶, 0⟩) = 0)
75	58, 74	mpdan 683	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (2nd ‘⟨𝐶, 0⟩) = 0)
76	75	sq0id 13412	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨𝐶, 0⟩)↑2) = 0)
77	73, 76	oveq12d 7039	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = (𝐶 + 0))
78		addid1 10672	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (𝐶 + 0) = 𝐶)
79	77, 78	eqtrd 2831	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶)
80		op1stg 7562	. . . . . . . . 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 7563	. . . . . . . . . . 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 7036	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = ((√‘(𝐶 − 1))↑2))
85	62	sqsqrtd 14638	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → ((√‘(𝐶 − 1))↑2) = (𝐶 − 1))
86	84, 85	eqtrd 2831	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = (𝐶 − 1))
87	81, 86	oveq12d 7039	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = (1 + (𝐶 − 1)))
88	61, 57	pncan3d 10853	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (1 + (𝐶 − 1)) = 𝐶)
89	87, 88	eqtrd 2831	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶)
90	79, 89	jca 512	. . . . 5 ⊢ (𝐶 ∈ ℂ → (((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶 ∧ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
91	90	adantr 481	. . . 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 6543	. . . . . . 7 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (1st ‘𝑝) = (1st ‘⟨𝐶, 0⟩))
93		fveq2 6543	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝐶, 0⟩))
94	93	oveq1d 7036	. . . . . . 7 ⊢ (𝑝 = ⟨𝐶, 0⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨𝐶, 0⟩)↑2))
95	92, 94	oveq12d 7039	. . . . . 6 ⊢ (𝑝 = ⟨𝐶, 0⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)))
96	95	eqeq1d 2797	. . . . 5 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶))
97		fveq2 6543	. . . . . . 7 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (1st ‘𝑝) = (1st ‘⟨1, (√‘(𝐶 − 1))⟩))
98		fveq2 6543	. . . . . . . 8 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (2nd ‘𝑝) = (2nd ‘⟨1, (√‘(𝐶 − 1))⟩))
99	98	oveq1d 7036	. . . . . . 7 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2))
100	97, 99	oveq12d 7039	. . . . . 6 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)))
101	100	eqeq1d 2797	. . . . 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 4311	. . . 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 414	. 2 ⊢ (𝐶 ≠ 1 → (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶))
105	56, 104	pm2.61ine 3068	1 ⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  ∃!wreu 3107  ⟨cop 4482   × cxp 5446  ‘cfv 6230  (class class class)co 7021  1^st c1st 7548  2^nd c2nd 7549  ℂcc 10386  0cc0 10388  1c1 10389   + caddc 10391   − cmin 10722  -cneg 10723  2c2 11545  3c3 11546  4c4 11547  ↑cexp 13284  √csqrt 14431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465  ax-pre-sup 10466

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-er 8144  df-en 8363  df-dom 8364  df-sdom 8365  df-sup 8757  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-div 11151  df-nn 11492  df-2 11553  df-3 11554  df-4 11555  df-n0 11751  df-z 11835  df-uz 12099  df-rp 12245  df-seq 13225  df-exp 13285  df-cj 14297  df-re 14298  df-im 14299  df-sqrt 14433  df-abs 14434

This theorem is referenced by: (None)