Theorem addsqnreup 27607

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 27604, 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 2597 and 2eu4 2682.

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 27604). 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 27606). 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 4485 resp. 2eu4 2682). These two representations are equivalent (see opreu2reurex 6295). An analogon of this theorem using the latter variant is given in addsqn2reurex2 27609. 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 27625 and 2sqreuopb 27632). 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 49588). (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 11153	. . . . . . 7 ⊢ 1 ∈ ℂ
2		0cn 11193	. . . . . . 7 ⊢ 0 ∈ ℂ
3		opelxpi 5698	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨1, 0⟩ ∈ (ℂ × ℂ))
4	1, 2, 3	mp2an 704	. . . . . 6 ⊢ ⟨1, 0⟩ ∈ (ℂ × ℂ)
5		3cn 12317	. . . . . . . 8 ⊢ 3 ∈ ℂ
6	5	negcli 11521	. . . . . . 7 ⊢ -3 ∈ ℂ
7		2cn 12311	. . . . . . 7 ⊢ 2 ∈ ℂ
8		opelxpi 5698	. . . . . . 7 ⊢ ((-3 ∈ ℂ ∧ 2 ∈ ℂ) → ⟨-3, 2⟩ ∈ (ℂ × ℂ))
9	6, 7, 8	mp2an 704	. . . . . 6 ⊢ ⟨-3, 2⟩ ∈ (ℂ × ℂ)
10		0ne2 12445	. . . . . . . 8 ⊢ 0 ≠ 2
11	10	olci 879	. . . . . . 7 ⊢ (1 ≠ -3 ∨ 0 ≠ 2)
12		1ex 11198	. . . . . . . 8 ⊢ 1 ∈ V
13		c0ex 11195	. . . . . . . 8 ⊢ 0 ∈ V
14	12, 13	opthne 5464	. . . . . . 7 ⊢ (⟨1, 0⟩ ≠ ⟨-3, 2⟩ ↔ (1 ≠ -3 ∨ 0 ≠ 2))
15	11, 14	mpbir 234	. . . . . 6 ⊢ ⟨1, 0⟩ ≠ ⟨-3, 2⟩
16	4, 9, 15	3pm3.2i 1358	. . . . 5 ⊢ (⟨1, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨-3, 2⟩ ∈ (ℂ × ℂ) ∧ ⟨1, 0⟩ ≠ ⟨-3, 2⟩)
17	12, 13	op1st 7990	. . . . . . . 8 ⊢ (1st ‘⟨1, 0⟩) = 1
18	12, 13	op2nd 7991	. . . . . . . . . 10 ⊢ (2nd ‘⟨1, 0⟩) = 0
19	18	oveq1i 7420	. . . . . . . . 9 ⊢ ((2nd ‘⟨1, 0⟩)↑2) = (0↑2)
20		sq0 14224	. . . . . . . . 9 ⊢ (0↑2) = 0
21	19, 20	eqtri 2786	. . . . . . . 8 ⊢ ((2nd ‘⟨1, 0⟩)↑2) = 0
22	17, 21	oveq12i 7422	. . . . . . 7 ⊢ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = (1 + 0)
23		1p0e1 12358	. . . . . . 7 ⊢ (1 + 0) = 1
24	22, 23	eqtri 2786	. . . . . 6 ⊢ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1
25		negex 11450	. . . . . . . . 9 ⊢ -3 ∈ V
26		2ex 12313	. . . . . . . . 9 ⊢ 2 ∈ V
27	25, 26	op1st 7990	. . . . . . . 8 ⊢ (1st ‘⟨-3, 2⟩) = -3
28	25, 26	op2nd 7991	. . . . . . . . . 10 ⊢ (2nd ‘⟨-3, 2⟩) = 2
29	28	oveq1i 7420	. . . . . . . . 9 ⊢ ((2nd ‘⟨-3, 2⟩)↑2) = (2↑2)
30		sq2 14229	. . . . . . . . 9 ⊢ (2↑2) = 4
31	29, 30	eqtri 2786	. . . . . . . 8 ⊢ ((2nd ‘⟨-3, 2⟩)↑2) = 4
32	27, 31	oveq12i 7422	. . . . . . 7 ⊢ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = (-3 + 4)
33		4cn 12321	. . . . . . . 8 ⊢ 4 ∈ ℂ
34	33, 5	negsubi 11531	. . . . . . . . 9 ⊢ (4 + -3) = (4 − 3)
35		3p1e4 12380	. . . . . . . . . 10 ⊢ (3 + 1) = 4
36	33, 5, 1, 35	subaddrii 11542	. . . . . . . . 9 ⊢ (4 − 3) = 1
37	34, 36	eqtri 2786	. . . . . . . 8 ⊢ (4 + -3) = 1
38	33, 6, 37	addcomli 11397	. . . . . . 7 ⊢ (-3 + 4) = 1
39	32, 38	eqtri 2786	. . . . . 6 ⊢ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1
40	24, 39	pm3.2i 475	. . . . 5 ⊢ (((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1 ∧ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1)
41		fveq2 6881	. . . . . . . 8 ⊢ (𝑝 = ⟨1, 0⟩ → (1st ‘𝑝) = (1st ‘⟨1, 0⟩))
42		fveq2 6881	. . . . . . . . 9 ⊢ (𝑝 = ⟨1, 0⟩ → (2nd ‘𝑝) = (2nd ‘⟨1, 0⟩))
43	42	oveq1d 7425	. . . . . . . 8 ⊢ (𝑝 = ⟨1, 0⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨1, 0⟩)↑2))
44	41, 43	oveq12d 7428	. . . . . . 7 ⊢ (𝑝 = ⟨1, 0⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)))
45	44	eqeq1d 2765	. . . . . 6 ⊢ (𝑝 = ⟨1, 0⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1 ↔ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1))
46		fveq2 6881	. . . . . . . 8 ⊢ (𝑝 = ⟨-3, 2⟩ → (1st ‘𝑝) = (1st ‘⟨-3, 2⟩))
47		fveq2 6881	. . . . . . . . 9 ⊢ (𝑝 = ⟨-3, 2⟩ → (2nd ‘𝑝) = (2nd ‘⟨-3, 2⟩))
48	47	oveq1d 7425	. . . . . . . 8 ⊢ (𝑝 = ⟨-3, 2⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨-3, 2⟩)↑2))
49	46, 48	oveq12d 7428	. . . . . . 7 ⊢ (𝑝 = ⟨-3, 2⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)))
50	49	eqeq1d 2765	. . . . . 6 ⊢ (𝑝 = ⟨-3, 2⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1 ↔ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1))
51	45, 50	2nreu 4409	. . . . 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 2775	. . . . 5 ⊢ (𝐶 = 1 → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1))
54	53	reubidv 3385	. . . 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 330	. . 3 ⊢ (𝐶 = 1 → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶)
56	55	a1d 26	. 2 ⊢ (𝐶 = 1 → (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶))
57		id 23	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 𝐶 ∈ ℂ)
58		0cnd 11194	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 0 ∈ ℂ)
59	57, 58	opelxpd 5700	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
60	59	adantr 485	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
61		1cnd 11197	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ)
62		peano2cnm 11519	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ)
63	62	sqrtcld 15487	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (√‘(𝐶 − 1)) ∈ ℂ)
64	61, 63	opelxpd 5700	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
65	64	adantr 485	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
66		animorrl 996	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1))))
67		0cnd 11194	. . . . . . 7 ⊢ (𝐶 ≠ 1 → 0 ∈ ℂ)
68		opthneg 5463	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
69	67, 68	sylan2 604	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
70	66, 69	mpbird 260	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩)
71	60, 65, 70	3jca 1146	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ) ∧ ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩))
72		op1stg 7994	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (1st ‘⟨𝐶, 0⟩) = 𝐶)
73	58, 72	mpdan 699	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (1st ‘⟨𝐶, 0⟩) = 𝐶)
74		op2ndg 7995	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (2nd ‘⟨𝐶, 0⟩) = 0)
75	58, 74	mpdan 699	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (2nd ‘⟨𝐶, 0⟩) = 0)
76	75	sq0id 14226	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨𝐶, 0⟩)↑2) = 0)
77	73, 76	oveq12d 7428	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = (𝐶 + 0))
78		addrid 11385	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (𝐶 + 0) = 𝐶)
79	77, 78	eqtrd 2798	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶)
80		op1stg 7994	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
81	61, 63, 80	syl2anc 595	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
82		op2ndg 7995	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
83	61, 63, 82	syl2anc 595	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℂ → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
84	83	oveq1d 7425	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = ((√‘(𝐶 − 1))↑2))
85	62	sqsqrtd 15489	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → ((√‘(𝐶 − 1))↑2) = (𝐶 − 1))
86	84, 85	eqtrd 2798	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = (𝐶 − 1))
87	81, 86	oveq12d 7428	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = (1 + (𝐶 − 1)))
88	61, 57	pncan3d 11567	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (1 + (𝐶 − 1)) = 𝐶)
89	87, 88	eqtrd 2798	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶)
90	79, 89	jca 520	. . . . 5 ⊢ (𝐶 ∈ ℂ → (((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶 ∧ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
91	90	adantr 485	. . . 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 6881	. . . . . . 7 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (1st ‘𝑝) = (1st ‘⟨𝐶, 0⟩))
93		fveq2 6881	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝐶, 0⟩))
94	93	oveq1d 7425	. . . . . . 7 ⊢ (𝑝 = ⟨𝐶, 0⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨𝐶, 0⟩)↑2))
95	92, 94	oveq12d 7428	. . . . . 6 ⊢ (𝑝 = ⟨𝐶, 0⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)))
96	95	eqeq1d 2765	. . . . 5 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶))
97		fveq2 6881	. . . . . . 7 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (1st ‘𝑝) = (1st ‘⟨1, (√‘(𝐶 − 1))⟩))
98		fveq2 6881	. . . . . . . 8 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (2nd ‘𝑝) = (2nd ‘⟨1, (√‘(𝐶 − 1))⟩))
99	98	oveq1d 7425	. . . . . . 7 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2))
100	97, 99	oveq12d 7428	. . . . . 6 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)))
101	100	eqeq1d 2765	. . . . 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 4409	. . . 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 66	. . 3 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶)
104	103	expcom 418	. 2 ⊢ (𝐶 ≠ 1 → (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶))
105	56, 104	pm2.61ine 3041	1 ⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃!wreu 3367  ⟨cop 4595   × cxp 5659  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  ℂcc 11093  0cc0 11095  1c1 11096   + caddc 11098   − cmin 11436  -cneg 11437  2c2 12290  3c3 12291  4c4 12292  ↑cexp 14093  √csqrt 15280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9398  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-n0 12500  df-z 12587  df-uz 12858  df-rp 13012  df-seq 14034  df-exp 14094  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283

This theorem is referenced by: (None)