Theorem addsqnreup 27427

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 27424, 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 2570 and 2eu4 2656.

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 27424). 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 27426). 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 4479 resp. 2eu4 2656). These two representations are equivalent (see opreu2reurex 6262). An analogon of this theorem using the latter variant is given in addsqn2reurex2 27429. 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 27445 and 2sqreuopb 27452). 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 49177). (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 11098	. . . . . . 7 ⊢ 1 ∈ ℂ
2		0cn 11138	. . . . . . 7 ⊢ 0 ∈ ℂ
3		opelxpi 5671	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨1, 0⟩ ∈ (ℂ × ℂ))
4	1, 2, 3	mp2an 693	. . . . . 6 ⊢ ⟨1, 0⟩ ∈ (ℂ × ℂ)
5		3cn 12240	. . . . . . . 8 ⊢ 3 ∈ ℂ
6	5	negcli 11463	. . . . . . 7 ⊢ -3 ∈ ℂ
7		2cn 12234	. . . . . . 7 ⊢ 2 ∈ ℂ
8		opelxpi 5671	. . . . . . 7 ⊢ ((-3 ∈ ℂ ∧ 2 ∈ ℂ) → ⟨-3, 2⟩ ∈ (ℂ × ℂ))
9	6, 7, 8	mp2an 693	. . . . . 6 ⊢ ⟨-3, 2⟩ ∈ (ℂ × ℂ)
10		0ne2 12361	. . . . . . . 8 ⊢ 0 ≠ 2
11	10	olci 867	. . . . . . 7 ⊢ (1 ≠ -3 ∨ 0 ≠ 2)
12		1ex 11142	. . . . . . . 8 ⊢ 1 ∈ V
13		c0ex 11140	. . . . . . . 8 ⊢ 0 ∈ V
14	12, 13	opthne 5440	. . . . . . 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 1341	. . . . 5 ⊢ (⟨1, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨-3, 2⟩ ∈ (ℂ × ℂ) ∧ ⟨1, 0⟩ ≠ ⟨-3, 2⟩)
17	12, 13	op1st 7953	. . . . . . . 8 ⊢ (1st ‘⟨1, 0⟩) = 1
18	12, 13	op2nd 7954	. . . . . . . . . 10 ⊢ (2nd ‘⟨1, 0⟩) = 0
19	18	oveq1i 7380	. . . . . . . . 9 ⊢ ((2nd ‘⟨1, 0⟩)↑2) = (0↑2)
20		sq0 14129	. . . . . . . . 9 ⊢ (0↑2) = 0
21	19, 20	eqtri 2760	. . . . . . . 8 ⊢ ((2nd ‘⟨1, 0⟩)↑2) = 0
22	17, 21	oveq12i 7382	. . . . . . 7 ⊢ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = (1 + 0)
23		1p0e1 12278	. . . . . . 7 ⊢ (1 + 0) = 1
24	22, 23	eqtri 2760	. . . . . 6 ⊢ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1
25		negex 11392	. . . . . . . . 9 ⊢ -3 ∈ V
26		2ex 12236	. . . . . . . . 9 ⊢ 2 ∈ V
27	25, 26	op1st 7953	. . . . . . . 8 ⊢ (1st ‘⟨-3, 2⟩) = -3
28	25, 26	op2nd 7954	. . . . . . . . . 10 ⊢ (2nd ‘⟨-3, 2⟩) = 2
29	28	oveq1i 7380	. . . . . . . . 9 ⊢ ((2nd ‘⟨-3, 2⟩)↑2) = (2↑2)
30		sq2 14134	. . . . . . . . 9 ⊢ (2↑2) = 4
31	29, 30	eqtri 2760	. . . . . . . 8 ⊢ ((2nd ‘⟨-3, 2⟩)↑2) = 4
32	27, 31	oveq12i 7382	. . . . . . 7 ⊢ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = (-3 + 4)
33		4cn 12244	. . . . . . . 8 ⊢ 4 ∈ ℂ
34	33, 5	negsubi 11473	. . . . . . . . 9 ⊢ (4 + -3) = (4 − 3)
35		3p1e4 12299	. . . . . . . . . 10 ⊢ (3 + 1) = 4
36	33, 5, 1, 35	subaddrii 11484	. . . . . . . . 9 ⊢ (4 − 3) = 1
37	34, 36	eqtri 2760	. . . . . . . 8 ⊢ (4 + -3) = 1
38	33, 6, 37	addcomli 11339	. . . . . . 7 ⊢ (-3 + 4) = 1
39	32, 38	eqtri 2760	. . . . . 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 6844	. . . . . . . 8 ⊢ (𝑝 = ⟨1, 0⟩ → (1st ‘𝑝) = (1st ‘⟨1, 0⟩))
42		fveq2 6844	. . . . . . . . 9 ⊢ (𝑝 = ⟨1, 0⟩ → (2nd ‘𝑝) = (2nd ‘⟨1, 0⟩))
43	42	oveq1d 7385	. . . . . . . 8 ⊢ (𝑝 = ⟨1, 0⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨1, 0⟩)↑2))
44	41, 43	oveq12d 7388	. . . . . . 7 ⊢ (𝑝 = ⟨1, 0⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)))
45	44	eqeq1d 2739	. . . . . 6 ⊢ (𝑝 = ⟨1, 0⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1 ↔ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1))
46		fveq2 6844	. . . . . . . 8 ⊢ (𝑝 = ⟨-3, 2⟩ → (1st ‘𝑝) = (1st ‘⟨-3, 2⟩))
47		fveq2 6844	. . . . . . . . 9 ⊢ (𝑝 = ⟨-3, 2⟩ → (2nd ‘𝑝) = (2nd ‘⟨-3, 2⟩))
48	47	oveq1d 7385	. . . . . . . 8 ⊢ (𝑝 = ⟨-3, 2⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨-3, 2⟩)↑2))
49	46, 48	oveq12d 7388	. . . . . . 7 ⊢ (𝑝 = ⟨-3, 2⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)))
50	49	eqeq1d 2739	. . . . . 6 ⊢ (𝑝 = ⟨-3, 2⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1 ↔ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1))
51	45, 50	2nreu 4398	. . . . 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 2749	. . . . 5 ⊢ (𝐶 = 1 → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 1))
54	53	reubidv 3368	. . . 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 11139	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 0 ∈ ℂ)
59	57, 58	opelxpd 5673	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
60	59	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
61		1cnd 11141	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ)
62		peano2cnm 11461	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ)
63	62	sqrtcld 15377	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (√‘(𝐶 − 1)) ∈ ℂ)
64	61, 63	opelxpd 5673	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
65	64	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
66		animorrl 983	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1))))
67		0cnd 11139	. . . . . . 7 ⊢ (𝐶 ≠ 1 → 0 ∈ ℂ)
68		opthneg 5439	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
69	67, 68	sylan2 594	. . . . . 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 1129	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ) ∧ ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩))
72		op1stg 7957	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (1st ‘⟨𝐶, 0⟩) = 𝐶)
73	58, 72	mpdan 688	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (1st ‘⟨𝐶, 0⟩) = 𝐶)
74		op2ndg 7958	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (2nd ‘⟨𝐶, 0⟩) = 0)
75	58, 74	mpdan 688	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (2nd ‘⟨𝐶, 0⟩) = 0)
76	75	sq0id 14131	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨𝐶, 0⟩)↑2) = 0)
77	73, 76	oveq12d 7388	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = (𝐶 + 0))
78		addrid 11327	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (𝐶 + 0) = 𝐶)
79	77, 78	eqtrd 2772	. . . . . 6 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶)
80		op1stg 7957	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
81	61, 63, 80	syl2anc 585	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
82		op2ndg 7958	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
83	61, 63, 82	syl2anc 585	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℂ → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
84	83	oveq1d 7385	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = ((√‘(𝐶 − 1))↑2))
85	62	sqsqrtd 15379	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → ((√‘(𝐶 − 1))↑2) = (𝐶 − 1))
86	84, 85	eqtrd 2772	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = (𝐶 − 1))
87	81, 86	oveq12d 7388	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = (1 + (𝐶 − 1)))
88	61, 57	pncan3d 11509	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (1 + (𝐶 − 1)) = 𝐶)
89	87, 88	eqtrd 2772	. . . . . 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 6844	. . . . . . 7 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (1st ‘𝑝) = (1st ‘⟨𝐶, 0⟩))
93		fveq2 6844	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝐶, 0⟩))
94	93	oveq1d 7385	. . . . . . 7 ⊢ (𝑝 = ⟨𝐶, 0⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨𝐶, 0⟩)↑2))
95	92, 94	oveq12d 7388	. . . . . 6 ⊢ (𝑝 = ⟨𝐶, 0⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)))
96	95	eqeq1d 2739	. . . . 5 ⊢ (𝑝 = ⟨𝐶, 0⟩ → (((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶))
97		fveq2 6844	. . . . . . 7 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (1st ‘𝑝) = (1st ‘⟨1, (√‘(𝐶 − 1))⟩))
98		fveq2 6844	. . . . . . . 8 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (2nd ‘𝑝) = (2nd ‘⟨1, (√‘(𝐶 − 1))⟩))
99	98	oveq1d 7385	. . . . . . 7 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((2nd ‘𝑝)↑2) = ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2))
100	97, 99	oveq12d 7388	. . . . . 6 ⊢ (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)))
101	100	eqeq1d 2739	. . . . 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 4398	. . . 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 3016	1 ⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st ‘𝑝) + ((2nd ‘𝑝)↑2)) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃!wreu 3350  ⟨cop 4588   × cxp 5632  ‘cfv 6502  (class class class)co 7370  1^st c1st 7943  2^nd c2nd 7944  ℂcc 11038  0cc0 11040  1c1 11041   + caddc 11043   − cmin 11378  -cneg 11379  2c2 12214  3c3 12215  4c4 12216  ↑cexp 13998  √csqrt 15170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117  ax-pre-sup 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-sup 9359  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-div 11809  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-n0 12416  df-z 12503  df-uz 12766  df-rp 12920  df-seq 13939  df-exp 13999  df-cj 15036  df-re 15037  df-im 15038  df-sqrt 15172  df-abs 15173

This theorem is referenced by: (None)