Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  addsqnreup Structured version   Visualization version   GIF version

 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 26128, 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 2588 and 2eu4 2675. 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 26128). 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 26130). 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 4422 resp. 2eu4 2675). These two representations are equivalent (see opreu2reurex 6127). An analogon of this theorem using the latter variant is given in addsqn2reurex2 26133. 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 26149 and 2sqreuopb 26156). Alternatively, (proper) unordered pairs can be used: ∃!𝑝𝑒𝒫 𝐴((♯‘𝑝) = 2 ∧ 𝜑), or, using the definition of proper pairs: ∃!𝑝 ∈ (Pairsproper‘𝐴)𝜑 (see, for example, inlinecirc02preu 45595). (Contributed by AV, 21-Jun-2023.)
Assertion
Ref Expression
addsqnreup (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶)
Distinct variable group:   𝐶,𝑝

StepHypRef Expression
1 ax-1cn 10638 . . . . . . 7 1 ∈ ℂ
2 0cn 10676 . . . . . . 7 0 ∈ ℂ
3 opelxpi 5564 . . . . . . 7 ((1 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨1, 0⟩ ∈ (ℂ × ℂ))
41, 2, 3mp2an 691 . . . . . 6 ⟨1, 0⟩ ∈ (ℂ × ℂ)
5 3cn 11760 . . . . . . . 8 3 ∈ ℂ
65negcli 10997 . . . . . . 7 -3 ∈ ℂ
7 2cn 11754 . . . . . . 7 2 ∈ ℂ
8 opelxpi 5564 . . . . . . 7 ((-3 ∈ ℂ ∧ 2 ∈ ℂ) → ⟨-3, 2⟩ ∈ (ℂ × ℂ))
96, 7, 8mp2an 691 . . . . . 6 ⟨-3, 2⟩ ∈ (ℂ × ℂ)
10 0ne2 11886 . . . . . . . 8 0 ≠ 2
1110olci 863 . . . . . . 7 (1 ≠ -3 ∨ 0 ≠ 2)
12 1ex 10680 . . . . . . . 8 1 ∈ V
13 c0ex 10678 . . . . . . . 8 0 ∈ V
1412, 13opthne 5345 . . . . . . 7 (⟨1, 0⟩ ≠ ⟨-3, 2⟩ ↔ (1 ≠ -3 ∨ 0 ≠ 2))
1511, 14mpbir 234 . . . . . 6 ⟨1, 0⟩ ≠ ⟨-3, 2⟩
164, 9, 153pm3.2i 1336 . . . . 5 (⟨1, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨-3, 2⟩ ∈ (ℂ × ℂ) ∧ ⟨1, 0⟩ ≠ ⟨-3, 2⟩)
1712, 13op1st 7706 . . . . . . . 8 (1st ‘⟨1, 0⟩) = 1
1812, 13op2nd 7707 . . . . . . . . . 10 (2nd ‘⟨1, 0⟩) = 0
1918oveq1i 7165 . . . . . . . . 9 ((2nd ‘⟨1, 0⟩)↑2) = (0↑2)
20 sq0 13610 . . . . . . . . 9 (0↑2) = 0
2119, 20eqtri 2781 . . . . . . . 8 ((2nd ‘⟨1, 0⟩)↑2) = 0
2217, 21oveq12i 7167 . . . . . . 7 ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = (1 + 0)
23 1p0e1 11803 . . . . . . 7 (1 + 0) = 1
2422, 23eqtri 2781 . . . . . 6 ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1
25 negex 10927 . . . . . . . . 9 -3 ∈ V
26 2ex 11756 . . . . . . . . 9 2 ∈ V
2725, 26op1st 7706 . . . . . . . 8 (1st ‘⟨-3, 2⟩) = -3
2825, 26op2nd 7707 . . . . . . . . . 10 (2nd ‘⟨-3, 2⟩) = 2
2928oveq1i 7165 . . . . . . . . 9 ((2nd ‘⟨-3, 2⟩)↑2) = (2↑2)
30 sq2 13615 . . . . . . . . 9 (2↑2) = 4
3129, 30eqtri 2781 . . . . . . . 8 ((2nd ‘⟨-3, 2⟩)↑2) = 4
3227, 31oveq12i 7167 . . . . . . 7 ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = (-3 + 4)
33 4cn 11764 . . . . . . . 8 4 ∈ ℂ
3433, 5negsubi 11007 . . . . . . . . 9 (4 + -3) = (4 − 3)
35 3p1e4 11824 . . . . . . . . . 10 (3 + 1) = 4
3633, 5, 1, 35subaddrii 11018 . . . . . . . . 9 (4 − 3) = 1
3734, 36eqtri 2781 . . . . . . . 8 (4 + -3) = 1
3833, 6, 37addcomli 10875 . . . . . . 7 (-3 + 4) = 1
3932, 38eqtri 2781 . . . . . 6 ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1
4024, 39pm3.2i 474 . . . . 5 (((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1 ∧ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1)
41 fveq2 6662 . . . . . . . 8 (𝑝 = ⟨1, 0⟩ → (1st𝑝) = (1st ‘⟨1, 0⟩))
42 fveq2 6662 . . . . . . . . 9 (𝑝 = ⟨1, 0⟩ → (2nd𝑝) = (2nd ‘⟨1, 0⟩))
4342oveq1d 7170 . . . . . . . 8 (𝑝 = ⟨1, 0⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨1, 0⟩)↑2))
4441, 43oveq12d 7173 . . . . . . 7 (𝑝 = ⟨1, 0⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)))
4544eqeq1d 2760 . . . . . 6 (𝑝 = ⟨1, 0⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 1 ↔ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1))
46 fveq2 6662 . . . . . . . 8 (𝑝 = ⟨-3, 2⟩ → (1st𝑝) = (1st ‘⟨-3, 2⟩))
47 fveq2 6662 . . . . . . . . 9 (𝑝 = ⟨-3, 2⟩ → (2nd𝑝) = (2nd ‘⟨-3, 2⟩))
4847oveq1d 7170 . . . . . . . 8 (𝑝 = ⟨-3, 2⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨-3, 2⟩)↑2))
4946, 48oveq12d 7173 . . . . . . 7 (𝑝 = ⟨-3, 2⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)))
5049eqeq1d 2760 . . . . . 6 (𝑝 = ⟨-3, 2⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 1 ↔ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1))
5145, 502nreu 4341 . . . . 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)((1st𝑝) + ((2nd𝑝)↑2)) = 1))
5216, 40, 51mp2 9 . . . 4 ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 1
53 eqeq2 2770 . . . . 5 (𝐶 = 1 → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st𝑝) + ((2nd𝑝)↑2)) = 1))
5453reubidv 3307 . . . 4 (𝐶 = 1 → (∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 1))
5552, 54mtbiri 330 . . 3 (𝐶 = 1 → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶)
5655a1d 25 . 2 (𝐶 = 1 → (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶))
57 id 22 . . . . . . 7 (𝐶 ∈ ℂ → 𝐶 ∈ ℂ)
58 0cnd 10677 . . . . . . 7 (𝐶 ∈ ℂ → 0 ∈ ℂ)
5957, 58opelxpd 5565 . . . . . 6 (𝐶 ∈ ℂ → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
6059adantr 484 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
61 1cnd 10679 . . . . . . 7 (𝐶 ∈ ℂ → 1 ∈ ℂ)
62 peano2cnm 10995 . . . . . . . 8 (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ)
6362sqrtcld 14850 . . . . . . 7 (𝐶 ∈ ℂ → (√‘(𝐶 − 1)) ∈ ℂ)
6461, 63opelxpd 5565 . . . . . 6 (𝐶 ∈ ℂ → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
6564adantr 484 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
66 animorrl 978 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1))))
67 0cnd 10677 . . . . . . 7 (𝐶 ≠ 1 → 0 ∈ ℂ)
68 opthneg 5344 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
6967, 68sylan2 595 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
7066, 69mpbird 260 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩)
7160, 65, 703jca 1125 . . . 4 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ) ∧ ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩))
72 op1stg 7710 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (1st ‘⟨𝐶, 0⟩) = 𝐶)
7358, 72mpdan 686 . . . . . . . 8 (𝐶 ∈ ℂ → (1st ‘⟨𝐶, 0⟩) = 𝐶)
74 op2ndg 7711 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (2nd ‘⟨𝐶, 0⟩) = 0)
7558, 74mpdan 686 . . . . . . . . 9 (𝐶 ∈ ℂ → (2nd ‘⟨𝐶, 0⟩) = 0)
7675sq0id 13612 . . . . . . . 8 (𝐶 ∈ ℂ → ((2nd ‘⟨𝐶, 0⟩)↑2) = 0)
7773, 76oveq12d 7173 . . . . . . 7 (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = (𝐶 + 0))
78 addid1 10863 . . . . . . 7 (𝐶 ∈ ℂ → (𝐶 + 0) = 𝐶)
7977, 78eqtrd 2793 . . . . . 6 (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶)
80 op1stg 7710 . . . . . . . . 9 ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
8161, 63, 80syl2anc 587 . . . . . . . 8 (𝐶 ∈ ℂ → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
82 op2ndg 7711 . . . . . . . . . . 11 ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
8361, 63, 82syl2anc 587 . . . . . . . . . 10 (𝐶 ∈ ℂ → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
8483oveq1d 7170 . . . . . . . . 9 (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = ((√‘(𝐶 − 1))↑2))
8562sqsqrtd 14852 . . . . . . . . 9 (𝐶 ∈ ℂ → ((√‘(𝐶 − 1))↑2) = (𝐶 − 1))
8684, 85eqtrd 2793 . . . . . . . 8 (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = (𝐶 − 1))
8781, 86oveq12d 7173 . . . . . . 7 (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = (1 + (𝐶 − 1)))
8861, 57pncan3d 11043 . . . . . . 7 (𝐶 ∈ ℂ → (1 + (𝐶 − 1)) = 𝐶)
8987, 88eqtrd 2793 . . . . . 6 (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶)
9079, 89jca 515 . . . . 5 (𝐶 ∈ ℂ → (((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶 ∧ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
9190adantr 484 . . . 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 6662 . . . . . . 7 (𝑝 = ⟨𝐶, 0⟩ → (1st𝑝) = (1st ‘⟨𝐶, 0⟩))
93 fveq2 6662 . . . . . . . 8 (𝑝 = ⟨𝐶, 0⟩ → (2nd𝑝) = (2nd ‘⟨𝐶, 0⟩))
9493oveq1d 7170 . . . . . . 7 (𝑝 = ⟨𝐶, 0⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨𝐶, 0⟩)↑2))
9592, 94oveq12d 7173 . . . . . 6 (𝑝 = ⟨𝐶, 0⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)))
9695eqeq1d 2760 . . . . 5 (𝑝 = ⟨𝐶, 0⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶))
97 fveq2 6662 . . . . . . 7 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (1st𝑝) = (1st ‘⟨1, (√‘(𝐶 − 1))⟩))
98 fveq2 6662 . . . . . . . 8 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (2nd𝑝) = (2nd ‘⟨1, (√‘(𝐶 − 1))⟩))
9998oveq1d 7170 . . . . . . 7 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2))
10097, 99oveq12d 7173 . . . . . 6 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)))
101100eqeq1d 2760 . . . . 5 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
10296, 1012nreu 4341 . . . 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)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶))
10371, 91, 102sylc 65 . . 3 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶)
104103expcom 417 . 2 (𝐶 ≠ 1 → (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶))
10556, 104pm2.61ine 3034 1 (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∃!wreu 3072  ⟨cop 4531   × cxp 5525  ‘cfv 6339  (class class class)co 7155  1st c1st 7696  2nd c2nd 7697  ℂcc 10578  0cc0 10580  1c1 10581   + caddc 10583   − cmin 10913  -cneg 10914  2c2 11734  3c3 11735  4c4 11736  ↑cexp 13484  √csqrt 14645 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657  ax-pre-sup 10658 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-er 8304  df-en 8533  df-dom 8534  df-sdom 8535  df-sup 8944  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-div 11341  df-nn 11680  df-2 11742  df-3 11743  df-4 11744  df-n0 11940  df-z 12026  df-uz 12288  df-rp 12436  df-seq 13424  df-exp 13485  df-cj 14511  df-re 14512  df-im 14513  df-sqrt 14647  df-abs 14648 This theorem is referenced by: (None)
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