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Theorem addsqnreup 26791
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 26788, is a real life example (about a numerical property) showing that the pattern ∃!𝑎𝐴∃!𝑏𝐵𝜑 does not necessarily mean "There are unique sets 𝑎 and 𝑏 fulfilling 𝜑"). See also comments for df-eu 2567 and 2eu4 2654.

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 26788). 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 26790). 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 (∃!𝑥𝐴𝑦𝐵𝜑 ∧ ∃!𝑦𝐵𝑥𝐴𝜑) (see 2reu4 4484 resp. 2eu4 2654). These two representations are equivalent (see opreu2reurex 6246). An analogon of this theorem using the latter variant is given in addsqn2reurex2 26793. In some cases, however, the variant with (ordered!) pairs may be possible only for ordered sets (like or ) and claiming that the first component is less than or equal to the second component (see, for example, 2sqreunnltb 26809 and 2sqreuopb 26816). Alternatively, (proper) unordered pairs can be used: ∃!𝑝𝑒𝒫 𝐴((♯‘𝑝) = 2 ∧ 𝜑), or, using the definition of proper pairs: ∃!𝑝 ∈ (Pairsproper𝐴)𝜑 (see, for example, inlinecirc02preu 46864). (Contributed by AV, 21-Jun-2023.)

Assertion
Ref Expression
addsqnreup (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶)
Distinct variable group:   𝐶,𝑝

Proof of Theorem addsqnreup
StepHypRef Expression
1 ax-1cn 11109 . . . . . . 7 1 ∈ ℂ
2 0cn 11147 . . . . . . 7 0 ∈ ℂ
3 opelxpi 5670 . . . . . . 7 ((1 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨1, 0⟩ ∈ (ℂ × ℂ))
41, 2, 3mp2an 690 . . . . . 6 ⟨1, 0⟩ ∈ (ℂ × ℂ)
5 3cn 12234 . . . . . . . 8 3 ∈ ℂ
65negcli 11469 . . . . . . 7 -3 ∈ ℂ
7 2cn 12228 . . . . . . 7 2 ∈ ℂ
8 opelxpi 5670 . . . . . . 7 ((-3 ∈ ℂ ∧ 2 ∈ ℂ) → ⟨-3, 2⟩ ∈ (ℂ × ℂ))
96, 7, 8mp2an 690 . . . . . 6 ⟨-3, 2⟩ ∈ (ℂ × ℂ)
10 0ne2 12360 . . . . . . . 8 0 ≠ 2
1110olci 864 . . . . . . 7 (1 ≠ -3 ∨ 0 ≠ 2)
12 1ex 11151 . . . . . . . 8 1 ∈ V
13 c0ex 11149 . . . . . . . 8 0 ∈ V
1412, 13opthne 5439 . . . . . . 7 (⟨1, 0⟩ ≠ ⟨-3, 2⟩ ↔ (1 ≠ -3 ∨ 0 ≠ 2))
1511, 14mpbir 230 . . . . . 6 ⟨1, 0⟩ ≠ ⟨-3, 2⟩
164, 9, 153pm3.2i 1339 . . . . 5 (⟨1, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨-3, 2⟩ ∈ (ℂ × ℂ) ∧ ⟨1, 0⟩ ≠ ⟨-3, 2⟩)
1712, 13op1st 7929 . . . . . . . 8 (1st ‘⟨1, 0⟩) = 1
1812, 13op2nd 7930 . . . . . . . . . 10 (2nd ‘⟨1, 0⟩) = 0
1918oveq1i 7367 . . . . . . . . 9 ((2nd ‘⟨1, 0⟩)↑2) = (0↑2)
20 sq0 14096 . . . . . . . . 9 (0↑2) = 0
2119, 20eqtri 2764 . . . . . . . 8 ((2nd ‘⟨1, 0⟩)↑2) = 0
2217, 21oveq12i 7369 . . . . . . 7 ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = (1 + 0)
23 1p0e1 12277 . . . . . . 7 (1 + 0) = 1
2422, 23eqtri 2764 . . . . . 6 ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1
25 negex 11399 . . . . . . . . 9 -3 ∈ V
26 2ex 12230 . . . . . . . . 9 2 ∈ V
2725, 26op1st 7929 . . . . . . . 8 (1st ‘⟨-3, 2⟩) = -3
2825, 26op2nd 7930 . . . . . . . . . 10 (2nd ‘⟨-3, 2⟩) = 2
2928oveq1i 7367 . . . . . . . . 9 ((2nd ‘⟨-3, 2⟩)↑2) = (2↑2)
30 sq2 14101 . . . . . . . . 9 (2↑2) = 4
3129, 30eqtri 2764 . . . . . . . 8 ((2nd ‘⟨-3, 2⟩)↑2) = 4
3227, 31oveq12i 7369 . . . . . . 7 ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = (-3 + 4)
33 4cn 12238 . . . . . . . 8 4 ∈ ℂ
3433, 5negsubi 11479 . . . . . . . . 9 (4 + -3) = (4 − 3)
35 3p1e4 12298 . . . . . . . . . 10 (3 + 1) = 4
3633, 5, 1, 35subaddrii 11490 . . . . . . . . 9 (4 − 3) = 1
3734, 36eqtri 2764 . . . . . . . 8 (4 + -3) = 1
3833, 6, 37addcomli 11347 . . . . . . 7 (-3 + 4) = 1
3932, 38eqtri 2764 . . . . . 6 ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1
4024, 39pm3.2i 471 . . . . 5 (((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1 ∧ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1)
41 fveq2 6842 . . . . . . . 8 (𝑝 = ⟨1, 0⟩ → (1st𝑝) = (1st ‘⟨1, 0⟩))
42 fveq2 6842 . . . . . . . . 9 (𝑝 = ⟨1, 0⟩ → (2nd𝑝) = (2nd ‘⟨1, 0⟩))
4342oveq1d 7372 . . . . . . . 8 (𝑝 = ⟨1, 0⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨1, 0⟩)↑2))
4441, 43oveq12d 7375 . . . . . . 7 (𝑝 = ⟨1, 0⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)))
4544eqeq1d 2738 . . . . . 6 (𝑝 = ⟨1, 0⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 1 ↔ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1))
46 fveq2 6842 . . . . . . . 8 (𝑝 = ⟨-3, 2⟩ → (1st𝑝) = (1st ‘⟨-3, 2⟩))
47 fveq2 6842 . . . . . . . . 9 (𝑝 = ⟨-3, 2⟩ → (2nd𝑝) = (2nd ‘⟨-3, 2⟩))
4847oveq1d 7372 . . . . . . . 8 (𝑝 = ⟨-3, 2⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨-3, 2⟩)↑2))
4946, 48oveq12d 7375 . . . . . . 7 (𝑝 = ⟨-3, 2⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)))
5049eqeq1d 2738 . . . . . 6 (𝑝 = ⟨-3, 2⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 1 ↔ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1))
5145, 502nreu 4401 . . . . 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 2748 . . . . 5 (𝐶 = 1 → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st𝑝) + ((2nd𝑝)↑2)) = 1))
5453reubidv 3371 . . . 4 (𝐶 = 1 → (∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 1))
5552, 54mtbiri 326 . . 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 11148 . . . . . . 7 (𝐶 ∈ ℂ → 0 ∈ ℂ)
5957, 58opelxpd 5671 . . . . . 6 (𝐶 ∈ ℂ → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
6059adantr 481 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
61 1cnd 11150 . . . . . . 7 (𝐶 ∈ ℂ → 1 ∈ ℂ)
62 peano2cnm 11467 . . . . . . . 8 (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ)
6362sqrtcld 15322 . . . . . . 7 (𝐶 ∈ ℂ → (√‘(𝐶 − 1)) ∈ ℂ)
6461, 63opelxpd 5671 . . . . . 6 (𝐶 ∈ ℂ → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
6564adantr 481 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
66 animorrl 979 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1))))
67 0cnd 11148 . . . . . . 7 (𝐶 ≠ 1 → 0 ∈ ℂ)
68 opthneg 5438 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
6967, 68sylan2 593 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩ ↔ (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1)))))
7066, 69mpbird 256 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩)
7160, 65, 703jca 1128 . . . 4 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ) ∧ ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩))
72 op1stg 7933 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (1st ‘⟨𝐶, 0⟩) = 𝐶)
7358, 72mpdan 685 . . . . . . . 8 (𝐶 ∈ ℂ → (1st ‘⟨𝐶, 0⟩) = 𝐶)
74 op2ndg 7934 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (2nd ‘⟨𝐶, 0⟩) = 0)
7558, 74mpdan 685 . . . . . . . . 9 (𝐶 ∈ ℂ → (2nd ‘⟨𝐶, 0⟩) = 0)
7675sq0id 14098 . . . . . . . 8 (𝐶 ∈ ℂ → ((2nd ‘⟨𝐶, 0⟩)↑2) = 0)
7773, 76oveq12d 7375 . . . . . . 7 (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = (𝐶 + 0))
78 addid1 11335 . . . . . . 7 (𝐶 ∈ ℂ → (𝐶 + 0) = 𝐶)
7977, 78eqtrd 2776 . . . . . 6 (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶)
80 op1stg 7933 . . . . . . . . 9 ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
8161, 63, 80syl2anc 584 . . . . . . . 8 (𝐶 ∈ ℂ → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
82 op2ndg 7934 . . . . . . . . . . 11 ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
8361, 63, 82syl2anc 584 . . . . . . . . . 10 (𝐶 ∈ ℂ → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
8483oveq1d 7372 . . . . . . . . 9 (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = ((√‘(𝐶 − 1))↑2))
8562sqsqrtd 15324 . . . . . . . . 9 (𝐶 ∈ ℂ → ((√‘(𝐶 − 1))↑2) = (𝐶 − 1))
8684, 85eqtrd 2776 . . . . . . . 8 (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = (𝐶 − 1))
8781, 86oveq12d 7375 . . . . . . 7 (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = (1 + (𝐶 − 1)))
8861, 57pncan3d 11515 . . . . . . 7 (𝐶 ∈ ℂ → (1 + (𝐶 − 1)) = 𝐶)
8987, 88eqtrd 2776 . . . . . 6 (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶)
9079, 89jca 512 . . . . 5 (𝐶 ∈ ℂ → (((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶 ∧ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
9190adantr 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 6842 . . . . . . 7 (𝑝 = ⟨𝐶, 0⟩ → (1st𝑝) = (1st ‘⟨𝐶, 0⟩))
93 fveq2 6842 . . . . . . . 8 (𝑝 = ⟨𝐶, 0⟩ → (2nd𝑝) = (2nd ‘⟨𝐶, 0⟩))
9493oveq1d 7372 . . . . . . 7 (𝑝 = ⟨𝐶, 0⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨𝐶, 0⟩)↑2))
9592, 94oveq12d 7375 . . . . . 6 (𝑝 = ⟨𝐶, 0⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)))
9695eqeq1d 2738 . . . . 5 (𝑝 = ⟨𝐶, 0⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶))
97 fveq2 6842 . . . . . . 7 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (1st𝑝) = (1st ‘⟨1, (√‘(𝐶 − 1))⟩))
98 fveq2 6842 . . . . . . . 8 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (2nd𝑝) = (2nd ‘⟨1, (√‘(𝐶 − 1))⟩))
9998oveq1d 7372 . . . . . . 7 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2))
10097, 99oveq12d 7375 . . . . . 6 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)))
101100eqeq1d 2738 . . . . 5 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
10296, 1012nreu 4401 . . . 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 414 . 2 (𝐶 ≠ 1 → (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶))
10556, 104pm2.61ine 3028 1 (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  ∃!wreu 3351  cop 4592   × cxp 5631  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  cc 11049  0cc0 11051  1c1 11052   + caddc 11054  cmin 11385  -cneg 11386  2c2 12208  3c3 12209  4c4 12210  cexp 13967  csqrt 15118
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121
This theorem is referenced by: (None)
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