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Theorem addsqnreup 26935
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 26932, 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 2563 and 2eu4 2650.

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 26932). 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 26934). 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 4525 resp. 2eu4 2650). These two representations are equivalent (see opreu2reurex 6290). An analogon of this theorem using the latter variant is given in addsqn2reurex2 26937. 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 26953 and 2sqreuopb 26960). 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 47427). (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 11164 . . . . . . 7 1 ∈ ℂ
2 0cn 11202 . . . . . . 7 0 ∈ ℂ
3 opelxpi 5712 . . . . . . 7 ((1 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨1, 0⟩ ∈ (ℂ × ℂ))
41, 2, 3mp2an 690 . . . . . 6 ⟨1, 0⟩ ∈ (ℂ × ℂ)
5 3cn 12289 . . . . . . . 8 3 ∈ ℂ
65negcli 11524 . . . . . . 7 -3 ∈ ℂ
7 2cn 12283 . . . . . . 7 2 ∈ ℂ
8 opelxpi 5712 . . . . . . 7 ((-3 ∈ ℂ ∧ 2 ∈ ℂ) → ⟨-3, 2⟩ ∈ (ℂ × ℂ))
96, 7, 8mp2an 690 . . . . . 6 ⟨-3, 2⟩ ∈ (ℂ × ℂ)
10 0ne2 12415 . . . . . . . 8 0 ≠ 2
1110olci 864 . . . . . . 7 (1 ≠ -3 ∨ 0 ≠ 2)
12 1ex 11206 . . . . . . . 8 1 ∈ V
13 c0ex 11204 . . . . . . . 8 0 ∈ V
1412, 13opthne 5481 . . . . . . 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 7979 . . . . . . . 8 (1st ‘⟨1, 0⟩) = 1
1812, 13op2nd 7980 . . . . . . . . . 10 (2nd ‘⟨1, 0⟩) = 0
1918oveq1i 7415 . . . . . . . . 9 ((2nd ‘⟨1, 0⟩)↑2) = (0↑2)
20 sq0 14152 . . . . . . . . 9 (0↑2) = 0
2119, 20eqtri 2760 . . . . . . . 8 ((2nd ‘⟨1, 0⟩)↑2) = 0
2217, 21oveq12i 7417 . . . . . . 7 ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = (1 + 0)
23 1p0e1 12332 . . . . . . 7 (1 + 0) = 1
2422, 23eqtri 2760 . . . . . 6 ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1
25 negex 11454 . . . . . . . . 9 -3 ∈ V
26 2ex 12285 . . . . . . . . 9 2 ∈ V
2725, 26op1st 7979 . . . . . . . 8 (1st ‘⟨-3, 2⟩) = -3
2825, 26op2nd 7980 . . . . . . . . . 10 (2nd ‘⟨-3, 2⟩) = 2
2928oveq1i 7415 . . . . . . . . 9 ((2nd ‘⟨-3, 2⟩)↑2) = (2↑2)
30 sq2 14157 . . . . . . . . 9 (2↑2) = 4
3129, 30eqtri 2760 . . . . . . . 8 ((2nd ‘⟨-3, 2⟩)↑2) = 4
3227, 31oveq12i 7417 . . . . . . 7 ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = (-3 + 4)
33 4cn 12293 . . . . . . . 8 4 ∈ ℂ
3433, 5negsubi 11534 . . . . . . . . 9 (4 + -3) = (4 − 3)
35 3p1e4 12353 . . . . . . . . . 10 (3 + 1) = 4
3633, 5, 1, 35subaddrii 11545 . . . . . . . . 9 (4 − 3) = 1
3734, 36eqtri 2760 . . . . . . . 8 (4 + -3) = 1
3833, 6, 37addcomli 11402 . . . . . . 7 (-3 + 4) = 1
3932, 38eqtri 2760 . . . . . 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 6888 . . . . . . . 8 (𝑝 = ⟨1, 0⟩ → (1st𝑝) = (1st ‘⟨1, 0⟩))
42 fveq2 6888 . . . . . . . . 9 (𝑝 = ⟨1, 0⟩ → (2nd𝑝) = (2nd ‘⟨1, 0⟩))
4342oveq1d 7420 . . . . . . . 8 (𝑝 = ⟨1, 0⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨1, 0⟩)↑2))
4441, 43oveq12d 7423 . . . . . . 7 (𝑝 = ⟨1, 0⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)))
4544eqeq1d 2734 . . . . . 6 (𝑝 = ⟨1, 0⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 1 ↔ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1))
46 fveq2 6888 . . . . . . . 8 (𝑝 = ⟨-3, 2⟩ → (1st𝑝) = (1st ‘⟨-3, 2⟩))
47 fveq2 6888 . . . . . . . . 9 (𝑝 = ⟨-3, 2⟩ → (2nd𝑝) = (2nd ‘⟨-3, 2⟩))
4847oveq1d 7420 . . . . . . . 8 (𝑝 = ⟨-3, 2⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨-3, 2⟩)↑2))
4946, 48oveq12d 7423 . . . . . . 7 (𝑝 = ⟨-3, 2⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)))
5049eqeq1d 2734 . . . . . 6 (𝑝 = ⟨-3, 2⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 1 ↔ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1))
5145, 502nreu 4440 . . . . 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 2744 . . . . 5 (𝐶 = 1 → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st𝑝) + ((2nd𝑝)↑2)) = 1))
5453reubidv 3394 . . . 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 11203 . . . . . . 7 (𝐶 ∈ ℂ → 0 ∈ ℂ)
5957, 58opelxpd 5713 . . . . . 6 (𝐶 ∈ ℂ → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
6059adantr 481 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
61 1cnd 11205 . . . . . . 7 (𝐶 ∈ ℂ → 1 ∈ ℂ)
62 peano2cnm 11522 . . . . . . . 8 (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ)
6362sqrtcld 15380 . . . . . . 7 (𝐶 ∈ ℂ → (√‘(𝐶 − 1)) ∈ ℂ)
6461, 63opelxpd 5713 . . . . . 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 11203 . . . . . . 7 (𝐶 ≠ 1 → 0 ∈ ℂ)
68 opthneg 5480 . . . . . . 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 7983 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (1st ‘⟨𝐶, 0⟩) = 𝐶)
7358, 72mpdan 685 . . . . . . . 8 (𝐶 ∈ ℂ → (1st ‘⟨𝐶, 0⟩) = 𝐶)
74 op2ndg 7984 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (2nd ‘⟨𝐶, 0⟩) = 0)
7558, 74mpdan 685 . . . . . . . . 9 (𝐶 ∈ ℂ → (2nd ‘⟨𝐶, 0⟩) = 0)
7675sq0id 14154 . . . . . . . 8 (𝐶 ∈ ℂ → ((2nd ‘⟨𝐶, 0⟩)↑2) = 0)
7773, 76oveq12d 7423 . . . . . . 7 (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = (𝐶 + 0))
78 addrid 11390 . . . . . . 7 (𝐶 ∈ ℂ → (𝐶 + 0) = 𝐶)
7977, 78eqtrd 2772 . . . . . 6 (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶)
80 op1stg 7983 . . . . . . . . 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 7984 . . . . . . . . . . 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 7420 . . . . . . . . 9 (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = ((√‘(𝐶 − 1))↑2))
8562sqsqrtd 15382 . . . . . . . . 9 (𝐶 ∈ ℂ → ((√‘(𝐶 − 1))↑2) = (𝐶 − 1))
8684, 85eqtrd 2772 . . . . . . . 8 (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = (𝐶 − 1))
8781, 86oveq12d 7423 . . . . . . 7 (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = (1 + (𝐶 − 1)))
8861, 57pncan3d 11570 . . . . . . 7 (𝐶 ∈ ℂ → (1 + (𝐶 − 1)) = 𝐶)
8987, 88eqtrd 2772 . . . . . 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 6888 . . . . . . 7 (𝑝 = ⟨𝐶, 0⟩ → (1st𝑝) = (1st ‘⟨𝐶, 0⟩))
93 fveq2 6888 . . . . . . . 8 (𝑝 = ⟨𝐶, 0⟩ → (2nd𝑝) = (2nd ‘⟨𝐶, 0⟩))
9493oveq1d 7420 . . . . . . 7 (𝑝 = ⟨𝐶, 0⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨𝐶, 0⟩)↑2))
9592, 94oveq12d 7423 . . . . . 6 (𝑝 = ⟨𝐶, 0⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)))
9695eqeq1d 2734 . . . . 5 (𝑝 = ⟨𝐶, 0⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶))
97 fveq2 6888 . . . . . . 7 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (1st𝑝) = (1st ‘⟨1, (√‘(𝐶 − 1))⟩))
98 fveq2 6888 . . . . . . . 8 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (2nd𝑝) = (2nd ‘⟨1, (√‘(𝐶 − 1))⟩))
9998oveq1d 7420 . . . . . . 7 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2))
10097, 99oveq12d 7423 . . . . . 6 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)))
101100eqeq1d 2734 . . . . 5 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
10296, 1012nreu 4440 . . . 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 3025 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 2940  ∃!wreu 3374  cop 4633   × cxp 5673  cfv 6540  (class class class)co 7405  1st c1st 7969  2nd c2nd 7970  cc 11104  0cc0 11106  1c1 11107   + caddc 11109  cmin 11440  -cneg 11441  2c2 12263  3c3 12264  4c4 12265  cexp 14023  csqrt 15176
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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179
This theorem is referenced by: (None)
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