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Theorem addsqnreup 27182
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 27179, 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 2561 and 2eu4 2648.

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 27179). 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 27181). 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 2648). These two representations are equivalent (see opreu2reurex 6292). An analogon of this theorem using the latter variant is given in addsqn2reurex2 27184. 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 27200 and 2sqreuopb 27207). 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 47561). (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 11170 . . . . . . 7 1 ∈ ℂ
2 0cn 11210 . . . . . . 7 0 ∈ ℂ
3 opelxpi 5712 . . . . . . 7 ((1 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨1, 0⟩ ∈ (ℂ × ℂ))
41, 2, 3mp2an 688 . . . . . 6 ⟨1, 0⟩ ∈ (ℂ × ℂ)
5 3cn 12297 . . . . . . . 8 3 ∈ ℂ
65negcli 11532 . . . . . . 7 -3 ∈ ℂ
7 2cn 12291 . . . . . . 7 2 ∈ ℂ
8 opelxpi 5712 . . . . . . 7 ((-3 ∈ ℂ ∧ 2 ∈ ℂ) → ⟨-3, 2⟩ ∈ (ℂ × ℂ))
96, 7, 8mp2an 688 . . . . . 6 ⟨-3, 2⟩ ∈ (ℂ × ℂ)
10 0ne2 12423 . . . . . . . 8 0 ≠ 2
1110olci 862 . . . . . . 7 (1 ≠ -3 ∨ 0 ≠ 2)
12 1ex 11214 . . . . . . . 8 1 ∈ V
13 c0ex 11212 . . . . . . . 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 1337 . . . . 5 (⟨1, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨-3, 2⟩ ∈ (ℂ × ℂ) ∧ ⟨1, 0⟩ ≠ ⟨-3, 2⟩)
1712, 13op1st 7985 . . . . . . . 8 (1st ‘⟨1, 0⟩) = 1
1812, 13op2nd 7986 . . . . . . . . . 10 (2nd ‘⟨1, 0⟩) = 0
1918oveq1i 7421 . . . . . . . . 9 ((2nd ‘⟨1, 0⟩)↑2) = (0↑2)
20 sq0 14160 . . . . . . . . 9 (0↑2) = 0
2119, 20eqtri 2758 . . . . . . . 8 ((2nd ‘⟨1, 0⟩)↑2) = 0
2217, 21oveq12i 7423 . . . . . . 7 ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = (1 + 0)
23 1p0e1 12340 . . . . . . 7 (1 + 0) = 1
2422, 23eqtri 2758 . . . . . 6 ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1
25 negex 11462 . . . . . . . . 9 -3 ∈ V
26 2ex 12293 . . . . . . . . 9 2 ∈ V
2725, 26op1st 7985 . . . . . . . 8 (1st ‘⟨-3, 2⟩) = -3
2825, 26op2nd 7986 . . . . . . . . . 10 (2nd ‘⟨-3, 2⟩) = 2
2928oveq1i 7421 . . . . . . . . 9 ((2nd ‘⟨-3, 2⟩)↑2) = (2↑2)
30 sq2 14165 . . . . . . . . 9 (2↑2) = 4
3129, 30eqtri 2758 . . . . . . . 8 ((2nd ‘⟨-3, 2⟩)↑2) = 4
3227, 31oveq12i 7423 . . . . . . 7 ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = (-3 + 4)
33 4cn 12301 . . . . . . . 8 4 ∈ ℂ
3433, 5negsubi 11542 . . . . . . . . 9 (4 + -3) = (4 − 3)
35 3p1e4 12361 . . . . . . . . . 10 (3 + 1) = 4
3633, 5, 1, 35subaddrii 11553 . . . . . . . . 9 (4 − 3) = 1
3734, 36eqtri 2758 . . . . . . . 8 (4 + -3) = 1
3833, 6, 37addcomli 11410 . . . . . . 7 (-3 + 4) = 1
3932, 38eqtri 2758 . . . . . 6 ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1
4024, 39pm3.2i 469 . . . . 5 (((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1 ∧ ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)) = 1)
41 fveq2 6890 . . . . . . . 8 (𝑝 = ⟨1, 0⟩ → (1st𝑝) = (1st ‘⟨1, 0⟩))
42 fveq2 6890 . . . . . . . . 9 (𝑝 = ⟨1, 0⟩ → (2nd𝑝) = (2nd ‘⟨1, 0⟩))
4342oveq1d 7426 . . . . . . . 8 (𝑝 = ⟨1, 0⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨1, 0⟩)↑2))
4441, 43oveq12d 7429 . . . . . . 7 (𝑝 = ⟨1, 0⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)))
4544eqeq1d 2732 . . . . . 6 (𝑝 = ⟨1, 0⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 1 ↔ ((1st ‘⟨1, 0⟩) + ((2nd ‘⟨1, 0⟩)↑2)) = 1))
46 fveq2 6890 . . . . . . . 8 (𝑝 = ⟨-3, 2⟩ → (1st𝑝) = (1st ‘⟨-3, 2⟩))
47 fveq2 6890 . . . . . . . . 9 (𝑝 = ⟨-3, 2⟩ → (2nd𝑝) = (2nd ‘⟨-3, 2⟩))
4847oveq1d 7426 . . . . . . . 8 (𝑝 = ⟨-3, 2⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨-3, 2⟩)↑2))
4946, 48oveq12d 7429 . . . . . . 7 (𝑝 = ⟨-3, 2⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨-3, 2⟩) + ((2nd ‘⟨-3, 2⟩)↑2)))
5049eqeq1d 2732 . . . . . 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 2742 . . . . 5 (𝐶 = 1 → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st𝑝) + ((2nd𝑝)↑2)) = 1))
5453reubidv 3392 . . . 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 11211 . . . . . . 7 (𝐶 ∈ ℂ → 0 ∈ ℂ)
5957, 58opelxpd 5714 . . . . . 6 (𝐶 ∈ ℂ → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
6059adantr 479 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨𝐶, 0⟩ ∈ (ℂ × ℂ))
61 1cnd 11213 . . . . . . 7 (𝐶 ∈ ℂ → 1 ∈ ℂ)
62 peano2cnm 11530 . . . . . . . 8 (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ)
6362sqrtcld 15388 . . . . . . 7 (𝐶 ∈ ℂ → (√‘(𝐶 − 1)) ∈ ℂ)
6461, 63opelxpd 5714 . . . . . 6 (𝐶 ∈ ℂ → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
6564adantr 479 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ))
66 animorrl 977 . . . . . 6 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (𝐶 ≠ 1 ∨ 0 ≠ (√‘(𝐶 − 1))))
67 0cnd 11211 . . . . . . 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 591 . . . . . 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 1126 . . . 4 ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 1) → (⟨𝐶, 0⟩ ∈ (ℂ × ℂ) ∧ ⟨1, (√‘(𝐶 − 1))⟩ ∈ (ℂ × ℂ) ∧ ⟨𝐶, 0⟩ ≠ ⟨1, (√‘(𝐶 − 1))⟩))
72 op1stg 7989 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (1st ‘⟨𝐶, 0⟩) = 𝐶)
7358, 72mpdan 683 . . . . . . . 8 (𝐶 ∈ ℂ → (1st ‘⟨𝐶, 0⟩) = 𝐶)
74 op2ndg 7990 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → (2nd ‘⟨𝐶, 0⟩) = 0)
7558, 74mpdan 683 . . . . . . . . 9 (𝐶 ∈ ℂ → (2nd ‘⟨𝐶, 0⟩) = 0)
7675sq0id 14162 . . . . . . . 8 (𝐶 ∈ ℂ → ((2nd ‘⟨𝐶, 0⟩)↑2) = 0)
7773, 76oveq12d 7429 . . . . . . 7 (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = (𝐶 + 0))
78 addrid 11398 . . . . . . 7 (𝐶 ∈ ℂ → (𝐶 + 0) = 𝐶)
7977, 78eqtrd 2770 . . . . . 6 (𝐶 ∈ ℂ → ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶)
80 op1stg 7989 . . . . . . . . 9 ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
8161, 63, 80syl2anc 582 . . . . . . . 8 (𝐶 ∈ ℂ → (1st ‘⟨1, (√‘(𝐶 − 1))⟩) = 1)
82 op2ndg 7990 . . . . . . . . . . 11 ((1 ∈ ℂ ∧ (√‘(𝐶 − 1)) ∈ ℂ) → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
8361, 63, 82syl2anc 582 . . . . . . . . . 10 (𝐶 ∈ ℂ → (2nd ‘⟨1, (√‘(𝐶 − 1))⟩) = (√‘(𝐶 − 1)))
8483oveq1d 7426 . . . . . . . . 9 (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = ((√‘(𝐶 − 1))↑2))
8562sqsqrtd 15390 . . . . . . . . 9 (𝐶 ∈ ℂ → ((√‘(𝐶 − 1))↑2) = (𝐶 − 1))
8684, 85eqtrd 2770 . . . . . . . 8 (𝐶 ∈ ℂ → ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2) = (𝐶 − 1))
8781, 86oveq12d 7429 . . . . . . 7 (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = (1 + (𝐶 − 1)))
8861, 57pncan3d 11578 . . . . . . 7 (𝐶 ∈ ℂ → (1 + (𝐶 − 1)) = 𝐶)
8987, 88eqtrd 2770 . . . . . 6 (𝐶 ∈ ℂ → ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶)
9079, 89jca 510 . . . . 5 (𝐶 ∈ ℂ → (((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶 ∧ ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)) = 𝐶))
9190adantr 479 . . . 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 6890 . . . . . . 7 (𝑝 = ⟨𝐶, 0⟩ → (1st𝑝) = (1st ‘⟨𝐶, 0⟩))
93 fveq2 6890 . . . . . . . 8 (𝑝 = ⟨𝐶, 0⟩ → (2nd𝑝) = (2nd ‘⟨𝐶, 0⟩))
9493oveq1d 7426 . . . . . . 7 (𝑝 = ⟨𝐶, 0⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨𝐶, 0⟩)↑2))
9592, 94oveq12d 7429 . . . . . 6 (𝑝 = ⟨𝐶, 0⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)))
9695eqeq1d 2732 . . . . 5 (𝑝 = ⟨𝐶, 0⟩ → (((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶 ↔ ((1st ‘⟨𝐶, 0⟩) + ((2nd ‘⟨𝐶, 0⟩)↑2)) = 𝐶))
97 fveq2 6890 . . . . . . 7 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (1st𝑝) = (1st ‘⟨1, (√‘(𝐶 − 1))⟩))
98 fveq2 6890 . . . . . . . 8 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → (2nd𝑝) = (2nd ‘⟨1, (√‘(𝐶 − 1))⟩))
9998oveq1d 7426 . . . . . . 7 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((2nd𝑝)↑2) = ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2))
10097, 99oveq12d 7429 . . . . . 6 (𝑝 = ⟨1, (√‘(𝐶 − 1))⟩ → ((1st𝑝) + ((2nd𝑝)↑2)) = ((1st ‘⟨1, (√‘(𝐶 − 1))⟩) + ((2nd ‘⟨1, (√‘(𝐶 − 1))⟩)↑2)))
101100eqeq1d 2732 . . . . 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 412 . 2 (𝐶 ≠ 1 → (𝐶 ∈ ℂ → ¬ ∃!𝑝 ∈ (ℂ × ℂ)((1st𝑝) + ((2nd𝑝)↑2)) = 𝐶))
10556, 104pm2.61ine 3023 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 394  wo 843  w3a 1085   = wceq 1539  wcel 2104  wne 2938  ∃!wreu 3372  cop 4633   × cxp 5673  cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  cc 11110  0cc0 11112  1c1 11113   + caddc 11115  cmin 11448  -cneg 11449  2c2 12271  3c3 12272  4c4 12273  cexp 14031  csqrt 15184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187
This theorem is referenced by: (None)
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