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Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcjsub 10901 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (βˆ—β€˜(𝐴 βˆ’ 𝐡)) = ((βˆ—β€˜π΄) βˆ’ (βˆ—β€˜π΅)))
 
Theoremcjexp 10902 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•0) β†’ (βˆ—β€˜(𝐴↑𝑁)) = ((βˆ—β€˜π΄)↑𝑁))
 
Theoremimval2 10903 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ β„‚ β†’ (β„‘β€˜π΄) = ((𝐴 βˆ’ (βˆ—β€˜π΄)) / (2 Β· i)))
 
Theoremre0 10904 The real part of zero. (Contributed by NM, 27-Jul-1999.)
(β„œβ€˜0) = 0
 
Theoremim0 10905 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
(β„‘β€˜0) = 0
 
Theoremre1 10906 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(β„œβ€˜1) = 1
 
Theoremim1 10907 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(β„‘β€˜1) = 0
 
Theoremrei 10908 The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(β„œβ€˜i) = 0
 
Theoremimi 10909 The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(β„‘β€˜i) = 1
 
Theoremcj0 10910 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
(βˆ—β€˜0) = 0
 
Theoremcji 10911 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(βˆ—β€˜i) = -i
 
Theoremcjreim 10912 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (βˆ—β€˜(𝐴 + (i Β· 𝐡))) = (𝐴 βˆ’ (i Β· 𝐡)))
 
Theoremcjreim2 10913 The conjugate of the representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (βˆ—β€˜(𝐴 βˆ’ (i Β· 𝐡))) = (𝐴 + (i Β· 𝐡)))
 
Theoremcj11 10914 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((βˆ—β€˜π΄) = (βˆ—β€˜π΅) ↔ 𝐴 = 𝐡))
 
Theoremcjap 10915 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((βˆ—β€˜π΄) # (βˆ—β€˜π΅) ↔ 𝐴 # 𝐡))
 
Theoremcjap0 10916 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
(𝐴 ∈ β„‚ β†’ (𝐴 # 0 ↔ (βˆ—β€˜π΄) # 0))
 
Theoremcjne0 10917 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10916 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)
(𝐴 ∈ β„‚ β†’ (𝐴 β‰  0 ↔ (βˆ—β€˜π΄) β‰  0))
 
Theoremcjdivap 10918 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐡 # 0) β†’ (βˆ—β€˜(𝐴 / 𝐡)) = ((βˆ—β€˜π΄) / (βˆ—β€˜π΅)))
 
Theoremcnrecnv 10919* The inverse to the canonical bijection from (ℝ Γ— ℝ) to β„‚ from cnref1o 9650. (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐹 = (π‘₯ ∈ ℝ, 𝑦 ∈ ℝ ↦ (π‘₯ + (i Β· 𝑦)))    β‡’   β—‘𝐹 = (𝑧 ∈ β„‚ ↦ ⟨(β„œβ€˜π‘§), (β„‘β€˜π‘§)⟩)
 
Theoremrecli 10920 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ β„‚    β‡’   (β„œβ€˜π΄) ∈ ℝ
 
Theoremimcli 10921 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ β„‚    β‡’   (β„‘β€˜π΄) ∈ ℝ
 
Theoremcjcli 10922 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
𝐴 ∈ β„‚    β‡’   (βˆ—β€˜π΄) ∈ β„‚
 
Theoremreplimi 10923 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
𝐴 ∈ β„‚    β‡’   π΄ = ((β„œβ€˜π΄) + (i Β· (β„‘β€˜π΄)))
 
Theoremcjcji 10924 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 11-May-1999.)
𝐴 ∈ β„‚    β‡’   (βˆ—β€˜(βˆ—β€˜π΄)) = 𝐴
 
Theoremreim0bi 10925 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
𝐴 ∈ β„‚    β‡’   (𝐴 ∈ ℝ ↔ (β„‘β€˜π΄) = 0)
 
Theoremrerebi 10926 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
𝐴 ∈ β„‚    β‡’   (𝐴 ∈ ℝ ↔ (β„œβ€˜π΄) = 𝐴)
 
Theoremcjrebi 10927 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.)
𝐴 ∈ β„‚    β‡’   (𝐴 ∈ ℝ ↔ (βˆ—β€˜π΄) = 𝐴)
 
Theoremrecji 10928 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ β„‚    β‡’   (β„œβ€˜(βˆ—β€˜π΄)) = (β„œβ€˜π΄)
 
Theoremimcji 10929 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ β„‚    β‡’   (β„‘β€˜(βˆ—β€˜π΄)) = -(β„‘β€˜π΄)
 
Theoremcjmulrcli 10930 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
𝐴 ∈ β„‚    β‡’   (𝐴 Β· (βˆ—β€˜π΄)) ∈ ℝ
 
Theoremcjmulvali 10931 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ β„‚    β‡’   (𝐴 Β· (βˆ—β€˜π΄)) = (((β„œβ€˜π΄)↑2) + ((β„‘β€˜π΄)↑2))
 
Theoremcjmulge0i 10932 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
𝐴 ∈ β„‚    β‡’   0 ≀ (𝐴 Β· (βˆ—β€˜π΄))
 
Theoremrenegi 10933 Real part of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ β„‚    β‡’   (β„œβ€˜-𝐴) = -(β„œβ€˜π΄)
 
Theoremimnegi 10934 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ β„‚    β‡’   (β„‘β€˜-𝐴) = -(β„‘β€˜π΄)
 
Theoremcjnegi 10935 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ β„‚    β‡’   (βˆ—β€˜-𝐴) = -(βˆ—β€˜π΄)
 
Theoremaddcji 10936 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ β„‚    β‡’   (𝐴 + (βˆ—β€˜π΄)) = (2 Β· (β„œβ€˜π΄))
 
Theoremreaddi 10937 Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (β„œβ€˜(𝐴 + 𝐡)) = ((β„œβ€˜π΄) + (β„œβ€˜π΅))
 
Theoremimaddi 10938 Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (β„‘β€˜(𝐴 + 𝐡)) = ((β„‘β€˜π΄) + (β„‘β€˜π΅))
 
Theoremremuli 10939 Real part of a product. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (β„œβ€˜(𝐴 Β· 𝐡)) = (((β„œβ€˜π΄) Β· (β„œβ€˜π΅)) βˆ’ ((β„‘β€˜π΄) Β· (β„‘β€˜π΅)))
 
Theoremimmuli 10940 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (β„‘β€˜(𝐴 Β· 𝐡)) = (((β„œβ€˜π΄) Β· (β„‘β€˜π΅)) + ((β„‘β€˜π΄) Β· (β„œβ€˜π΅)))
 
Theoremcjaddi 10941 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (βˆ—β€˜(𝐴 + 𝐡)) = ((βˆ—β€˜π΄) + (βˆ—β€˜π΅))
 
Theoremcjmuli 10942 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (βˆ—β€˜(𝐴 Β· 𝐡)) = ((βˆ—β€˜π΄) Β· (βˆ—β€˜π΅))
 
Theoremipcni 10943 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (β„œβ€˜(𝐴 Β· (βˆ—β€˜π΅))) = (((β„œβ€˜π΄) Β· (β„œβ€˜π΅)) + ((β„‘β€˜π΄) Β· (β„‘β€˜π΅)))
 
Theoremcjdivapi 10944 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   (𝐡 # 0 β†’ (βˆ—β€˜(𝐴 / 𝐡)) = ((βˆ—β€˜π΄) / (βˆ—β€˜π΅)))
 
Theoremcrrei 10945 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
𝐴 ∈ ℝ    &   π΅ ∈ ℝ    β‡’   (β„œβ€˜(𝐴 + (i Β· 𝐡))) = 𝐴
 
Theoremcrimi 10946 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
𝐴 ∈ ℝ    &   π΅ ∈ ℝ    β‡’   (β„‘β€˜(𝐴 + (i Β· 𝐡))) = 𝐡
 
Theoremrecld 10947 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„œβ€˜π΄) ∈ ℝ)
 
Theoremimcld 10948 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„‘β€˜π΄) ∈ ℝ)
 
Theoremcjcld 10949 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (βˆ—β€˜π΄) ∈ β„‚)
 
Theoremreplimd 10950 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ 𝐴 = ((β„œβ€˜π΄) + (i Β· (β„‘β€˜π΄))))
 
Theoremremimd 10951 Value of the conjugate of a complex number. The value is the real part minus i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (βˆ—β€˜π΄) = ((β„œβ€˜π΄) βˆ’ (i Β· (β„‘β€˜π΄))))
 
Theoremcjcjd 10952 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (βˆ—β€˜(βˆ—β€˜π΄)) = 𝐴)
 
Theoremreim0bd 10953 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ (β„‘β€˜π΄) = 0)    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ)
 
Theoremrerebd 10954 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ (β„œβ€˜π΄) = 𝐴)    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ)
 
Theoremcjrebd 10955 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ (βˆ—β€˜π΄) = 𝐴)    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ)
 
Theoremcjne0d 10956 A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 10957 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 β‰  0)    β‡’   (πœ‘ β†’ (βˆ—β€˜π΄) β‰  0)
 
Theoremcjap0d 10957 A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 # 0)    β‡’   (πœ‘ β†’ (βˆ—β€˜π΄) # 0)
 
Theoremrecjd 10958 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„œβ€˜(βˆ—β€˜π΄)) = (β„œβ€˜π΄))
 
Theoremimcjd 10959 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„‘β€˜(βˆ—β€˜π΄)) = -(β„‘β€˜π΄))
 
Theoremcjmulrcld 10960 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (𝐴 Β· (βˆ—β€˜π΄)) ∈ ℝ)
 
Theoremcjmulvald 10961 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (𝐴 Β· (βˆ—β€˜π΄)) = (((β„œβ€˜π΄)↑2) + ((β„‘β€˜π΄)↑2)))
 
Theoremcjmulge0d 10962 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ 0 ≀ (𝐴 Β· (βˆ—β€˜π΄)))
 
Theoremrenegd 10963 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„œβ€˜-𝐴) = -(β„œβ€˜π΄))
 
Theoremimnegd 10964 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„‘β€˜-𝐴) = -(β„‘β€˜π΄))
 
Theoremcjnegd 10965 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (βˆ—β€˜-𝐴) = -(βˆ—β€˜π΄))
 
Theoremaddcjd 10966 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (𝐴 + (βˆ—β€˜π΄)) = (2 Β· (β„œβ€˜π΄)))
 
Theoremcjexpd 10967 Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (βˆ—β€˜(𝐴↑𝑁)) = ((βˆ—β€˜π΄)↑𝑁))
 
Theoremreaddd 10968 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„œβ€˜(𝐴 + 𝐡)) = ((β„œβ€˜π΄) + (β„œβ€˜π΅)))
 
Theoremimaddd 10969 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„‘β€˜(𝐴 + 𝐡)) = ((β„‘β€˜π΄) + (β„‘β€˜π΅)))
 
Theoremresubd 10970 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„œβ€˜(𝐴 βˆ’ 𝐡)) = ((β„œβ€˜π΄) βˆ’ (β„œβ€˜π΅)))
 
Theoremimsubd 10971 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„‘β€˜(𝐴 βˆ’ 𝐡)) = ((β„‘β€˜π΄) βˆ’ (β„‘β€˜π΅)))
 
Theoremremuld 10972 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„œβ€˜(𝐴 Β· 𝐡)) = (((β„œβ€˜π΄) Β· (β„œβ€˜π΅)) βˆ’ ((β„‘β€˜π΄) Β· (β„‘β€˜π΅))))
 
Theoremimmuld 10973 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„‘β€˜(𝐴 Β· 𝐡)) = (((β„œβ€˜π΄) Β· (β„‘β€˜π΅)) + ((β„‘β€˜π΄) Β· (β„œβ€˜π΅))))
 
Theoremcjaddd 10974 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (βˆ—β€˜(𝐴 + 𝐡)) = ((βˆ—β€˜π΄) + (βˆ—β€˜π΅)))
 
Theoremcjmuld 10975 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (βˆ—β€˜(𝐴 Β· 𝐡)) = ((βˆ—β€˜π΄) Β· (βˆ—β€˜π΅)))
 
Theoremipcnd 10976 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„œβ€˜(𝐴 Β· (βˆ—β€˜π΅))) = (((β„œβ€˜π΄) Β· (β„œβ€˜π΅)) + ((β„‘β€˜π΄) Β· (β„‘β€˜π΅))))
 
Theoremcjdivapd 10977 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 # 0)    β‡’   (πœ‘ β†’ (βˆ—β€˜(𝐴 / 𝐡)) = ((βˆ—β€˜π΄) / (βˆ—β€˜π΅)))
 
Theoremrered 10978 A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (β„œβ€˜π΄) = 𝐴)
 
Theoremreim0d 10979 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (β„‘β€˜π΄) = 0)
 
Theoremcjred 10980 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (βˆ—β€˜π΄) = 𝐴)
 
Theoremremul2d 10981 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„œβ€˜(𝐴 Β· 𝐡)) = (𝐴 Β· (β„œβ€˜π΅)))
 
Theoremimmul2d 10982 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„‘β€˜(𝐴 Β· 𝐡)) = (𝐴 Β· (β„‘β€˜π΅)))
 
Theoremredivapd 10983 Real part of a division. Related to remul2 10882. (Contributed by Jim Kingdon, 15-Jun-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 # 0)    β‡’   (πœ‘ β†’ (β„œβ€˜(𝐡 / 𝐴)) = ((β„œβ€˜π΅) / 𝐴))
 
Theoremimdivapd 10984 Imaginary part of a division. Related to remul2 10882. (Contributed by Jim Kingdon, 15-Jun-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 # 0)    β‡’   (πœ‘ β†’ (β„‘β€˜(𝐡 / 𝐴)) = ((β„‘β€˜π΅) / 𝐴))
 
Theoremcrred 10985 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (β„œβ€˜(𝐴 + (i Β· 𝐡))) = 𝐴)
 
Theoremcrimd 10986 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (β„‘β€˜(𝐴 + (i Β· 𝐡))) = 𝐡)
 
Theoremcnreim 10987 Complex apartness in terms of real and imaginary parts. See also apreim 8560 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 # 𝐡 ↔ ((β„œβ€˜π΄) # (β„œβ€˜π΅) ∨ (β„‘β€˜π΄) # (β„‘β€˜π΅))))
 
4.7.3  Sequence convergence
 
Theoremcaucvgrelemrec 10988* Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) β†’ (β„©π‘Ÿ ∈ ℝ (𝐴 Β· π‘Ÿ) = 1) = (1 / 𝐴))
 
Theoremcaucvgrelemcau 10989* Lemma for caucvgre 10990. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)
(πœ‘ β†’ 𝐹:β„•βŸΆβ„)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘›) < ((πΉβ€˜π‘˜) + (1 / 𝑛)) ∧ (πΉβ€˜π‘˜) < ((πΉβ€˜π‘›) + (1 / 𝑛))))    β‡’   (πœ‘ β†’ βˆ€π‘› ∈ β„• βˆ€π‘˜ ∈ β„• (𝑛 <ℝ π‘˜ β†’ ((πΉβ€˜π‘›) <ℝ ((πΉβ€˜π‘˜) + (β„©π‘Ÿ ∈ ℝ (𝑛 Β· π‘Ÿ) = 1)) ∧ (πΉβ€˜π‘˜) <ℝ ((πΉβ€˜π‘›) + (β„©π‘Ÿ ∈ ℝ (𝑛 Β· π‘Ÿ) = 1)))))
 
Theoremcaucvgre 10990* Convergence of real sequences.

A Cauchy sequence (as defined here, which has a rate of convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / 𝑛 of the nth term.

(Contributed by Jim Kingdon, 19-Jul-2021.)

(πœ‘ β†’ 𝐹:β„•βŸΆβ„)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘›) < ((πΉβ€˜π‘˜) + (1 / 𝑛)) ∧ (πΉβ€˜π‘˜) < ((πΉβ€˜π‘›) + (1 / 𝑛))))    β‡’   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ β„• βˆ€π‘– ∈ (β„€β‰₯β€˜π‘—)((πΉβ€˜π‘–) < (𝑦 + π‘₯) ∧ 𝑦 < ((πΉβ€˜π‘–) + π‘₯)))
 
Theoremcvg1nlemcxze 10991 Lemma for cvg1n 10995. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)
(πœ‘ β†’ 𝐢 ∈ ℝ+)    &   (πœ‘ β†’ 𝑋 ∈ ℝ+)    &   (πœ‘ β†’ 𝑍 ∈ β„•)    &   (πœ‘ β†’ 𝐸 ∈ β„•)    &   (πœ‘ β†’ 𝐴 ∈ β„•)    &   (πœ‘ β†’ ((((𝐢 Β· 2) / 𝑋) / 𝑍) + 𝐴) < 𝐸)    β‡’   (πœ‘ β†’ (𝐢 / (𝐸 Β· 𝑍)) < (𝑋 / 2))
 
Theoremcvg1nlemf 10992* Lemma for cvg1n 10995. The modified sequence 𝐺 is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)
(πœ‘ β†’ 𝐹:β„•βŸΆβ„)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘›) < ((πΉβ€˜π‘˜) + (𝐢 / 𝑛)) ∧ (πΉβ€˜π‘˜) < ((πΉβ€˜π‘›) + (𝐢 / 𝑛))))    &   πΊ = (𝑗 ∈ β„• ↦ (πΉβ€˜(𝑗 Β· 𝑍)))    &   (πœ‘ β†’ 𝑍 ∈ β„•)    &   (πœ‘ β†’ 𝐢 < 𝑍)    β‡’   (πœ‘ β†’ 𝐺:β„•βŸΆβ„)
 
Theoremcvg1nlemcau 10993* Lemma for cvg1n 10995. By selecting spaced out terms for the modified sequence 𝐺, the terms are within 1 / 𝑛 (without the constant 𝐢). (Contributed by Jim Kingdon, 1-Aug-2021.)
(πœ‘ β†’ 𝐹:β„•βŸΆβ„)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘›) < ((πΉβ€˜π‘˜) + (𝐢 / 𝑛)) ∧ (πΉβ€˜π‘˜) < ((πΉβ€˜π‘›) + (𝐢 / 𝑛))))    &   πΊ = (𝑗 ∈ β„• ↦ (πΉβ€˜(𝑗 Β· 𝑍)))    &   (πœ‘ β†’ 𝑍 ∈ β„•)    &   (πœ‘ β†’ 𝐢 < 𝑍)    β‡’   (πœ‘ β†’ βˆ€π‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((πΊβ€˜π‘›) < ((πΊβ€˜π‘˜) + (1 / 𝑛)) ∧ (πΊβ€˜π‘˜) < ((πΊβ€˜π‘›) + (1 / 𝑛))))
 
Theoremcvg1nlemres 10994* Lemma for cvg1n 10995. The original sequence 𝐹 has a limit (turns out it is the same as the limit of the modified sequence 𝐺). (Contributed by Jim Kingdon, 1-Aug-2021.)
(πœ‘ β†’ 𝐹:β„•βŸΆβ„)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘›) < ((πΉβ€˜π‘˜) + (𝐢 / 𝑛)) ∧ (πΉβ€˜π‘˜) < ((πΉβ€˜π‘›) + (𝐢 / 𝑛))))    &   πΊ = (𝑗 ∈ β„• ↦ (πΉβ€˜(𝑗 Β· 𝑍)))    &   (πœ‘ β†’ 𝑍 ∈ β„•)    &   (πœ‘ β†’ 𝐢 < 𝑍)    β‡’   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ β„• βˆ€π‘– ∈ (β„€β‰₯β€˜π‘—)((πΉβ€˜π‘–) < (𝑦 + π‘₯) ∧ 𝑦 < ((πΉβ€˜π‘–) + π‘₯)))
 
Theoremcvg1n 10995* Convergence of real sequences.

This is a version of caucvgre 10990 with a constant multiplier 𝐢 on the rate of convergence. That is, all terms after the nth term must be within 𝐢 / 𝑛 of the nth term.

(Contributed by Jim Kingdon, 1-Aug-2021.)

(πœ‘ β†’ 𝐹:β„•βŸΆβ„)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘›) < ((πΉβ€˜π‘˜) + (𝐢 / 𝑛)) ∧ (πΉβ€˜π‘˜) < ((πΉβ€˜π‘›) + (𝐢 / 𝑛))))    β‡’   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ β„• βˆ€π‘– ∈ (β„€β‰₯β€˜π‘—)((πΉβ€˜π‘–) < (𝑦 + π‘₯) ∧ 𝑦 < ((πΉβ€˜π‘–) + π‘₯)))
 
Theoremuzin2 10996 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
((𝐴 ∈ ran β„€β‰₯ ∧ 𝐡 ∈ ran β„€β‰₯) β†’ (𝐴 ∩ 𝐡) ∈ ran β„€β‰₯)
 
Theoremrexanuz 10997* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)
(βˆƒπ‘— ∈ β„€ βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)(πœ‘ ∧ πœ“) ↔ (βˆƒπ‘— ∈ β„€ βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)πœ‘ ∧ βˆƒπ‘— ∈ β„€ βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)πœ“))
 
Theoremrexfiuz 10998* Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)
(𝐴 ∈ Fin β†’ (βˆƒπ‘— ∈ β„€ βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)βˆ€π‘› ∈ 𝐴 πœ‘ ↔ βˆ€π‘› ∈ 𝐴 βˆƒπ‘— ∈ β„€ βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)πœ‘))
 
Theoremrexuz3 10999* Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (β„€β‰₯β€˜π‘€)    β‡’   (𝑀 ∈ β„€ β†’ (βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)πœ‘ ↔ βˆƒπ‘— ∈ β„€ βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)πœ‘))
 
Theoremrexanuz2 11000* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (β„€β‰₯β€˜π‘€)    β‡’   (βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)(πœ‘ ∧ πœ“) ↔ (βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)πœ‘ ∧ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)πœ“))
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