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Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremixi 8001 i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(i · i) = -1
 
Theoreminelr 8002 The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
¬ i ∈ ℝ
 
Theoremrimul 8003 A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → 𝐴 = 0)
 
Theoremrereim 8004 Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐴 = (𝐵 + (i · 𝐶)))) → (𝐵 = 𝐴𝐶 = 0))
 
Theoremapreap 8005 Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵𝐴 # 𝐵))
 
Theoremreaplt 8006 Real apartness in terms of less than. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
 
Theoremreapltxor 8007 Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
 
Theorem1ap0 8008 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
1 # 0
 
Theoremltmul1a 8009 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) ∧ 𝐴 < 𝐵) → (𝐴 · 𝐶) < (𝐵 · 𝐶))
 
Theoremltmul1 8010 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
 
Theoremlemul1 8011 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
 
Theoremreapmul1lem 8012 Lemma for reapmul1 8013. (Contributed by Jim Kingdon, 8-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
 
Theoremreapmul1 8013 Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8194. (Contributed by Jim Kingdon, 8-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
 
Theoremreapadd1 8014 Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶)))
 
Theoremreapneg 8015 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵))
 
Theoremreapcotr 8016 Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 → (𝐴 # 𝐶𝐵 # 𝐶)))
 
Theoremremulext1 8017 Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵))
 
Theoremremulext2 8018 Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵))
 
Theoremapsqgt0 8019 The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴 · 𝐴))
 
Theoremcru 8020 The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) = (𝐶 + (i · 𝐷)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theoremapreim 8021 Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) # (𝐶 + (i · 𝐷)) ↔ (𝐴 # 𝐶𝐵 # 𝐷)))
 
Theoremmulreim 8022 Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) · (𝐶 + (i · 𝐷))) = (((𝐴 · 𝐶) + -(𝐵 · 𝐷)) + (i · ((𝐶 · 𝐵) + (𝐷 · 𝐴)))))
 
Theoremapirr 8023 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
(𝐴 ∈ ℂ → ¬ 𝐴 # 𝐴)
 
Theoremapsym 8024 Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵𝐵 # 𝐴))
 
Theoremapcotr 8025 Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 → (𝐴 # 𝐶𝐵 # 𝐶)))
 
Theoremapadd1 8026 Addition respects apartness. Analogue of addcan 7606 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶)))
 
Theoremapadd2 8027 Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐶 + 𝐴) # (𝐶 + 𝐵)))
 
Theoremaddext 8028 Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5622. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) # (𝐶 + 𝐷) → (𝐴 # 𝐶𝐵 # 𝐷)))
 
Theoremapneg 8029 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵))
 
Theoremmulext1 8030 Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵))
 
Theoremmulext2 8031 Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵))
 
Theoremmulext 8032 Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5622. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) # (𝐶 · 𝐷) → (𝐴 # 𝐶𝐵 # 𝐷)))
 
Theoremmulap0r 8033 A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 · 𝐵) # 0) → (𝐴 # 0 ∧ 𝐵 # 0))
 
Theoremmsqge0 8034 A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℝ → 0 ≤ (𝐴 · 𝐴))
 
Theoremmsqge0i 8035 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ       0 ≤ (𝐴 · 𝐴)
 
Theoremmsqge0d 8036 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → 0 ≤ (𝐴 · 𝐴))
 
Theoremmulge0 8037 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 · 𝐵))
 
Theoremmulge0i 8038 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → 0 ≤ (𝐴 · 𝐵))
 
Theoremmulge0d 8039 The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → 0 ≤ (𝐴 · 𝐵))
 
Theoremapti 8040 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
 
Theoremapne 8041 Apartness implies negated equality. We cannot in general prove the converse, which is the whole point of having separate notations for apartness and negated equality. (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵𝐴𝐵))
 
Theoremleltap 8042 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (𝐴 < 𝐵𝐵 # 𝐴))
 
Theoremgt0ap0 8043 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 # 0)
 
Theoremgt0ap0i 8044 Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℝ       (0 < 𝐴𝐴 # 0)
 
Theoremgt0ap0ii 8045 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 # 0
 
Theoremgt0ap0d 8046 Positive implies apart from zero. Because of the way we define #, 𝐴 must be an element of , not just *. (Contributed by Jim Kingdon, 27-Feb-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)       (𝜑𝐴 # 0)
 
Theoremnegap0 8047 A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
(𝐴 ∈ ℂ → (𝐴 # 0 ↔ -𝐴 # 0))
 
Theoremltleap 8048 Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴𝐵𝐴 # 𝐵)))
 
Theoremltap 8049 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 # 𝐴)
 
Theoremgtapii 8050 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵       𝐵 # 𝐴
 
Theoremltapii 8051 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵       𝐴 # 𝐵
 
Theoremltapi 8052 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵𝐵 # 𝐴)
 
Theoremgtapd 8053 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵 # 𝐴)
 
Theoremltapd 8054 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴 # 𝐵)
 
Theoremleltapd 8055 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 < 𝐵𝐵 # 𝐴))
 
Theoremap0gt0 8056 A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 # 0 ↔ 0 < 𝐴))
 
Theoremap0gt0d 8057 A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 # 0)       (𝜑 → 0 < 𝐴)
 
Theoremsubap0d 8058 Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 𝐵)       (𝜑 → (𝐴𝐵) # 0)
 
3.3.7  Reciprocals
 
Theoremrecextlem1 8059 Lemma for recexap 8061. (Contributed by Eric Schmidt, 23-May-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵)))
 
Theoremrecexaplem2 8060 Lemma for recexap 8061. (Contributed by Jim Kingdon, 20-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) # 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) # 0)
 
Theoremrecexap 8061* Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1)
 
Theoremmulap0 8062 The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 · 𝐵) # 0)
 
Theoremmulap0b 8063 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0))
 
Theoremmulap0i 8064 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 # 0    &   𝐵 # 0       (𝐴 · 𝐵) # 0
 
Theoremmulap0bd 8065 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0))
 
Theoremmulap0d 8066 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 · 𝐵) # 0)
 
Theoremmulap0bad 8067 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 8066 and consequence of mulap0bd 8065. (Contributed by Jim Kingdon, 24-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) # 0)       (𝜑𝐴 # 0)
 
Theoremmulap0bbd 8068 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 8066 and consequence of mulap0bd 8065. (Contributed by Jim Kingdon, 24-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) # 0)       (𝜑𝐵 # 0)
 
Theoremmulcanapd 8069 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremmulcanap2d 8070 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremmulcanapad 8071 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 8069. (Contributed by Jim Kingdon, 21-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)    &   (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremmulcanap2ad 8072 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 8070. (Contributed by Jim Kingdon, 21-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)    &   (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶))       (𝜑𝐴 = 𝐵)
 
Theoremmulcanap 8073 Cancellation law for multiplication (full theorem form). (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremmulcanap2 8074 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremmulcanapi 8075 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0       ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)
 
Theoremmuleqadd 8076 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = (𝐴 + 𝐵) ↔ ((𝐴 − 1) · (𝐵 − 1)) = 1))
 
Theoremreceuap 8077* Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)
 
3.3.8  Division
 
Syntaxcdiv 8078 Extend class notation to include division.
class /
 
Definitiondf-div 8079* Define division. Theorem divmulap 8081 relates it to multiplication, and divclap 8084 and redivclap 8137 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
/ = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
 
Theoremdivvalap 8080* Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
 
Theoremdivmulap 8081 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴))
 
Theoremdivmulap2 8082 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐶 · 𝐵)))
 
Theoremdivmulap3 8083 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐵 · 𝐶)))
 
Theoremdivclap 8084 Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ)
 
Theoremrecclap 8085 Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ)
 
Theoremdivcanap2 8086 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
Theoremdivcanap1 8087 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
Theoremdiveqap0 8088 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))
 
Theoremdivap0b 8089 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0))
 
Theoremdivap0 8090 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / 𝐵) # 0)
 
Theoremrecap0 8091 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) # 0)
 
Theoremrecidap 8092 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremrecidap2 8093 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1)
 
Theoremdivrecap 8094 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
 
Theoremdivrecap2 8095 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))
 
Theoremdivassap 8096 An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
 
Theoremdiv23ap 8097 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵))
 
Theoremdiv32ap 8098 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵)))
 
Theoremdiv13ap 8099 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴))
 
Theoremdiv12ap 8100 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶)))
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