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Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrplcan 18101 Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))
 
Theoremgrpasscan1 18102 An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑋) + 𝑌)) = 𝑌)
 
Theoremgrpasscan2 18103 An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)
 
Theoremgrpidrcan 18104 If right adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) = 𝑋𝑍 = 0 ))
 
Theoremgrpidlcan 18105 If left adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑍 + 𝑋) = 𝑋𝑍 = 0 ))
 
Theoremgrpinvinv 18106 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁‘(𝑁𝑋)) = 𝑋)
 
Theoremgrpinvcnv 18107 The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       (𝐺 ∈ Grp → 𝑁 = 𝑁)
 
Theoremgrpinv11 18108 The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) = (𝑁𝑌) ↔ 𝑋 = 𝑌))
 
Theoremgrpinvf1o 18109 The group inverse is a one-to-one onto function. (Contributed by NM, 22-Oct-2014.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Grp)       (𝜑𝑁:𝐵1-1-onto𝐵)
 
Theoremgrpinvnz 18110 The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑋0 ) → (𝑁𝑋) ≠ 0 )
 
Theoremgrpinvnzcl 18111 The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → (𝑁𝑋) ∈ (𝐵 ∖ { 0 }))
 
Theoremgrpsubinv 18112 Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 (𝑁𝑌)) = (𝑋 + 𝑌))
 
Theoremgrplmulf1o 18113* Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐹 = (𝑥𝐵 ↦ (𝑋 + 𝑥))       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → 𝐹:𝐵1-1-onto𝐵)
 
Theoremgrpinvpropd 18114* If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (invg𝐾) = (invg𝐿))
 
Theoremgrpidssd 18115* If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then both groups have the same identity element. (Contributed by AV, 15-Mar-2019.)
(𝜑𝑀 ∈ Grp)    &   (𝜑𝑆 ∈ Grp)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐵 ⊆ (Base‘𝑀))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (0g𝑀) = (0g𝑆))
 
Theoremgrpinvssd 18116* If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the elements of the first group have the same inverses in both groups. (Contributed by AV, 15-Mar-2019.)
(𝜑𝑀 ∈ Grp)    &   (𝜑𝑆 ∈ Grp)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐵 ⊆ (Base‘𝑀))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (𝑋𝐵 → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
 
Theoremgrpinvadd 18117 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)))
 
Theoremgrpsubf 18118 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       (𝐺 ∈ Grp → :(𝐵 × 𝐵)⟶𝐵)
 
Theoremgrpsubcl 18119 Closure of group subtraction. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
 
Theoremgrpsubrcan 18120 Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))
 
Theoremgrpinvsub 18121 Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
 
Theoremgrpinvval2 18122 A df-neg 10862-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   𝑁 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) = ( 0 𝑋))
 
Theoremgrpsubid 18123 Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = 0 )
 
Theoremgrpsubid1 18124 Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 0 ) = 𝑋)
 
Theoremgrpsubeq0 18125 If the difference between two group elements is zero, they are equal. (subeq0 10901 analog.) (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) = 0𝑋 = 𝑌))
 
Theoremgrpsubadd0sub 18126 Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ( 0 𝑌)))
 
Theoremgrpsubadd 18127 Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))
 
Theoremgrpsubsub 18128 Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑋 + (𝑍 𝑌)))
 
Theoremgrpaddsubass 18129 Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = (𝑋 + (𝑌 𝑍)))
 
Theoremgrppncan 18130 Cancellation law for subtraction (pncan 10881 analog). (Contributed by NM, 16-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑌) = 𝑋)
 
Theoremgrpnpcan 18131 Cancellation law for subtraction (npcan 10884 analog). (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)
 
Theoremgrpsubsub4 18132 Double group subtraction (subsub4 10908 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))
 
Theoremgrppnpcan2 18133 Cancellation law for mixed addition and subtraction. (pnpcan2 10915 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) (𝑌 + 𝑍)) = (𝑋 𝑌))
 
Theoremgrpnpncan 18134 Cancellation law for group subtraction. (npncan 10896 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) + (𝑌 𝑍)) = (𝑋 𝑍))
 
Theoremgrpnpncan0 18135 Cancellation law for group subtraction (npncan2 10902 analog). (Contributed by AV, 24-Nov-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = 0 )
 
Theoremgrpnnncan2 18136 Cancellation law for group subtraction. (nnncan2 10912 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 𝑌))
 
Theoremdfgrp3lem 18137* Lemma for dfgrp3 18138. (Contributed by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
 
Theoremdfgrp3 18138* Alternate definition of a group as semigroup (with at least one element) which is also a quasigroup, i.e. a magma in which solutions 𝑥 and 𝑦 of the equations (𝑎 + 𝑥) = 𝑏 and (𝑥 + 𝑎) = 𝑏 exist. Theorem 3.2 of [Bruck] p. 28. (Contributed by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)))
 
Theoremdfgrp3e 18139* Alternate definition of a group as a set with a closed, associative operation, for which solutions 𝑥 and 𝑦 of the equations (𝑎 + 𝑥) = 𝑏 and (𝑥 + 𝑎) = 𝑏 exist. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Grp ↔ (𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦))))
 
Theoremgrplactfval 18140* The left group action of element 𝐴 of group 𝐺. (Contributed by Paul Chapman, 18-Mar-2008.)
𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))    &   𝑋 = (Base‘𝐺)       (𝐴𝑋 → (𝐹𝐴) = (𝑎𝑋 ↦ (𝐴 + 𝑎)))
 
Theoremgrplactval 18141* The value of the left group action of element 𝐴 of group 𝐺 at 𝐵. (Contributed by Paul Chapman, 18-Mar-2008.)
𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))    &   𝑋 = (Base‘𝐺)       ((𝐴𝑋𝐵𝑋) → ((𝐹𝐴)‘𝐵) = (𝐴 + 𝐵))
 
Theoremgrplactcnv 18142* The left group action of element 𝐴 of group 𝐺 maps the underlying set 𝑋 of 𝐺 one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝐹𝐴):𝑋1-1-onto𝑋(𝐹𝐴) = (𝐹‘(𝐼𝐴))))
 
Theoremgrplactf1o 18143* The left group action of element 𝐴 of group 𝐺 maps the underlying set 𝑋 of 𝐺 one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹𝐴):𝑋1-1-onto𝑋)
 
Theoremgrpsubpropd 18144 Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)
(𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   (𝜑 → (+g𝐺) = (+g𝐻))       (𝜑 → (-g𝐺) = (-g𝐻))
 
Theoremgrpsubpropd2 18145* Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐵 = (Base‘𝐻))    &   (𝜑𝐺 ∈ Grp)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))       (𝜑 → (-g𝐺) = (-g𝐻))
 
Theoremgrp1 18146 The (smallest) structure representing a trivial group. According to Wikipedia ("Trivial group", 28-Apr-2019, https://en.wikipedia.org/wiki/Trivial_group) "In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element". (Contributed by AV, 28-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉𝑀 ∈ Grp)
 
Theoremgrp1inv 18147 The inverse function of the trivial group. (Contributed by FL, 21-Jun-2010.) (Revised by AV, 26-Aug-2021.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉 → (invg𝑀) = ( I ↾ {𝐼}))
 
Theoremprdsinvlem 18148* Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Grp)    &   (𝜑𝐹𝐵)    &    0 = (0g𝑅)    &   𝑁 = (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝐹𝑦)))       (𝜑 → (𝑁𝐵 ∧ (𝑁 + 𝐹) = 0 ))
 
Theoremprdsgrpd 18149 The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)       (𝜑𝑌 ∈ Grp)
 
Theoremprdsinvgd 18150* Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)    &   𝐵 = (Base‘𝑌)    &   𝑁 = (invg𝑌)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = (𝑥𝐼 ↦ ((invg‘(𝑅𝑥))‘(𝑋𝑥))))
 
Theorempwsgrp 18151 The product of a family of groups is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Grp ∧ 𝐼𝑉) → 𝑌 ∈ Grp)
 
Theorempwsinvg 18152 Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑀 = (invg𝑅)    &   𝑁 = (invg𝑌)       ((𝑅 ∈ Grp ∧ 𝐼𝑉𝑋𝐵) → (𝑁𝑋) = (𝑀𝑋))
 
Theorempwssub 18153 Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑀 = (-g𝑅)    &    = (-g𝑌)       (((𝑅 ∈ Grp ∧ 𝐼𝑉) ∧ (𝐹𝐵𝐺𝐵)) → (𝐹 𝐺) = (𝐹f 𝑀𝐺))
 
Theoremimasgrp2 18154* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅𝑊)    &   ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))    &   (𝜑0𝑉)    &   ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))    &   ((𝜑𝑥𝑉) → 𝑁𝑉)    &   ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹0 ))       (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
 
Theoremimasgrp 18155* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅 ∈ Grp)    &    0 = (0g𝑅)       (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
 
Theoremimasgrpf1 18156 The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑈 = (𝐹s 𝑅)    &   𝑉 = (Base‘𝑅)       ((𝐹:𝑉1-1𝐵𝑅 ∈ Grp) → 𝑈 ∈ Grp)
 
Theoremqusgrp2 18157* Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑 Er 𝑉)    &   (𝜑𝑅𝑋)    &   (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 + 𝑏) (𝑝 + 𝑞)))    &   ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) + 𝑧) (𝑥 + (𝑦 + 𝑧)))    &   (𝜑0𝑉)    &   ((𝜑𝑥𝑉) → ( 0 + 𝑥) 𝑥)    &   ((𝜑𝑥𝑉) → 𝑁𝑉)    &   ((𝜑𝑥𝑉) → (𝑁 + 𝑥) 0 )       (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] = (0g𝑈)))
 
Theoremxpsgrp 18158 The binary product of groups is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)       ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 ∈ Grp)
 
Theoremmhmlem 18159* Lemma for mhmmnd 18161 and ghmgrp 18163. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)
((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵)))
 
Theoremmhmid 18160* A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.)
((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Mnd)    &    0 = (0g𝐺)       (𝜑 → (𝐹0 ) = (0g𝐻))
 
Theoremmhmmnd 18161* The image of a monoid 𝐺 under a monoid homomorphism 𝐹 is a monoid. (Contributed by Thierry Arnoux, 25-Jan-2020.)
((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Mnd)       (𝜑𝐻 ∈ Mnd)
 
Theoremmhmfmhm 18162* The function fulfilling the conditions of mhmmnd 18161 is a monoid homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Mnd)       (𝜑𝐹 ∈ (𝐺 MndHom 𝐻))
 
Theoremghmgrp 18163* The image of a group 𝐺 under a group homomorphism 𝐹 is a group. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator 𝑂 in our model) need not have any of the properties of a group as a prerequisite. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)
((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Grp)       (𝜑𝐻 ∈ Grp)
 
10.2.2  Group multiple operation

The "group multiple" operation (if the group is multiplicative, also called "group power" or "group exponentiation" operation), can be defined for arbitrary magmas, if the multiplier/exponent is a nonnegative integer. See also the definition in [Lang] p. 6, where an element 𝑥(of a monoid) to the power of a nonnegative integer 𝑛 is defined and denoted by 𝑥𝑛. Definition df-mulg 18165, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires the definition df-minusg 18047 of the inverse operation invg.

 
Syntaxcmg 18164 Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group.
class .g
 
Definitiondf-mulg 18165* Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g = (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g𝑔), seq1((+g𝑔), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑔)‘(𝑠‘-𝑛))))))
 
Theoremmulgfval 18166* Group multiple (exponentiation) operation. For a shorter proof using ax-rep 5182, see mulgfvalALT 18167. (Contributed by Mario Carneiro, 11-Dec-2014.) Remove dependency on ax-rep 5182. (Revised by Rohan Ridenour, 17-Aug-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐼 = (invg𝐺)    &    · = (.g𝐺)        · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
 
TheoremmulgfvalALT 18167* Shorter proof of mulgfval 18166 using ax-rep 5182. (Contributed by Mario Carneiro, 11-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐼 = (invg𝐺)    &    · = (.g𝐺)        · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
 
Theoremmulgval 18168 Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐼 = (invg𝐺)    &    · = (.g𝐺)    &   𝑆 = seq1( + , (ℕ × {𝑋}))       ((𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))))
 
Theoremmulgfn 18169 Functionality of the group multiple operation. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)        · Fn (ℤ × 𝐵)
 
Theoremmulgfvi 18170 The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.)
· = (.g𝐺)        · = (.g‘( I ‘𝐺))
 
Theoremmulg0 18171 Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)       (𝑋𝐵 → (0 · 𝑋) = 0 )
 
Theoremmulgnn 18172 Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    · = (.g𝐺)    &   𝑆 = seq1( + , (ℕ × {𝑋}))       ((𝑁 ∈ ℕ ∧ 𝑋𝐵) → (𝑁 · 𝑋) = (𝑆𝑁))
 
Theoremmulgnngsum 18173* Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)       ((𝑁 ∈ ℕ ∧ 𝑋𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
 
Theoremmulgnn0gsum 18174* Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)       ((𝑁 ∈ ℕ0𝑋𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
 
Theoremmulg1 18175 Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       (𝑋𝐵 → (1 · 𝑋) = 𝑋)
 
Theoremmulgnnp1 18176 Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝑁 ∈ ℕ ∧ 𝑋𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))
 
Theoremmulg2 18177 Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       (𝑋𝐵 → (2 · 𝑋) = (𝑋 + 𝑋))
 
Theoremmulgnegnn 18178 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝑁 ∈ ℕ ∧ 𝑋𝐵) → (-𝑁 · 𝑋) = (𝐼‘(𝑁 · 𝑋)))
 
Theoremmulgnn0p1 18179 Group multiple (exponentiation) operation at a successor, extended to 0. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0𝑋𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))
 
Theoremmulgnnsubcl 18180* Closure of the group multiple (exponentiation) operation in a subsemigroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑆𝐵)    &   ((𝜑𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)       ((𝜑𝑁 ∈ ℕ ∧ 𝑋𝑆) → (𝑁 · 𝑋) ∈ 𝑆)
 
Theoremmulgnn0subcl 18181* Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑆𝐵)    &   ((𝜑𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &    0 = (0g𝐺)    &   (𝜑0𝑆)       ((𝜑𝑁 ∈ ℕ0𝑋𝑆) → (𝑁 · 𝑋) ∈ 𝑆)
 
Theoremmulgsubcl 18182* Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑆𝐵)    &   ((𝜑𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &    0 = (0g𝐺)    &   (𝜑0𝑆)    &   𝐼 = (invg𝐺)    &   ((𝜑𝑥𝑆) → (𝐼𝑥) ∈ 𝑆)       ((𝜑𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) ∈ 𝑆)
 
Theoremmulgnncl 18183 Closure of the group multiple (exponentiation) operation for a positive multiplier in a magma. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵) → (𝑁 · 𝑋) ∈ 𝐵)
 
Theoremmulgnn0cl 18184 Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝑁 · 𝑋) ∈ 𝐵)
 
Theoremmulgcl 18185 Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝑁 · 𝑋) ∈ 𝐵)
 
Theoremmulgneg 18186 Group multiple (exponentiation) operation at a negative integer. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (-𝑁 · 𝑋) = (𝐼‘(𝑁 · 𝑋)))
 
Theoremmulgnegneg 18187 The inverse of a negative group multiple is the positive group multiple. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝐼‘(-𝑁 · 𝑋)) = (𝑁 · 𝑋))
 
Theoremmulgm1 18188 Group multiple (exponentiation) operation at negative one. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 20-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (-1 · 𝑋) = (𝐼𝑋))
 
Theoremmulgcld 18189 Deduction associated with mulgcl 18185. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁 · 𝑋) ∈ 𝐵)
 
Theoremmulgaddcomlem 18190 Lemma for mulgaddcom 18191. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋𝐵) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋)))
 
Theoremmulgaddcom 18191 The group multiple operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋)))
 
Theoremmulginvcom 18192 The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝑁 · (𝐼𝑋)) = (𝐼‘(𝑁 · 𝑋)))
 
Theoremmulginvinv 18193 The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝐼‘(𝑁 · (𝐼𝑋))) = (𝑁 · 𝑋))
 
Theoremmulgnn0z 18194 A group multiple of the identity, for nonnegative multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 )
 
Theoremmulgz 18195 A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝑁 · 0 ) = 0 )
 
Theoremmulgnndir 18196 Sum of group multiples, for positive multiples. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Smgrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))
 
Theoremmulgnn0dir 18197 Sum of group multiples, generalized to 0. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑋𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))
 
Theoremmulgdirlem 18198 Lemma for mulgdir 18199. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) ∧ (𝑀 + 𝑁) ∈ ℕ0) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))
 
Theoremmulgdir 18199 Sum of group multiples, generalized to . (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))
 
Theoremmulgp1 18200 Group multiple (exponentiation) operation at a successor, extended to . (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))
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