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Theorem List for Metamath Proof Explorer - 17201-17300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrpinvnz 17201 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 17202 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 17203 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 17204* 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 17205* 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 17206* 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 17207* 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 17208 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 17209 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       (𝐺 ∈ Grp → :(𝐵 × 𝐵)⟶𝐵)
 
Theoremgrpsubcl 17210 Closure of group subtraction. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
 
Theoremgrpsubrcan 17211 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 17212 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 17213 A df-neg 10020-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 17214 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 17215 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 17216 If the difference between two group elements is zero, they are equal. (subeq0 10058 analog.) (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) = 0𝑋 = 𝑌))
 
Theoremgrpsubadd0sub 17217 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 17218 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 17219 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 17220 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 17221 Cancellation law for subtraction (pncan 10038 analog). (Contributed by NM, 16-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑌) = 𝑋)
 
Theoremgrpnpcan 17222 Cancellation law for subtraction (npcan 10041 analog). (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)
 
Theoremgrpsubsub4 17223 Double group subtraction (subsub4 10065 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))
 
Theoremgrppnpcan2 17224 Cancellation law for mixed addition and subtraction. (pnpcan2 10072 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 17225 Cancellation law for group subtraction. (npncan 10053 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 17226 Cancellation law for group subtraction (npncan2 10059 analog). (Contributed by AV, 24-Nov-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = 0 )
 
Theoremgrpnnncan2 17227 Cancellation law for group subtraction. (nnncan2 10069 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 17228* Lemma for dfgrp3 17229. (Contributed by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
 
Theoremdfgrp3 17229* 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 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)))
 
Theoremdfgrp3e 17230* 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 17231* 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 17232* 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 17233* 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 17234* 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 17235 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 17236* 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 17237 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 17238 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 17239* 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 17240 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 17241* 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 17242 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 17243 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 17244 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 17245* 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 17246* 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 17247 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 17248* 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 17249 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 17250* Lemma for mhmmnd 17252 and ghmgrp 17254. (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 17251* 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 17252* 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 17253* The function fulfilling the conditions of mhmmnd 17252 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 17254* 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. df-mulg 17256, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires the definition df-minusg 17141 of the inverse operation invg.

 
Syntaxcmg 17255 Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group.
class .g
 
Definitiondf-mulg 17256* 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 17257* Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
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 17258 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 17259 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 17260 The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.)
· = (.g𝐺)        · = (.g‘( I ‘𝐺))
 
Theoremmulg0 17261 Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)       (𝑋𝐵 → (0 · 𝑋) = 0 )
 
Theoremmulgnn 17262 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))
 
Theoremmulg1 17263 Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       (𝑋𝐵 → (1 · 𝑋) = 𝑋)
 
Theoremmulgnnp1 17264 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 17265 Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       (𝑋𝐵 → (2 · 𝑋) = (𝑋 + 𝑋))
 
Theoremmulgnegnn 17266 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 17267 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 17268* 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 17269* 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 17270* 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 17271 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)
 
TheoremmulgnnclOLD 17272 Obsolete proof of mulgnncl 17271 as of 29-Aug-2021. Closure of the group multiple (exponentiation) operation. TODO: This can be generalized to a magma if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵) → (𝑁 · 𝑋) ∈ 𝐵)
 
Theoremmulgnn0cl 17273 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 17274 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 17275 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 17276 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 17277 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))
 
Theoremmulgaddcomlem 17278 Lemma for mulgaddcom 17279. (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 17279 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 17280 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 17281 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 17282 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 17283 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 17284 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 ∈ SGrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))
 
TheoremmulgnndirOLD 17285 Obsolete proof of mulgnndir 17284 as of 29-Aug-2021. Sum of group multiples, for positive multiples. TODO: This can be generalized to a semigroup if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))
 
Theoremmulgnn0dir 17286 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 17287 Lemma for mulgdir 17288. (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 17288 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 17289 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))
 
Theoremmulgneg2 17290 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (-𝑁 · 𝑋) = (𝑁 · (𝐼𝑋)))
 
Theoremmulgnnass 17291 Product of group multiples, for positive multiples in a semigroup. (Contributed by Mario Carneiro, 13-Dec-2014.) (Revised by AV, 29-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ SGrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))
 
TheoremmulgnnassOLD 17292 Obsolete proof of mulgnnass 17291 as of 29-Aug-2021. Product of group multiples, for positive multiples. TODO: This can be generalized to a semigroup if/when we introduce them. (Contributed by Mario Carneiro, 13-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))
 
Theoremmulgnn0ass 17293 Product of group multiples, generalized to 0. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑋𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))
 
Theoremmulgass 17294 Product of group multiples, generalized to . (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))
 
Theoremmulgassr 17295 Reversed product of group multiples. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑁 · 𝑀) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))
 
Theoremmulgmodid 17296 Casting out multiples of the identity element leaves the group multiple unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ (𝑋𝐵 ∧ (𝑀 · 𝑋) = 0 )) → ((𝑁 mod 𝑀) · 𝑋) = (𝑁 · 𝑋))
 
Theoremmulgsubdir 17297 Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑀𝑁) · 𝑋) = ((𝑀 · 𝑋) (𝑁 · 𝑋)))
 
Theoremmhmmulg 17298 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    × = (.g𝐻)       ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))
 
Theoremmulgpropd 17299* Two structures with the same group-nature have the same group multiple function. 𝐾 is expected to either be V (when strong equality is available) or 𝐵 (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
· = (.g𝐺)    &    × = (.g𝐻)    &   (𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐵 = (Base‘𝐻))    &   (𝜑𝐵𝐾)    &   ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) ∈ 𝐾)    &   ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))       (𝜑· = × )
 
Theoremsubmmulgcl 17300 Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.)
= (.g𝐺)       ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0𝑋𝑆) → (𝑁 𝑋) ∈ 𝑆)
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