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Theorem List for Metamath Proof Explorer - 17301-17400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgrpasscan1 17301 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 17302 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 17303 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 17304 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 17305 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 17306 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 17307 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 17308 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 17309 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 17310 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 17311 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 17312* 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 17313* 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 17314* 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 17315* 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 17316 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 17317 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       (𝐺 ∈ Grp → :(𝐵 × 𝐵)⟶𝐵)

Theoremgrpsubcl 17318 Closure of group subtraction. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)

Theoremgrpsubrcan 17319 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 17320 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 17321 A df-neg 10148-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 17322 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 17323 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 17324 If the difference between two group elements is zero, they are equal. (subeq0 10186 analog.) (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) = 0𝑋 = 𝑌))

Theoremgrpsubadd0sub 17325 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 17326 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 17327 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 17328 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 17329 Cancellation law for subtraction (pncan 10166 analog). (Contributed by NM, 16-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑌) = 𝑋)

Theoremgrpnpcan 17330 Cancellation law for subtraction (npcan 10169 analog). (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)

Theoremgrpsubsub4 17331 Double group subtraction (subsub4 10193 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))

Theoremgrppnpcan2 17332 Cancellation law for mixed addition and subtraction. (pnpcan2 10200 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 17333 Cancellation law for group subtraction. (npncan 10181 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 17334 Cancellation law for group subtraction (npncan2 10187 analog). (Contributed by AV, 24-Nov-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = 0 )

Theoremgrpnnncan2 17335 Cancellation law for group subtraction. (nnncan2 10197 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 17336* Lemma for dfgrp3 17337. (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 17337* 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 17338* 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 17339* 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 17340* 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 17341* 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 17342* 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 17343 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 17344* 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 17345 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 17346 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 17347* 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 17348 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 17349* 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 17350 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 17351 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 17352 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 17353* 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 17354* 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 17355 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 17356* 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 17357 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 17358* Lemma for mhmmnd 17360 and ghmgrp 17362. (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 17359* 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 17360* 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 17361* The function fulfilling the conditions of mhmmnd 17360 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 17362* 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 17364, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires the definition df-minusg 17249 of the inverse operation invg.

Syntaxcmg 17363 Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group.
class .g

Definitiondf-mulg 17364* 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 17365* 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 17366 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 17367 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 17368 The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.)
· = (.g𝐺)        · = (.g‘( I ‘𝐺))

Theoremmulg0 17369 Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)       (𝑋𝐵 → (0 · 𝑋) = 0 )

Theoremmulgnn 17370 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 17371 Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       (𝑋𝐵 → (1 · 𝑋) = 𝑋)

Theoremmulgnnp1 17372 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 17373 Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       (𝑋𝐵 → (2 · 𝑋) = (𝑋 + 𝑋))

Theoremmulgnegnn 17374 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 17375 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 17376* 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 17377* 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 17378* 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 17379 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 17380 Obsolete proof of mulgnncl 17379 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 17381 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 17382 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 17383 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 17384 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 17385 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 17386 Lemma for mulgaddcom 17387. (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 17387 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 17388 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 17389 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 17390 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 17391 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 17392 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 17393 Obsolete proof of mulgnndir 17392 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 17394 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 17395 Lemma for mulgdir 17396. (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 17396 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 17397 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 17398 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 17399 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 17400 Obsolete proof of mulgnnass 17399 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)))

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