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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | grplcan 18101 | Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | grpasscan1 18102 | An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑋) + 𝑌)) = 𝑌) | ||
Theorem | grpasscan2 18103 | An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋) | ||
Theorem | grpidrcan 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) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 + 𝑍) = 𝑋 ↔ 𝑍 = 0 )) | ||
Theorem | grpidlcan 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) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = 𝑋 ↔ 𝑍 = 0 )) | ||
Theorem | grpinvinv 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 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) | ||
Theorem | grpinvcnv 18107 | The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁) | ||
Theorem | grpinv11 18108 | The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | grpinvf1o 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 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵) | ||
Theorem | grpinvnz 18110 | The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑁‘𝑋) ≠ 0 ) | ||
Theorem | grpinvnzcl 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 = (0g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → (𝑁‘𝑋) ∈ (𝐵 ∖ { 0 })) | ||
Theorem | grpsubinv 18112 | Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + 𝑌)) | ||
Theorem | grplmulf1o 18113* | Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑋 + 𝑥)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵) | ||
Theorem | grpinvpropd 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.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿)) | ||
Theorem | grpidssd 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.) |
⊢ (𝜑 → 𝑀 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ Grp) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑀)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦)) ⇒ ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆)) | ||
Theorem | grpinvssd 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.) |
⊢ (𝜑 → 𝑀 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ Grp) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑀)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦)) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 → ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋))) | ||
Theorem | grpinvadd 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) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑁‘𝑋))) | ||
Theorem | grpsubf 18118 | Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → − :(𝐵 × 𝐵)⟶𝐵) | ||
Theorem | grpsubcl 18119 | Closure of group subtraction. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) ∈ 𝐵) | ||
Theorem | grpsubrcan 18120 | Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑍) = (𝑌 − 𝑍) ↔ 𝑋 = 𝑌)) | ||
Theorem | grpinvsub 18121 | Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) | ||
Theorem | grpinvval2 18122 | A df-neg 10862-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ( 0 − 𝑋)) | ||
Theorem | grpsubid 18123 | Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = 0 ) | ||
Theorem | grpsubid1 18124 | Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 0 ) = 𝑋) | ||
Theorem | grpsubeq0 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 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) | ||
Theorem | grpsubadd0sub 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 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ( 0 − 𝑌))) | ||
Theorem | grpsubadd 18127 | Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋)) | ||
Theorem | grpsubsub 18128 | Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 − (𝑌 − 𝑍)) = (𝑋 + (𝑍 − 𝑌))) | ||
Theorem | grpaddsubass 18129 | Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑍) = (𝑋 + (𝑌 − 𝑍))) | ||
Theorem | grppncan 18130 | Cancellation law for subtraction (pncan 10881 analog). (Contributed by NM, 16-Apr-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑌) = 𝑋) | ||
Theorem | grpnpcan 18131 | Cancellation law for subtraction (npcan 10884 analog). (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) + 𝑌) = 𝑋) | ||
Theorem | grpsubsub4 18132 | Double group subtraction (subsub4 10908 analog). (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑍 + 𝑌))) | ||
Theorem | grppnpcan2 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) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) − (𝑌 + 𝑍)) = (𝑋 − 𝑌)) | ||
Theorem | grpnpncan 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) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) + (𝑌 − 𝑍)) = (𝑋 − 𝑍)) | ||
Theorem | grpnpncan0 18135 | Cancellation law for group subtraction (npncan2 10902 analog). (Contributed by AV, 24-Nov-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 − 𝑌) + (𝑌 − 𝑋)) = 0 ) | ||
Theorem | grpnnncan2 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) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑍) − (𝑌 − 𝑍)) = (𝑋 − 𝑌)) | ||
Theorem | dfgrp3lem 18137* | Lemma for dfgrp3 18138. (Contributed by AV, 28-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑢)) | ||
Theorem | dfgrp3 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) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))) | ||
Theorem | dfgrp3e 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) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))) | ||
Theorem | grplactfval 18140* | The left group action of element 𝐴 of group 𝐺. (Contributed by Paul Chapman, 18-Mar-2008.) |
⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))) | ||
Theorem | grplactval 18141* | The value of the left group action of element 𝐴 of group 𝐺 at 𝐵. (Contributed by Paul Chapman, 18-Mar-2008.) |
⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐹‘𝐴)‘𝐵) = (𝐴 + 𝐵)) | ||
Theorem | grplactcnv 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 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡(𝐹‘𝐴) = (𝐹‘(𝐼‘𝐴)))) | ||
Theorem | grplactf1o 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 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴):𝑋–1-1-onto→𝑋) | ||
Theorem | grpsubpropd 18144 | Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.) |
⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) ⇒ ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻)) | ||
Theorem | grpsubpropd2 18145* | Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐻)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦)) ⇒ ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻)) | ||
Theorem | grp1 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〉}〉} ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) | ||
Theorem | grp1inv 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〉}〉} ⇒ ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀) = ( I ↾ {𝐼})) | ||
Theorem | prdsinvlem 18148* | Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 0 = (0g ∘ 𝑅) & ⊢ 𝑁 = (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝐹‘𝑦))) ⇒ ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ (𝑁 + 𝐹) = 0 )) | ||
Theorem | prdsgrpd 18149 | The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) ⇒ ⊢ (𝜑 → 𝑌 ∈ Grp) | ||
Theorem | prdsinvgd 18150* | Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑁 = (invg‘𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))) | ||
Theorem | pwsgrp 18151 | The product of a family of groups is a group. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Grp) | ||
Theorem | pwsinvg 18152 | Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑀 = (invg‘𝑅) & ⊢ 𝑁 = (invg‘𝑌) ⇒ ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑀 ∘ 𝑋)) | ||
Theorem | pwssub 18153 | Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑀 = (-g‘𝑅) & ⊢ − = (-g‘𝑌) ⇒ ⊢ (((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉) ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝐹 − 𝐺) = (𝐹 ∘f 𝑀𝐺)) | ||
Theorem | imasgrp2 18154* | The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧)))) & ⊢ (𝜑 → 0 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑁 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘ 0 )) ⇒ ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) | ||
Theorem | imasgrp 18155* | The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) | ||
Theorem | imasgrpf1 18156 | The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑈 = (𝐹 “s 𝑅) & ⊢ 𝑉 = (Base‘𝑅) ⇒ ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → 𝑈 ∈ Grp) | ||
Theorem | qusgrp2 18157* | Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑋) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) ∼ (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 0 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) ∼ 𝑥) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑁 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑁 + 𝑥) ∼ 0 ) ⇒ ⊢ (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0g‘𝑈))) | ||
Theorem | xpsgrp 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) | ||
Theorem | mhmlem 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.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵))) | ||
Theorem | mhmid 18160* | A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝐻)) | ||
Theorem | mhmmnd 18161* | The image of a monoid 𝐺 under a monoid homomorphism 𝐹 is a monoid. (Contributed by Thierry Arnoux, 25-Jan-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐻 ∈ Mnd) | ||
Theorem | mhmfmhm 18162* | The function fulfilling the conditions of mhmmnd 18161 is a monoid homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐺 MndHom 𝐻)) | ||
Theorem | ghmgrp 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.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → 𝐻 ∈ Grp) | ||
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 𝑥↑𝑛. 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. | ||
Syntax | cmg 18164 | Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group. |
class .g | ||
Definition | df-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, (0g‘𝑔), ⦋seq1((+g‘𝑔), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑔)‘(𝑠‘-𝑛)))))) | ||
Theorem | mulgfval 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) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) | ||
Theorem | mulgfvalALT 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) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) | ||
Theorem | mulgval 18168 | Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) ⇒ ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))))) | ||
Theorem | mulgfn 18169 | Functionality of the group multiple operation. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ · Fn (ℤ × 𝐵) | ||
Theorem | mulgfvi 18170 | The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ · = (.g‘𝐺) ⇒ ⊢ · = (.g‘( I ‘𝐺)) | ||
Theorem | mulg0 18171 | Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 ) | ||
Theorem | mulgnn 18172 | Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝑆‘𝑁)) | ||
Theorem | mulgnngsum 18173* | Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) | ||
Theorem | mulgnn0gsum 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) | ||
Theorem | mulg1 18175 | Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) | ||
Theorem | mulgnnp1 18176 | Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) | ||
Theorem | mulg2 18177 | Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝐵 → (2 · 𝑋) = (𝑋 + 𝑋)) | ||
Theorem | mulgnegnn 18178 | Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (-𝑁 · 𝑋) = (𝐼‘(𝑁 · 𝑋))) | ||
Theorem | mulgnn0p1 18179 | Group multiple (exponentiation) operation at a successor, extended to ℕ0. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) | ||
Theorem | mulgnnsubcl 18180* | Closure of the group multiple (exponentiation) operation in a subsemigroup. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) | ||
Theorem | mulgnn0subcl 18181* | Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 0 ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) | ||
Theorem | mulgsubcl 18182* | Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 0 ∈ 𝑆) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐼‘𝑥) ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) | ||
Theorem | mulgnncl 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) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) | ||
Theorem | mulgnn0cl 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) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) | ||
Theorem | mulgcl 18185 | Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) | ||
Theorem | mulgneg 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-𝑁 · 𝑋) = (𝐼‘(𝑁 · 𝑋))) | ||
Theorem | mulgnegneg 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(-𝑁 · 𝑋)) = (𝑁 · 𝑋)) | ||
Theorem | mulgm1 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (-1 · 𝑋) = (𝐼‘𝑋)) | ||
Theorem | mulgcld 18189 | Deduction associated with mulgcl 18185. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵) | ||
Theorem | mulgaddcomlem 18190 | Lemma for mulgaddcom 18191. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋))) | ||
Theorem | mulgaddcom 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) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋))) | ||
Theorem | mulginvcom 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · (𝐼‘𝑋)) = (𝐼‘(𝑁 · 𝑋))) | ||
Theorem | mulginvinv 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝑁 · (𝐼‘𝑋))) = (𝑁 · 𝑋)) | ||
Theorem | mulgnn0z 18194 | A group multiple of the identity, for nonnegative multiple. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) | ||
Theorem | mulgz 18195 | A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝑁 · 0 ) = 0 ) | ||
Theorem | mulgnndir 18196 | Sum of group multiples, for positive multiples. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Smgrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋))) | ||
Theorem | mulgnn0dir 18197 | Sum of group multiples, generalized to ℕ0. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋))) | ||
Theorem | mulgdirlem 18198 | Lemma for mulgdir 18199. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ (𝑀 + 𝑁) ∈ ℕ0) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋))) | ||
Theorem | mulgdir 18199 | Sum of group multiples, generalized to ℤ. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋))) | ||
Theorem | mulgp1 18200 | Group multiple (exponentiation) operation at a successor, extended to ℤ. (Contributed by Mario Carneiro, 11-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
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