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Mirrors > Home > MPE Home > Th. List > mndrid | Structured version Visualization version GIF version |
Description: The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.) |
Ref | Expression |
---|---|
mndlrid.b | ⊢ 𝐵 = (Base‘𝐺) |
mndlrid.p | ⊢ + = (+g‘𝐺) |
mndlrid.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
mndrid | ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndlrid.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mndlrid.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | mndlrid.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
4 | 1, 2, 3 | mndlrid 17924 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
5 | 4 | simprd 498 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Mndcmnd 17905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-riota 7108 df-ov 7153 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 |
This theorem is referenced by: mndpfo 17928 issubmnd 17932 ress0g 17933 submnd0 17934 mndinvmod 17935 prdsidlem 17937 imasmnd 17943 mndind 17986 gsumccatOLD 17999 gsumccat 18000 grprid 18128 mhmid 18214 mhmmnd 18215 mulgnn0dir 18251 cntzsubm 18460 oppgmnd 18476 lsmub1x 18765 gsumval3 19021 gsumzsplit 19041 srgbinomlem3 19286 mndvrid 20999 mndifsplit 21239 gsummatr01 21262 smadiadet 21273 pmatcollpw3fi1lem1 21388 chfacfscmulgsum 21462 chfacfpmmulgsum 21466 tsmssplit 22754 tsmsxp 22757 gsummptres 30685 cntzsnid 30691 slmd0vrid 30846 |
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