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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 18766 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
| 5 | 4 | simprd 495 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 |
| This theorem is referenced by: mndpfo 18770 issubmnd 18774 ress0g 18775 submnd0 18776 mndinvmod 18777 prdsidlem 18782 imasmnd 18788 xpsmnd0 18791 mndvrid 18813 mndind 18841 gsumccat 18854 grprid 18986 mhmid 19081 mhmmnd 19082 mulgnn0dir 19122 cntzsubm 19356 oppgmnd 19373 lsmub1x 19664 gsumval3 19925 gsumzsplit 19945 srgbinomlem3 20225 mndifsplit 22642 gsummatr01 22665 smadiadet 22676 pmatcollpw3fi1lem1 22792 chfacfscmulgsum 22866 chfacfpmmulgsum 22870 tsmssplit 24160 tsmsxp 24163 mndlrinv 33029 mndractf1 33033 mndractfo 33034 mndlactf1o 33035 mndractf1o 33036 gsummptres 33055 gsummptres2 33056 cntzsnid 33072 slmd0vrid 33229 mndmolinv 42096 primrootscoprbij 42103 aks6d1c1 42117 aks6d1c2lem3 42127 mndtccatid 49184 |
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