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Mirrors > Home > MPE Home > Th. List > mndlid | Structured version Visualization version GIF version |
Description: The identity element of a monoid is a left 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 |
---|---|
mndlid | ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 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 17517 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
5 | 4 | simpld 482 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ‘cfv 6030 (class class class)co 6795 Basecbs 16063 +gcplusg 16148 0gc0g 16307 Mndcmnd 17501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-iota 5993 df-fun 6032 df-fv 6038 df-riota 6756 df-ov 6798 df-0g 16309 df-mgm 17449 df-sgrp 17491 df-mnd 17502 |
This theorem is referenced by: issubmnd 17525 ress0g 17526 submnd0 17527 prdsidlem 17529 imasmnd 17535 0mhm 17565 mrcmndind 17573 gsumccat 17585 dfgrp2 17654 grplid 17659 dfgrp3 17721 mhmid 17743 mhmmnd 17744 mulgnn0p1 17759 mulgnn0z 17774 mulgnn0dir 17778 cntzsubm 17974 oppgmnd 17990 odmodnn0 18165 lsmub2x 18268 mulgnn0di 18437 gsumval3 18514 gsumzaddlem 18527 gsumzsplit 18533 srgbinomlem4 18750 dsmmacl 20301 mndvlid 20415 dmatmul 20520 mndifsplit 20659 tsmssplit 22174 omndmul2 30051 omndmul3 30052 slmd0vlid 30114 c0mgm 42434 c0mhm 42435 c0snmgmhm 42439 cznrng 42480 mndpsuppss 42677 |
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