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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 18192 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) |
5 | 4 | simpld 498 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +gcplusg 16802 0gc0g 16944 Mndcmnd 18173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-riota 7170 df-ov 7216 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 |
This theorem is referenced by: issubmnd 18200 ress0g 18201 submnd0 18202 mndinvmod 18203 prdsidlem 18205 imasmnd 18211 0subm 18244 0mhm 18246 mndind 18254 gsumccatOLD 18267 gsumccat 18268 dfgrp2 18392 grplid 18397 dfgrp3 18462 mhmid 18484 mhmmnd 18485 mulgnn0p1 18503 mulgnn0z 18518 mulgnn0dir 18521 cntzsubm 18730 oppgmnd 18746 odmodnn0 18932 lsmub2x 19036 mulgnn0di 19211 gsumval3 19292 gsumzaddlem 19306 gsumzsplit 19312 srgbinomlem4 19558 dsmmacl 20703 mndvlid 21292 dmatmul 21394 mndifsplit 21533 tsmssplit 23049 cntzsnid 31040 omndmul2 31057 omndmul3 31058 slmd0vlid 31194 c0mgm 45140 c0mhm 45141 c0snmgmhm 45145 cznrng 45186 mndpsuppss 45380 mndtccatid 46045 |
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