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Theorem grpridd 18951
Description: The identity element of a group is a right identity. Deduction associated with grprid 18949. (Contributed by SN, 29-Jan-2025.)
Hypotheses
Ref Expression
grpbn0.b 𝐵 = (Base‘𝐺)
grplid.p + = (+g𝐺)
grplid.o 0 = (0g𝐺)
grplidd.g (𝜑𝐺 ∈ Grp)
grplidd.1 (𝜑𝑋𝐵)
Assertion
Ref Expression
grpridd (𝜑 → (𝑋 + 0 ) = 𝑋)

Proof of Theorem grpridd
StepHypRef Expression
1 grplidd.g . 2 (𝜑𝐺 ∈ Grp)
2 grplidd.1 . 2 (𝜑𝑋𝐵)
3 grpbn0.b . . 3 𝐵 = (Base‘𝐺)
4 grplid.p . . 3 + = (+g𝐺)
5 grplid.o . . 3 0 = (0g𝐺)
63, 4, 5grprid 18949 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + 0 ) = 𝑋)
71, 2, 6syl2anc 582 1 (𝜑 → (𝑋 + 0 ) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6549  (class class class)co 7419  Basecbs 17199  +gcplusg 17252  0gc0g 17440  Grpcgrp 18914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-riota 7375  df-ov 7422  df-0g 17442  df-mgm 18619  df-sgrp 18698  df-mnd 18714  df-grp 18917
This theorem is referenced by:  grprcan  18954  ghmqusnsglem1  19260  ablsubaddsub  19798  rnglidlmcl  21141  rloccring  33081  evl1deg1  33405  evl1deg3  33406  irredminply  33534  primrootscoprmpow  41721  primrootscoprbij  41724
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