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Theorem grpridd 18891
Description: The identity element of a group is a right identity. Deduction associated with grprid 18889. (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 18889 . 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 1539  wcel 2104  cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  0gc0g 17389  Grpcgrp 18855
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-riota 7367  df-ov 7414  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858
This theorem is referenced by:  grprcan  18894  ablsubaddsub  19723  rnglidlmcl  20982
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