MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpridd Structured version   Visualization version   GIF version

Theorem grpridd 19001
Description: The identity element of a group is a right identity. Deduction associated with grprid 18999. (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 18999 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + 0 ) = 𝑋)
71, 2, 6syl2anc 584 1 (𝜑 → (𝑋 + 0 ) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967
This theorem is referenced by:  grprcan  19004  ghmqusnsglem1  19311  ablsubaddsub  19847  rnglidlmcl  21244  rloccring  33257  evl1deg1  33581  evl1deg2  33582  evl1deg3  33583  irredminply  33722  rtelextdg2lem  33732  primrootscoprmpow  42081  primrootscoprbij  42084
  Copyright terms: Public domain W3C validator