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Theorem grpridd 18854
Description: The identity element of a group is a right identity. Deduction associated with grprid 18852. (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 18852 . 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 1541  wcel 2106  cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  0gc0g 17384  Grpcgrp 18818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-riota 7364  df-ov 7411  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821
This theorem is referenced by:  grprcan  18857  ablsubaddsub  19681  rnglidlmcl  46738
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