Theorem grplidd 18945

Description: The identity element of a group is a left identity. Deduction associated with grplid 18943. (Contributed by SN, 29-Jan-2025.)

Hypotheses

Ref	Expression
grpbn0.b	⊢ 𝐵 = (Base‘𝐺)
grplid.p	⊢ + = (+g‘𝐺)
grplid.o	⊢ 0 = (0g‘𝐺)
grplidd.g	⊢ (𝜑 → 𝐺 ∈ Grp)
grplidd.1	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
grplidd	⊢ (𝜑 → ( 0 + 𝑋) = 𝑋)

Proof of Theorem grplidd

Step	Hyp	Ref	Expression
1		grplidd.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		grplidd.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		grpbn0.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
4		grplid.p	. . 3 ⊢ + = (+g‘𝐺)
5		grplid.o	. . 3 ⊢ 0 = (0g‘𝐺)
6	3, 4, 5	grplid 18943	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
7	1, 2, 6	syl2anc 585	1 ⊢ (𝜑 → ( 0 + 𝑋) = 𝑋)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  Grpcgrp 18909

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-riota 7324  df-ov 7370  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912

This theorem is referenced by:  eqger  19153  conjnmz  19227  rngqiprngimfolem  21288  rngqiprngfulem5  21313  ofldchr  21556  mhpaddcl  22117  r1pid2  26127  conjga  33231  erler  33326  rlocaddval  33329  rlocmulval  33330  rloccring  33331  rloc0g  33332  qsnzr  33515  qsdrngilem  33554  ressply1evls1  33625  r1pid2OLD  33669  mplgsum  33697  esplyind  33719  dimkerim  33771  rtelextdg2lem  33870  primrootspoweq0  42545  aks6d1c6lem5  42616  grpcominv1  42953