Theorem grplidd 19009

Description: The identity element of a group is a left identity. Deduction associated with grplid 19007. (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 19007	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
7	1, 2, 6	syl2anc 583	1 ⊢ (𝜑 → ( 0 + 𝑋) = 𝑋)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499  Grpcgrp 18973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-riota 7404  df-ov 7451  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976

This theorem is referenced by:  eqger  19218  conjnmz  19292  rngqiprngimfolem  21323  rngqiprngfulem5  21348  mhpaddcl  22178  r1pid2  26221  erler  33237  rlocaddval  33240  rlocmulval  33241  rloccring  33242  rloc0g  33243  ofldchr  33309  qsnzr  33448  qsdrngilem  33487  r1pid2OLD  33594  dimkerim  33640  rtelextdg2lem  33717  primrootspoweq0  42063  aks6d1c6lem5  42134  grpcominv1  42463