Theorem grplidd 18907

Description: The identity element of a group is a left identity. Deduction associated with grplid 18905. (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 18905	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
7	1, 2, 6	syl2anc 584	1 ⊢ (𝜑 → ( 0 + 𝑋) = 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  +_gcplusg 17226  0_gc0g 17408  Grpcgrp 18871

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6515  df-fv 6521  df-riota 7346  df-ov 7392  df-0g 17410  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18874

This theorem is referenced by:  eqger  19116  conjnmz  19190  rngqiprngimfolem  21206  rngqiprngfulem5  21231  mhpaddcl  22044  r1pid2  26073  conjga  33133  erler  33222  rlocaddval  33225  rlocmulval  33226  rloccring  33227  rloc0g  33228  ofldchr  33298  qsnzr  33432  qsdrngilem  33471  ressply1evls1  33540  r1pid2OLD  33580  dimkerim  33629  rtelextdg2lem  33722  primrootspoweq0  42089  aks6d1c6lem5  42160  grpcominv1  42489