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

Theorem grpidlcan 18863
Description: If left adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)
Hypotheses
Ref Expression
grpidrcan.b 𝐵 = (Base‘𝐺)
grpidrcan.p + = (+g𝐺)
grpidrcan.o 0 = (0g𝐺)
Assertion
Ref Expression
grpidlcan ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑍 + 𝑋) = 𝑋𝑍 = 0 ))

Proof of Theorem grpidlcan
StepHypRef Expression
1 grpidrcan.b . . . . 5 𝐵 = (Base‘𝐺)
2 grpidrcan.p . . . . 5 + = (+g𝐺)
3 grpidrcan.o . . . . 5 0 = (0g𝐺)
41, 2, 3grplid 18827 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)
543adant3 1132 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ( 0 + 𝑋) = 𝑋)
65eqeq2d 2742 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ (𝑍 + 𝑋) = 𝑋))
7 simp1 1136 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝐺 ∈ Grp)
8 simp3 1138 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝑍𝐵)
91, 3grpidcl 18825 . . . 4 (𝐺 ∈ Grp → 0𝐵)
1093ad2ant1 1133 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 0𝐵)
11 simp2 1137 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝑋𝐵)
121, 2grprcan 18833 . . 3 ((𝐺 ∈ Grp ∧ (𝑍𝐵0𝐵𝑋𝐵)) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 ))
137, 8, 10, 11, 12syl13anc 1372 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 ))
146, 13bitr3d 280 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑍 + 𝑋) = 𝑋𝑍 = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1087   = wceq 1541  wcel 2106  cfv 6532  (class class class)co 7393  Basecbs 17126  +gcplusg 17179  0gc0g 17367  Grpcgrp 18794
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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-riota 7349  df-ov 7396  df-0g 17369  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-grp 18797
This theorem is referenced by:  grpidssd  18873
  Copyright terms: Public domain W3C validator