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Theorem grpidlcan 17304
 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 17275 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)
543adant3 1074 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ( 0 + 𝑋) = 𝑋)
65eqeq2d 2620 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ (𝑍 + 𝑋) = 𝑋))
7 simp1 1054 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝐺 ∈ Grp)
8 simp3 1056 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝑍𝐵)
91, 3grpidcl 17273 . . . 4 (𝐺 ∈ Grp → 0𝐵)
1093ad2ant1 1075 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 0𝐵)
11 simp2 1055 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝑋𝐵)
121, 2grprcan 17278 . . 3 ((𝐺 ∈ Grp ∧ (𝑍𝐵0𝐵𝑋𝐵)) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 ))
137, 8, 10, 11, 12syl13anc 1320 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 ))
146, 13bitr3d 269 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑍 + 𝑋) = 𝑋𝑍 = 0 ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248 This theorem is referenced by:  grpidssd  17314
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