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Theorem grpidrcan 19034
Description: If right 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
grpidrcan ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) = 𝑋𝑍 = 0 ))

Proof of Theorem grpidrcan
StepHypRef Expression
1 grpidrcan.b . . . . 5 𝐵 = (Base‘𝐺)
2 grpidrcan.p . . . . 5 + = (+g𝐺)
3 grpidrcan.o . . . . 5 0 = (0g𝐺)
41, 2, 3grprid 18999 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + 0 ) = 𝑋)
543adant3 1131 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 + 0 ) = 𝑋)
65eqeq2d 2746 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) = (𝑋 + 0 ) ↔ (𝑋 + 𝑍) = 𝑋))
7 simp1 1135 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝐺 ∈ Grp)
8 simp3 1137 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝑍𝐵)
91, 3grpidcl 18996 . . . 4 (𝐺 ∈ Grp → 0𝐵)
1093ad2ant1 1132 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 0𝐵)
11 simp2 1136 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝑋𝐵)
121, 2grplcan 19031 . . 3 ((𝐺 ∈ Grp ∧ (𝑍𝐵0𝐵𝑋𝐵)) → ((𝑋 + 𝑍) = (𝑋 + 0 ) ↔ 𝑍 = 0 ))
137, 8, 10, 11, 12syl13anc 1371 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) = (𝑋 + 0 ) ↔ 𝑍 = 0 ))
146, 13bitr3d 281 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) = 𝑋𝑍 = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968
This theorem is referenced by: (None)
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