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Theorem grpidrcan 12981
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 12948 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + 0 ) = 𝑋)
543adant3 1019 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 + 0 ) = 𝑋)
65eqeq2d 2201 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) = (𝑋 + 0 ) ↔ (𝑋 + 𝑍) = 𝑋))
7 simp1 999 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝐺 ∈ Grp)
8 simp3 1001 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝑍𝐵)
91, 3grpidcl 12945 . . . 4 (𝐺 ∈ Grp → 0𝐵)
1093ad2ant1 1020 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 0𝐵)
11 simp2 1000 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝑋𝐵)
121, 2grplcan 12978 . . 3 ((𝐺 ∈ Grp ∧ (𝑍𝐵0𝐵𝑋𝐵)) → ((𝑋 + 𝑍) = (𝑋 + 0 ) ↔ 𝑍 = 0 ))
137, 8, 10, 11, 12syl13anc 1251 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) = (𝑋 + 0 ) ↔ 𝑍 = 0 ))
146, 13bitr3d 190 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) = 𝑋𝑍 = 0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  w3a 980   = wceq 1364  wcel 2160  cfv 5231  (class class class)co 5891  Basecbs 12486  +gcplusg 12561  0gc0g 12733  Grpcgrp 12917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-inn 8939  df-2 8997  df-ndx 12489  df-slot 12490  df-base 12492  df-plusg 12574  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-grp 12920  df-minusg 12921
This theorem is referenced by: (None)
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