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Theorem grpidrcan 12961
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 12928 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + 0 ) = 𝑋)
543adant3 1018 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 + 0 ) = 𝑋)
65eqeq2d 2199 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) = (𝑋 + 0 ) ↔ (𝑋 + 𝑍) = 𝑋))
7 simp1 998 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝐺 ∈ Grp)
8 simp3 1000 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝑍𝐵)
91, 3grpidcl 12925 . . . 4 (𝐺 ∈ Grp → 0𝐵)
1093ad2ant1 1019 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 0𝐵)
11 simp2 999 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → 𝑋𝐵)
121, 2grplcan 12958 . . 3 ((𝐺 ∈ Grp ∧ (𝑍𝐵0𝐵𝑋𝐵)) → ((𝑋 + 𝑍) = (𝑋 + 0 ) ↔ 𝑍 = 0 ))
137, 8, 10, 11, 12syl13anc 1250 . 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 979   = wceq 1363  wcel 2158  cfv 5228  (class class class)co 5888  Basecbs 12475  +gcplusg 12550  0gc0g 12722  Grpcgrp 12898
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-minusg 12902
This theorem is referenced by: (None)
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