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Theorem grpsubrcan 18175
 Description: Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
grpsubcl.b 𝐵 = (Base‘𝐺)
grpsubcl.m = (-g𝐺)
Assertion
Ref Expression
grpsubrcan ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))

Proof of Theorem grpsubrcan
StepHypRef Expression
1 grpsubcl.b . . . . . 6 𝐵 = (Base‘𝐺)
2 eqid 2798 . . . . . 6 (+g𝐺) = (+g𝐺)
3 eqid 2798 . . . . . 6 (invg𝐺) = (invg𝐺)
4 grpsubcl.m . . . . . 6 = (-g𝐺)
51, 2, 3, 4grpsubval 18144 . . . . 5 ((𝑋𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
653adant2 1128 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
71, 2, 3, 4grpsubval 18144 . . . . 5 ((𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
873adant1 1127 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
96, 8eqeq12d 2814 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ (𝑋(+g𝐺)((invg𝐺)‘𝑍)) = (𝑌(+g𝐺)((invg𝐺)‘𝑍))))
109adantl 485 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ (𝑋(+g𝐺)((invg𝐺)‘𝑍)) = (𝑌(+g𝐺)((invg𝐺)‘𝑍))))
11 simpl 486 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
12 simpr1 1191 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
13 simpr2 1192 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
141, 3grpinvcl 18146 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
15143ad2antr3 1187 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
161, 2grprcan 18132 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋(+g𝐺)((invg𝐺)‘𝑍)) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)) ↔ 𝑋 = 𝑌))
1711, 12, 13, 15, 16syl13anc 1369 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(+g𝐺)((invg𝐺)‘𝑍)) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)) ↔ 𝑋 = 𝑌))
1810, 17bitrd 282 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  +gcplusg 16559  Grpcgrp 18097  invgcminusg 18098  -gcsg 18099 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7673  df-2nd 7674  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-sbg 18102 This theorem is referenced by:  abladdsub4  18930  ogrpsublt  30779
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