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Theorem grpsubrcan 17697
 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 2760 . . . . . 6 (+g𝐺) = (+g𝐺)
3 eqid 2760 . . . . . 6 (invg𝐺) = (invg𝐺)
4 grpsubcl.m . . . . . 6 = (-g𝐺)
51, 2, 3, 4grpsubval 17666 . . . . 5 ((𝑋𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
653adant2 1126 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 𝑍) = (𝑋(+g𝐺)((invg𝐺)‘𝑍)))
71, 2, 3, 4grpsubval 17666 . . . . 5 ((𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
873adant1 1125 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌 𝑍) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)))
96, 8eqeq12d 2775 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ (𝑋(+g𝐺)((invg𝐺)‘𝑍)) = (𝑌(+g𝐺)((invg𝐺)‘𝑍))))
109adantl 473 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ (𝑋(+g𝐺)((invg𝐺)‘𝑍)) = (𝑌(+g𝐺)((invg𝐺)‘𝑍))))
11 simpl 474 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
12 simpr1 1234 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
13 simpr2 1236 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
141, 3grpinvcl 17668 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
15143ad2antr3 1206 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
161, 2grprcan 17656 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋(+g𝐺)((invg𝐺)‘𝑍)) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)) ↔ 𝑋 = 𝑌))
1711, 12, 13, 15, 16syl13anc 1479 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(+g𝐺)((invg𝐺)‘𝑍)) = (𝑌(+g𝐺)((invg𝐺)‘𝑍)) ↔ 𝑋 = 𝑌))
1810, 17bitrd 268 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) = (𝑌 𝑍) ↔ 𝑋 = 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ‘cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  Grpcgrp 17623  invgcminusg 17624  -gcsg 17625 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-sbg 17628 This theorem is referenced by:  abladdsub4  18419  ogrpsublt  30031
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