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Theorem grpsubeq0 17862
Description: If the difference between two group elements is zero, they are equal. (subeq0 10635 analog.) (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
grpsubid.b 𝐵 = (Base‘𝐺)
grpsubid.o 0 = (0g𝐺)
grpsubid.m = (-g𝐺)
Assertion
Ref Expression
grpsubeq0 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) = 0𝑋 = 𝑌))

Proof of Theorem grpsubeq0
StepHypRef Expression
1 grpsubid.b . . . . 5 𝐵 = (Base‘𝐺)
2 eqid 2825 . . . . 5 (+g𝐺) = (+g𝐺)
3 eqid 2825 . . . . 5 (invg𝐺) = (invg𝐺)
4 grpsubid.m . . . . 5 = (-g𝐺)
51, 2, 3, 4grpsubval 17826 . . . 4 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)((invg𝐺)‘𝑌)))
653adant1 1164 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)((invg𝐺)‘𝑌)))
76eqeq1d 2827 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) = 0 ↔ (𝑋(+g𝐺)((invg𝐺)‘𝑌)) = 0 ))
8 simp1 1170 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Grp)
91, 3grpinvcl 17828 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘𝑌) ∈ 𝐵)
1093adant2 1165 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((invg𝐺)‘𝑌) ∈ 𝐵)
11 simp2 1171 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
12 grpsubid.o . . . 4 0 = (0g𝐺)
131, 2, 12, 3grpinvid2 17832 . . 3 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑌) ∈ 𝐵𝑋𝐵) → (((invg𝐺)‘((invg𝐺)‘𝑌)) = 𝑋 ↔ (𝑋(+g𝐺)((invg𝐺)‘𝑌)) = 0 ))
148, 10, 11, 13syl3anc 1494 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((invg𝐺)‘((invg𝐺)‘𝑌)) = 𝑋 ↔ (𝑋(+g𝐺)((invg𝐺)‘𝑌)) = 0 ))
151, 3grpinvinv 17843 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘((invg𝐺)‘𝑌)) = 𝑌)
16153adant2 1165 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((invg𝐺)‘((invg𝐺)‘𝑌)) = 𝑌)
1716eqeq1d 2827 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((invg𝐺)‘((invg𝐺)‘𝑌)) = 𝑋𝑌 = 𝑋))
18 eqcom 2832 . . 3 (𝑌 = 𝑋𝑋 = 𝑌)
1917, 18syl6bb 279 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((invg𝐺)‘((invg𝐺)‘𝑌)) = 𝑋𝑋 = 𝑌))
207, 14, 193bitr2d 299 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) = 0𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1111   = wceq 1656  wcel 2164  cfv 6127  (class class class)co 6910  Basecbs 16229  +gcplusg 16312  0gc0g 16460  Grpcgrp 17783  invgcminusg 17784  -gcsg 17785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-0g 16462  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-grp 17786  df-minusg 17787  df-sbg 17788
This theorem is referenced by:  ghmeqker  18045  ghmf1  18047  odcong  18326  subgdisj1  18462  dprdf11  18783  kerf1hrm  19106  lmodsubeq0  19285  lvecvscan2  19478  ip2eq  20367  mdetuni0  20802  tgphaus  22297  nrmmetd  22756  ply1divmo  24301  dvdsq1p  24326  dvdsr1p  24327  ply1remlem  24328  ig1peu  24337  dchr2sum  25418  eqlkr  35169  hdmap11  37918  hdmapinvlem4  37991  idomrootle  38611  lidldomn1  42782
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