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Theorem grpnpcan 19002
Description: Cancellation law for subtraction (npcan 11509 analog). (Contributed by NM, 19-Apr-2014.)
Hypotheses
Ref Expression
grpsubadd.b 𝐵 = (Base‘𝐺)
grpsubadd.p + = (+g𝐺)
grpsubadd.m = (-g𝐺)
Assertion
Ref Expression
grpnpcan ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)

Proof of Theorem grpnpcan
StepHypRef Expression
1 grpsubadd.b . . . . . 6 𝐵 = (Base‘𝐺)
2 eqid 2728 . . . . . 6 (invg𝐺) = (invg𝐺)
31, 2grpinvcl 18958 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘𝑌) ∈ 𝐵)
433adant2 1128 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((invg𝐺)‘𝑌) ∈ 𝐵)
5 grpsubadd.p . . . . 5 + = (+g𝐺)
61, 5grpcl 18912 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
74, 6syld3an3 1406 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
8 grpsubadd.m . . . 4 = (-g𝐺)
91, 5, 2, 8grpsubval 18956 . . 3 (((𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵) → ((𝑋 + ((invg𝐺)‘𝑌)) ((invg𝐺)‘𝑌)) = ((𝑋 + ((invg𝐺)‘𝑌)) + ((invg𝐺)‘((invg𝐺)‘𝑌))))
107, 4, 9syl2anc 582 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + ((invg𝐺)‘𝑌)) ((invg𝐺)‘𝑌)) = ((𝑋 + ((invg𝐺)‘𝑌)) + ((invg𝐺)‘((invg𝐺)‘𝑌))))
111, 5, 8grppncan 19001 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵) → ((𝑋 + ((invg𝐺)‘𝑌)) ((invg𝐺)‘𝑌)) = 𝑋)
124, 11syld3an3 1406 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + ((invg𝐺)‘𝑌)) ((invg𝐺)‘𝑌)) = 𝑋)
131, 5, 2, 8grpsubval 18956 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
14133adant1 1127 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
1514eqcomd 2734 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((invg𝐺)‘𝑌)) = (𝑋 𝑌))
161, 2grpinvinv 18976 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘((invg𝐺)‘𝑌)) = 𝑌)
17163adant2 1128 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((invg𝐺)‘((invg𝐺)‘𝑌)) = 𝑌)
1815, 17oveq12d 7444 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + ((invg𝐺)‘𝑌)) + ((invg𝐺)‘((invg𝐺)‘𝑌))) = ((𝑋 𝑌) + 𝑌))
1910, 12, 183eqtr3rd 2777 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  cfv 6553  (class class class)co 7426  Basecbs 17189  +gcplusg 17242  Grpcgrp 18904  invgcminusg 18905  -gcsg 18906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-0g 17432  df-mgm 18609  df-sgrp 18688  df-mnd 18704  df-grp 18907  df-minusg 18908  df-sbg 18909
This theorem is referenced by:  grpsubsub4  19003  grpnpncan  19005  grpnnncan2  19007  dfgrp3  19009  xpsgrpsub  19031  nsgconj  19128  conjghm  19217  conjnmz  19220  sylow2blem1  19589  ablpncan3  19785  lmodvnpcan  20813  ipsubdir  21588  ipsubdi  21589  coe1subfv  22204  mdetunilem9  22550  subgntr  24039  ghmcnp  24047  tgpt0  24051  r1pid  26124  archiabllem1a  32928  archiabllem2a  32931  ornglmulle  33052  orngrmulle  33053  kercvrlsm  42556  hbtlem5  42601
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