MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablsubsub23 Structured version   Visualization version   GIF version

Theorem ablsubsub23 19843
Description: Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.)
Hypotheses
Ref Expression
ablsubsub23.v 𝑉 = (Base‘𝐺)
ablsubsub23.m = (-g𝐺)
Assertion
Ref Expression
ablsubsub23 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐴 𝐶) = 𝐵))

Proof of Theorem ablsubsub23
StepHypRef Expression
1 simpl 482 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐺 ∈ Abel)
2 simpr3 1196 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
3 simpr2 1195 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐵𝑉)
4 ablsubsub23.v . . . . 5 𝑉 = (Base‘𝐺)
5 eqid 2736 . . . . 5 (+g𝐺) = (+g𝐺)
64, 5ablcom 19818 . . . 4 ((𝐺 ∈ Abel ∧ 𝐶𝑉𝐵𝑉) → (𝐶(+g𝐺)𝐵) = (𝐵(+g𝐺)𝐶))
71, 2, 3, 6syl3anc 1372 . . 3 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐶(+g𝐺)𝐵) = (𝐵(+g𝐺)𝐶))
87eqeq1d 2738 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐶(+g𝐺)𝐵) = 𝐴 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
9 ablgrp 19804 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
10 ablsubsub23.m . . . 4 = (-g𝐺)
114, 5, 10grpsubadd 19047 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐶(+g𝐺)𝐵) = 𝐴))
129, 11sylan 580 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐶(+g𝐺)𝐵) = 𝐴))
13 3ancomb 1098 . . . 4 ((𝐴𝑉𝐵𝑉𝐶𝑉) ↔ (𝐴𝑉𝐶𝑉𝐵𝑉))
1413biimpi 216 . . 3 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐴𝑉𝐶𝑉𝐵𝑉))
154, 5, 10grpsubadd 19047 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑉𝐶𝑉𝐵𝑉)) → ((𝐴 𝐶) = 𝐵 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
169, 14, 15syl2an 596 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐶) = 𝐵 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
178, 12, 163bitr4d 311 1 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐴 𝐶) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  Grpcgrp 18952  -gcsg 18954  Abelcabl 19800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956  df-sbg 18957  df-cmn 19801  df-abl 19802
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator