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Theorem ablsubsub23 18053
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 472 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐺 ∈ Abel)
2 simpr3 1062 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
3 simpr2 1061 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐵𝑉)
4 ablsubsub23.v . . . . 5 𝑉 = (Base‘𝐺)
5 eqid 2610 . . . . 5 (+g𝐺) = (+g𝐺)
64, 5ablcom 18033 . . . 4 ((𝐺 ∈ Abel ∧ 𝐶𝑉𝐵𝑉) → (𝐶(+g𝐺)𝐵) = (𝐵(+g𝐺)𝐶))
71, 2, 3, 6syl3anc 1318 . . 3 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐶(+g𝐺)𝐵) = (𝐵(+g𝐺)𝐶))
87eqeq1d 2612 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐶(+g𝐺)𝐵) = 𝐴 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
9 ablgrp 18021 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
10 ablsubsub23.m . . . 4 = (-g𝐺)
114, 5, 10grpsubadd 17326 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐶(+g𝐺)𝐵) = 𝐴))
129, 11sylan 487 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐶(+g𝐺)𝐵) = 𝐴))
13 3ancomb 1040 . . . 4 ((𝐴𝑉𝐵𝑉𝐶𝑉) ↔ (𝐴𝑉𝐶𝑉𝐵𝑉))
1413biimpi 205 . . 3 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐴𝑉𝐶𝑉𝐵𝑉))
154, 5, 10grpsubadd 17326 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑉𝐶𝑉𝐵𝑉)) → ((𝐴 𝐶) = 𝐵 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
169, 14, 15syl2an 493 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐶) = 𝐵 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
178, 12, 163bitr4d 299 1 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐴 𝐶) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Grpcgrp 17245  -gcsg 17247  Abelcabl 18017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-cmn 18018  df-abl 18019
This theorem is referenced by: (None)
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