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Theorem ablsubsub23 18945
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 486 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐺 ∈ Abel)
2 simpr3 1193 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
3 simpr2 1192 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐵𝑉)
4 ablsubsub23.v . . . . 5 𝑉 = (Base‘𝐺)
5 eqid 2824 . . . . 5 (+g𝐺) = (+g𝐺)
64, 5ablcom 18924 . . . 4 ((𝐺 ∈ Abel ∧ 𝐶𝑉𝐵𝑉) → (𝐶(+g𝐺)𝐵) = (𝐵(+g𝐺)𝐶))
71, 2, 3, 6syl3anc 1368 . . 3 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐶(+g𝐺)𝐵) = (𝐵(+g𝐺)𝐶))
87eqeq1d 2826 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐶(+g𝐺)𝐵) = 𝐴 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
9 ablgrp 18911 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
10 ablsubsub23.m . . . 4 = (-g𝐺)
114, 5, 10grpsubadd 18187 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐶(+g𝐺)𝐵) = 𝐴))
129, 11sylan 583 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐶(+g𝐺)𝐵) = 𝐴))
13 3ancomb 1096 . . . 4 ((𝐴𝑉𝐵𝑉𝐶𝑉) ↔ (𝐴𝑉𝐶𝑉𝐵𝑉))
1413biimpi 219 . . 3 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐴𝑉𝐶𝑉𝐵𝑉))
154, 5, 10grpsubadd 18187 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑉𝐶𝑉𝐵𝑉)) → ((𝐴 𝐶) = 𝐵 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
169, 14, 15syl2an 598 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐶) = 𝐵 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
178, 12, 163bitr4d 314 1 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐴 𝐶) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  Grpcgrp 18103  -gcsg 18105  Abelcabl 18907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-sbg 18108  df-cmn 18908  df-abl 18909
This theorem is referenced by: (None)
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