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Theorem ablsubsub23 19610
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 484 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐺 ∈ Abel)
2 simpr3 1197 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
3 simpr2 1196 . . . 4 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐵𝑉)
4 ablsubsub23.v . . . . 5 𝑉 = (Base‘𝐺)
5 eqid 2737 . . . . 5 (+g𝐺) = (+g𝐺)
64, 5ablcom 19588 . . . 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 2739 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐶(+g𝐺)𝐵) = 𝐴 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
9 ablgrp 19574 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
10 ablsubsub23.m . . . 4 = (-g𝐺)
114, 5, 10grpsubadd 18842 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐶(+g𝐺)𝐵) = 𝐴))
129, 11sylan 581 . 2 ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐶(+g𝐺)𝐵) = 𝐴))
13 3ancomb 1100 . . . 4 ((𝐴𝑉𝐵𝑉𝐶𝑉) ↔ (𝐴𝑉𝐶𝑉𝐵𝑉))
1413biimpi 215 . . 3 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐴𝑉𝐶𝑉𝐵𝑉))
154, 5, 10grpsubadd 18842 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑉𝐶𝑉𝐵𝑉)) → ((𝐴 𝐶) = 𝐵 ↔ (𝐵(+g𝐺)𝐶) = 𝐴))
169, 14, 15syl2an 597 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cfv 6501  (class class class)co 7362  Basecbs 17090  +gcplusg 17140  Grpcgrp 18755  -gcsg 18757  Abelcabl 19570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-sbg 18760  df-cmn 19571  df-abl 19572
This theorem is referenced by: (None)
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