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Theorem ablsub32 19558
Description: Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.)
Hypotheses
Ref Expression
ablnncan.b 𝐵 = (Base‘𝐺)
ablnncan.m = (-g𝐺)
ablnncan.g (𝜑𝐺 ∈ Abel)
ablnncan.x (𝜑𝑋𝐵)
ablnncan.y (𝜑𝑌𝐵)
ablsub32.z (𝜑𝑍𝐵)
Assertion
Ref Expression
ablsub32 (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))

Proof of Theorem ablsub32
StepHypRef Expression
1 ablnncan.g . . . 4 (𝜑𝐺 ∈ Abel)
2 ablnncan.y . . . 4 (𝜑𝑌𝐵)
3 ablsub32.z . . . 4 (𝜑𝑍𝐵)
4 ablnncan.b . . . . 5 𝐵 = (Base‘𝐺)
5 eqid 2737 . . . . 5 (+g𝐺) = (+g𝐺)
64, 5ablcom 19539 . . . 4 ((𝐺 ∈ Abel ∧ 𝑌𝐵𝑍𝐵) → (𝑌(+g𝐺)𝑍) = (𝑍(+g𝐺)𝑌))
71, 2, 3, 6syl3anc 1371 . . 3 (𝜑 → (𝑌(+g𝐺)𝑍) = (𝑍(+g𝐺)𝑌))
87oveq2d 7367 . 2 (𝜑 → (𝑋 (𝑌(+g𝐺)𝑍)) = (𝑋 (𝑍(+g𝐺)𝑌)))
9 ablnncan.m . . 3 = (-g𝐺)
10 ablnncan.x . . 3 (𝜑𝑋𝐵)
114, 5, 9, 1, 10, 2, 3ablsubsub4 19555 . 2 (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌(+g𝐺)𝑍)))
124, 5, 9, 1, 10, 3, 2ablsubsub4 19555 . 2 (𝜑 → ((𝑋 𝑍) 𝑌) = (𝑋 (𝑍(+g𝐺)𝑌)))
138, 11, 123eqtr4d 2787 1 (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6493  (class class class)co 7351  Basecbs 17042  +gcplusg 17092  -gcsg 18709  Abelcabl 19521
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-0g 17282  df-mgm 18456  df-sgrp 18505  df-mnd 18516  df-grp 18710  df-minusg 18711  df-sbg 18712  df-cmn 19522  df-abl 19523
This theorem is referenced by:  ablnnncan1  19560  baerlem5alem2  40105
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