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Theorem ablnnncan 19592
Description: Cancellation law for group subtraction. (nnncan 11432 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)
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
ablnnncan (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 𝑌))

Proof of Theorem ablnnncan
StepHypRef Expression
1 ablnncan.b . . 3 𝐵 = (Base‘𝐺)
2 eqid 2736 . . 3 (+g𝐺) = (+g𝐺)
3 ablnncan.m . . 3 = (-g𝐺)
4 ablnncan.g . . 3 (𝜑𝐺 ∈ Abel)
5 ablnncan.x . . 3 (𝜑𝑋𝐵)
6 ablgrp 19558 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
74, 6syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
8 ablnncan.y . . . 4 (𝜑𝑌𝐵)
9 ablsub32.z . . . 4 (𝜑𝑍𝐵)
101, 3grpsubcl 18818 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
117, 8, 9, 10syl3anc 1371 . . 3 (𝜑 → (𝑌 𝑍) ∈ 𝐵)
121, 2, 3, 4, 5, 11, 9ablsubsub4 19588 . 2 (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)))
131, 2ablcom 19572 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑌 𝑍) ∈ 𝐵𝑍𝐵) → ((𝑌 𝑍)(+g𝐺)𝑍) = (𝑍(+g𝐺)(𝑌 𝑍)))
144, 11, 9, 13syl3anc 1371 . . . 4 (𝜑 → ((𝑌 𝑍)(+g𝐺)𝑍) = (𝑍(+g𝐺)(𝑌 𝑍)))
151, 2, 3ablpncan3 19586 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑍𝐵𝑌𝐵)) → (𝑍(+g𝐺)(𝑌 𝑍)) = 𝑌)
164, 9, 8, 15syl12anc 835 . . . 4 (𝜑 → (𝑍(+g𝐺)(𝑌 𝑍)) = 𝑌)
1714, 16eqtrd 2776 . . 3 (𝜑 → ((𝑌 𝑍)(+g𝐺)𝑍) = 𝑌)
1817oveq2d 7369 . 2 (𝜑 → (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)) = (𝑋 𝑌))
1912, 18eqtrd 2776 1 (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6493  (class class class)co 7353  Basecbs 17075  +gcplusg 17125  Grpcgrp 18740  -gcsg 18742  Abelcabl 19554
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 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  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 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7917  df-2nd 7918  df-0g 17315  df-mgm 18489  df-sgrp 18538  df-mnd 18549  df-grp 18743  df-minusg 18744  df-sbg 18745  df-cmn 19555  df-abl 19556
This theorem is referenced by: (None)
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