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Theorem grpnnncan2 18460
Description: Cancellation law for group subtraction. (nnncan2 11115 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
grpnnncan2.b 𝐵 = (Base‘𝐺)
grpnnncan2.m = (-g𝐺)
Assertion
Ref Expression
grpnnncan2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 𝑌))

Proof of Theorem grpnnncan2
StepHypRef Expression
1 simpl 486 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
2 simpr1 1196 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
3 simpr3 1198 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
4 grpnnncan2.b . . . . 5 𝐵 = (Base‘𝐺)
5 grpnnncan2.m . . . . 5 = (-g𝐺)
64, 5grpsubcl 18443 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
763adant3r1 1184 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
8 eqid 2737 . . . 4 (+g𝐺) = (+g𝐺)
94, 8, 5grpsubsub4 18456 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵 ∧ (𝑌 𝑍) ∈ 𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)))
101, 2, 3, 7, 9syl13anc 1374 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)))
114, 8, 5grpnpcan 18455 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → ((𝑌 𝑍)(+g𝐺)𝑍) = 𝑌)
12113adant3r1 1184 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍)(+g𝐺)𝑍) = 𝑌)
1312oveq2d 7229 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)) = (𝑋 𝑌))
1410, 13eqtrd 2777 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  Grpcgrp 18365  -gcsg 18367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-minusg 18369  df-sbg 18370
This theorem is referenced by:  2idlcpbl  20272  nrmmetd  23472  ttgcontlem1  26976
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