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Theorem grpnnncan2 17277
Description: Cancellation law for group subtraction. (nnncan2 10165 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 471 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
2 simpr1 1059 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
3 simpr3 1061 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
4 grpnnncan2.b . . . . 5 𝐵 = (Base‘𝐺)
5 grpnnncan2.m . . . . 5 = (-g𝐺)
64, 5grpsubcl 17260 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
763adant3r1 1265 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
8 eqid 2605 . . . 4 (+g𝐺) = (+g𝐺)
94, 8, 5grpsubsub4 17273 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵 ∧ (𝑌 𝑍) ∈ 𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)))
101, 2, 3, 7, 9syl13anc 1319 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)))
114, 8, 5grpnpcan 17272 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → ((𝑌 𝑍)(+g𝐺)𝑍) = 𝑌)
12113adant3r1 1265 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍)(+g𝐺)𝑍) = 𝑌)
1312oveq2d 6539 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)) = (𝑋 𝑌))
1410, 13eqtrd 2639 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  cfv 5786  (class class class)co 6523  Basecbs 15637  +gcplusg 15710  Grpcgrp 17187  -gcsg 17189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-0g 15867  df-mgm 17007  df-sgrp 17049  df-mnd 17060  df-grp 17190  df-minusg 17191  df-sbg 17192
This theorem is referenced by:  2idlcpbl  18997  nrmmetd  22126  ttgcontlem1  25479
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