MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpnnncan2 Structured version   Visualization version   GIF version

Theorem grpnnncan2 17828
Description: Cancellation law for group subtraction. (nnncan2 10610 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 475 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
2 simpr1 1249 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
3 simpr3 1253 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
4 grpnnncan2.b . . . . 5 𝐵 = (Base‘𝐺)
5 grpnnncan2.m . . . . 5 = (-g𝐺)
64, 5grpsubcl 17811 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
763adant3r1 1234 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
8 eqid 2799 . . . 4 (+g𝐺) = (+g𝐺)
94, 8, 5grpsubsub4 17824 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵 ∧ (𝑌 𝑍) ∈ 𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)))
101, 2, 3, 7, 9syl13anc 1492 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)))
114, 8, 5grpnpcan 17823 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → ((𝑌 𝑍)(+g𝐺)𝑍) = 𝑌)
12113adant3r1 1234 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍)(+g𝐺)𝑍) = 𝑌)
1312oveq2d 6894 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)) = (𝑋 𝑌))
1410, 13eqtrd 2833 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍) (𝑌 𝑍)) = (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  cfv 6101  (class class class)co 6878  Basecbs 16184  +gcplusg 16267  Grpcgrp 17738  -gcsg 17740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-0g 16417  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-grp 17741  df-minusg 17742  df-sbg 17743
This theorem is referenced by:  2idlcpbl  19557  nrmmetd  22707  ttgcontlem1  26122
  Copyright terms: Public domain W3C validator