MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablnnncan1 Structured version   Visualization version   GIF version

Theorem ablnnncan1 19842
Description: Cancellation law for group subtraction. (nnncan1 11546 analog.) (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
ablnnncan1 (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑍 𝑌))

Proof of Theorem ablnnncan1
StepHypRef Expression
1 ablnncan.b . . 3 𝐵 = (Base‘𝐺)
2 ablnncan.m . . 3 = (-g𝐺)
3 ablnncan.g . . 3 (𝜑𝐺 ∈ Abel)
4 ablnncan.x . . 3 (𝜑𝑋𝐵)
5 ablnncan.y . . 3 (𝜑𝑌𝐵)
6 ablgrp 19804 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
73, 6syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
8 ablsub32.z . . . 4 (𝜑𝑍𝐵)
91, 2grpsubcl 19039 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
107, 4, 8, 9syl3anc 1372 . . 3 (𝜑 → (𝑋 𝑍) ∈ 𝐵)
111, 2, 3, 4, 5, 10ablsub32 19840 . 2 (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = ((𝑋 (𝑋 𝑍)) 𝑌))
121, 2, 3, 4, 8ablnncan 19839 . . 3 (𝜑 → (𝑋 (𝑋 𝑍)) = 𝑍)
1312oveq1d 7447 . 2 (𝜑 → ((𝑋 (𝑋 𝑍)) 𝑌) = (𝑍 𝑌))
1411, 13eqtrd 2776 1 (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑍 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cfv 6560  (class class class)co 7432  Basecbs 17248  Grpcgrp 18952  -gcsg 18954  Abelcabl 19800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956  df-sbg 18957  df-cmn 19801  df-abl 19802
This theorem is referenced by:  minveclem2  25461  ply1divmo  26176  baerlem3lem2  41713
  Copyright terms: Public domain W3C validator