MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablpncan3 Structured version   Visualization version   GIF version

Theorem ablpncan3 19416
Description: A cancellation law for commutative groups. (Contributed by NM, 23-Mar-2015.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
Assertion
Ref Expression
ablpncan3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = 𝑌)

Proof of Theorem ablpncan3
StepHypRef Expression
1 simpl 483 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → 𝐺 ∈ Abel)
2 simprl 768 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
3 ablgrp 19389 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
43adantr 481 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → 𝐺 ∈ Grp)
5 simprr 770 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
6 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
7 ablsubadd.m . . . . 5 = (-g𝐺)
86, 7grpsubcl 18653 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
94, 5, 2, 8syl3anc 1370 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑌 𝑋) ∈ 𝐵)
10 ablsubadd.p . . . 4 + = (+g𝐺)
116, 10ablcom 19402 . . 3 ((𝐺 ∈ Abel ∧ 𝑋𝐵 ∧ (𝑌 𝑋) ∈ 𝐵) → (𝑋 + (𝑌 𝑋)) = ((𝑌 𝑋) + 𝑋))
121, 2, 9, 11syl3anc 1370 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = ((𝑌 𝑋) + 𝑋))
136, 10, 7grpnpcan 18665 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑋𝐵) → ((𝑌 𝑋) + 𝑋) = 𝑌)
144, 5, 2, 13syl3anc 1370 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → ((𝑌 𝑋) + 𝑋) = 𝑌)
1512, 14eqtrd 2778 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  cfv 6435  (class class class)co 7277  Basecbs 16910  +gcplusg 16960  Grpcgrp 18575  -gcsg 18577  Abelcabl 19385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-fv 6443  df-riota 7234  df-ov 7280  df-oprab 7281  df-mpo 7282  df-1st 7831  df-2nd 7832  df-0g 17150  df-mgm 18324  df-sgrp 18373  df-mnd 18384  df-grp 18578  df-minusg 18579  df-sbg 18580  df-cmn 19386  df-abl 19387
This theorem is referenced by:  ablnnncan  19422  tsmsxplem2  23303  pjthlem2  24600
  Copyright terms: Public domain W3C validator