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Theorem ablpncan3 19849
Description: A cancellation law for Abelian 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 482 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → 𝐺 ∈ Abel)
2 simprl 771 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
3 ablgrp 19818 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
43adantr 480 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → 𝐺 ∈ Grp)
5 simprr 773 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
6 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
7 ablsubadd.m . . . . 5 = (-g𝐺)
86, 7grpsubcl 19051 . . . 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 19832 . . 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 19063 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑋𝐵) → ((𝑌 𝑋) + 𝑋) = 𝑌)
144, 5, 2, 13syl3anc 1370 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → ((𝑌 𝑋) + 𝑋) = 𝑌)
1512, 14eqtrd 2775 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  -gcsg 18966  Abelcabl 19814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-cmn 19815  df-abl 19816
This theorem is referenced by:  ablnnncan  19855  tsmsxplem2  24178  pjthlem2  25486
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