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Theorem ablpncan3 19834
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 19803 . . . . 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 19038 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
94, 5, 2, 8syl3anc 1373 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑌 𝑋) ∈ 𝐵)
10 ablsubadd.p . . . 4 + = (+g𝐺)
116, 10ablcom 19817 . . 3 ((𝐺 ∈ Abel ∧ 𝑋𝐵 ∧ (𝑌 𝑋) ∈ 𝐵) → (𝑋 + (𝑌 𝑋)) = ((𝑌 𝑋) + 𝑋))
121, 2, 9, 11syl3anc 1373 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = ((𝑌 𝑋) + 𝑋))
136, 10, 7grpnpcan 19050 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑋𝐵) → ((𝑌 𝑋) + 𝑋) = 𝑌)
144, 5, 2, 13syl3anc 1373 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → ((𝑌 𝑋) + 𝑋) = 𝑌)
1512, 14eqtrd 2777 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  -gcsg 18953  Abelcabl 19799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-cmn 19800  df-abl 19801
This theorem is referenced by:  ablnnncan  19840  tsmsxplem2  24162  pjthlem2  25472
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