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Theorem ablpncan3 19736
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 768 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
3 ablgrp 19705 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
43adantr 480 . . . 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 18948 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
94, 5, 2, 8syl3anc 1368 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑌 𝑋) ∈ 𝐵)
10 ablsubadd.p . . . 4 + = (+g𝐺)
116, 10ablcom 19719 . . 3 ((𝐺 ∈ Abel ∧ 𝑋𝐵 ∧ (𝑌 𝑋) ∈ 𝐵) → (𝑋 + (𝑌 𝑋)) = ((𝑌 𝑋) + 𝑋))
121, 2, 9, 11syl3anc 1368 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = ((𝑌 𝑋) + 𝑋))
136, 10, 7grpnpcan 18960 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑋𝐵) → ((𝑌 𝑋) + 𝑋) = 𝑌)
144, 5, 2, 13syl3anc 1368 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → ((𝑌 𝑋) + 𝑋) = 𝑌)
1512, 14eqtrd 2766 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  Grpcgrp 18863  -gcsg 18865  Abelcabl 19701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-sbg 18868  df-cmn 19702  df-abl 19703
This theorem is referenced by:  ablnnncan  19742  tsmsxplem2  24013  pjthlem2  25321
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