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Theorem ablpncan2 19666
Description: Cancellation law for subtraction in an Abelian group. (Contributed by NM, 2-Oct-2014.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
Assertion
Ref Expression
ablpncan2 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑋) = 𝑌)

Proof of Theorem ablpncan2
StepHypRef Expression
1 simp1 1137 . . 3 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Abel)
2 simp2 1138 . . 3 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 simp3 1139 . . 3 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
4 ablsubadd.b . . . 4 𝐵 = (Base‘𝐺)
5 ablsubadd.p . . . 4 + = (+g𝐺)
6 ablsubadd.m . . . 4 = (-g𝐺)
74, 5, 6abladdsub 19663 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑋𝐵)) → ((𝑋 + 𝑌) 𝑋) = ((𝑋 𝑋) + 𝑌))
81, 2, 3, 2, 7syl13anc 1373 . 2 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑋) = ((𝑋 𝑋) + 𝑌))
9 ablgrp 19637 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
101, 9syl 17 . . . 4 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Grp)
11 eqid 2733 . . . . 5 (0g𝐺) = (0g𝐺)
124, 11, 6grpsubid 18894 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
1310, 2, 12syl2anc 585 . . 3 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑋) = (0g𝐺))
1413oveq1d 7411 . 2 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑋) + 𝑌) = ((0g𝐺) + 𝑌))
154, 5, 11grplid 18839 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
1610, 3, 15syl2anc 585 . 2 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
178, 14, 163eqtrd 2777 1 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑋) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  cfv 6535  (class class class)co 7396  Basecbs 17131  +gcplusg 17184  0gc0g 17372  Grpcgrp 18806  -gcsg 18808  Abelcabl 19633
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7962  df-2nd 7963  df-0g 17374  df-mgm 18548  df-sgrp 18597  df-mnd 18613  df-grp 18809  df-minusg 18810  df-sbg 18811  df-cmn 19634  df-abl 19635
This theorem is referenced by:  lssvancl1  20532  lspprabs  20683  lsmcv  20731  ngpocelbl  24190  ttgcontlem1  28109
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