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Theorem ablpncan2 18575
Description: Cancellation law for subtraction. (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 1172 . . 3 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Abel)
2 simp2 1173 . . 3 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 simp3 1174 . . 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 18574 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑋𝐵)) → ((𝑋 + 𝑌) 𝑋) = ((𝑋 𝑋) + 𝑌))
81, 2, 3, 2, 7syl13anc 1497 . 2 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑋) = ((𝑋 𝑋) + 𝑌))
9 ablgrp 18552 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
101, 9syl 17 . . . 4 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Grp)
11 eqid 2826 . . . . 5 (0g𝐺) = (0g𝐺)
124, 11, 6grpsubid 17854 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
1310, 2, 12syl2anc 581 . . 3 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑋) = (0g𝐺))
1413oveq1d 6921 . 2 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑋) + 𝑌) = ((0g𝐺) + 𝑌))
154, 5, 11grplid 17807 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
1610, 3, 15syl2anc 581 . 2 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
178, 14, 163eqtrd 2866 1 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑋) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1113   = wceq 1658  wcel 2166  cfv 6124  (class class class)co 6906  Basecbs 16223  +gcplusg 16306  0gc0g 16454  Grpcgrp 17777  -gcsg 17779  Abelcabl 18548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-0g 16456  df-mgm 17596  df-sgrp 17638  df-mnd 17649  df-grp 17780  df-minusg 17781  df-sbg 17782  df-cmn 18549  df-abl 18550
This theorem is referenced by:  lssvancl1  19302  lspprabs  19455  lsmcv  19502  ngpocelbl  22879  ttgcontlem1  26185
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