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Theorem ablnncan 18049
 Description: Cancellation law for group subtraction. (nncan 10189 analog.) (Contributed by NM, 7-Apr-2015.)
Hypotheses
Ref Expression
ablnncan.b 𝐵 = (Base‘𝐺)
ablnncan.m = (-g𝐺)
ablnncan.g (𝜑𝐺 ∈ Abel)
ablnncan.x (𝜑𝑋𝐵)
ablnncan.y (𝜑𝑌𝐵)
Assertion
Ref Expression
ablnncan (𝜑 → (𝑋 (𝑋 𝑌)) = 𝑌)

Proof of Theorem ablnncan
StepHypRef Expression
1 ablnncan.b . . 3 𝐵 = (Base‘𝐺)
2 eqid 2610 . . 3 (+g𝐺) = (+g𝐺)
3 ablnncan.m . . 3 = (-g𝐺)
4 ablnncan.g . . 3 (𝜑𝐺 ∈ Abel)
5 ablnncan.x . . 3 (𝜑𝑋𝐵)
6 ablnncan.y . . 3 (𝜑𝑌𝐵)
71, 2, 3, 4, 5, 5, 6ablsubsub 18046 . 2 (𝜑 → (𝑋 (𝑋 𝑌)) = ((𝑋 𝑋)(+g𝐺)𝑌))
8 ablgrp 18021 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
94, 8syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
10 eqid 2610 . . . . 5 (0g𝐺) = (0g𝐺)
111, 10, 3grpsubid 17322 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
129, 5, 11syl2anc 691 . . 3 (𝜑 → (𝑋 𝑋) = (0g𝐺))
1312oveq1d 6564 . 2 (𝜑 → ((𝑋 𝑋)(+g𝐺)𝑌) = ((0g𝐺)(+g𝐺)𝑌))
141, 2, 10grplid 17275 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺)(+g𝐺)𝑌) = 𝑌)
159, 6, 14syl2anc 691 . 2 (𝜑 → ((0g𝐺)(+g𝐺)𝑌) = 𝑌)
167, 13, 153eqtrd 2648 1 (𝜑 → (𝑋 (𝑋 𝑌)) = 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245  -gcsg 17247  Abelcabl 18017 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-cmn 18018  df-abl 18019 This theorem is referenced by:  ablnnncan1  18052  pgpfac1lem3  18299  tsmsxplem1  21766  baerlem5blem2  36019
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