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Theorem grpnpncan0 17334
Description: Cancellation law for group subtraction (npncan2 10187 analog). (Contributed by AV, 24-Nov-2019.)
Hypotheses
Ref Expression
grpsubadd.b 𝐵 = (Base‘𝐺)
grpsubadd.p + = (+g𝐺)
grpsubadd.m = (-g𝐺)
grpnpncan0.0 0 = (0g𝐺)
Assertion
Ref Expression
grpnpncan0 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = 0 )

Proof of Theorem grpnpncan0
StepHypRef Expression
1 simpl 472 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝐺 ∈ Grp)
2 simprl 790 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
3 simprr 792 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
4 grpsubadd.b . . . 4 𝐵 = (Base‘𝐺)
5 grpsubadd.p . . . 4 + = (+g𝐺)
6 grpsubadd.m . . . 4 = (-g𝐺)
74, 5, 6grpnpncan 17333 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑋𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = (𝑋 𝑋))
81, 2, 3, 2, 7syl13anc 1320 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = (𝑋 𝑋))
9 grpnpncan0.0 . . . 4 0 = (0g𝐺)
104, 9, 6grpsubid 17322 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = 0 )
1110adantrr 749 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑋) = 0 )
128, 11eqtrd 2644 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245  -gcsg 17247
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
This theorem is referenced by:  cayhamlem1  20490
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