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Theorem grpnpncan0 18994
Description: Cancellation law for group subtraction (npncan2 11515 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 481 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝐺 ∈ Grp)
2 simprl 769 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
3 simprr 771 . . 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 18993 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑋𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = (𝑋 𝑋))
81, 2, 3, 2, 7syl13anc 1369 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = (𝑋 𝑋))
9 grpnpncan0.0 . . . 4 0 = (0g𝐺)
104, 9, 6grpsubid 18982 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = 0 )
1110adantrr 715 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑋) = 0 )
128, 11eqtrd 2765 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cfv 6542  (class class class)co 7415  Basecbs 17177  +gcplusg 17230  0gc0g 17418  Grpcgrp 18892  -gcsg 18894
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-0g 17420  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18895  df-minusg 18896  df-sbg 18897
This theorem is referenced by:  cayhamlem1  22784
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