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Theorem grpnpncan0 19054
Description: Cancellation law for group subtraction (npncan2 11536 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 482 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝐺 ∈ Grp)
2 simprl 771 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
3 simprr 773 . . 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 19053 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑋𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = (𝑋 𝑋))
81, 2, 3, 2, 7syl13anc 1374 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = (𝑋 𝑋))
9 grpnpncan0.0 . . . 4 0 = (0g𝐺)
104, 9, 6grpsubid 19042 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = 0 )
1110adantrr 717 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑋) = 0 )
128, 11eqtrd 2777 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  -gcsg 18953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956
This theorem is referenced by:  cayhamlem1  22872
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