MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpnpncan0 Structured version   Visualization version   GIF version

Theorem grpnpncan0 19067
Description: Cancellation law for group subtraction (npncan2 11534 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 19066 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑋𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = (𝑋 𝑋))
81, 2, 3, 2, 7syl13anc 1371 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = (𝑋 𝑋))
9 grpnpncan0.0 . . . 4 0 = (0g𝐺)
104, 9, 6grpsubid 19055 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = 0 )
1110adantrr 717 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑋) = 0 )
128, 11eqtrd 2775 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌) + (𝑌 𝑋)) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964  -gcsg 18966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969
This theorem is referenced by:  cayhamlem1  22888
  Copyright terms: Public domain W3C validator