MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpinvval2 Structured version   Visualization version   GIF version

Theorem grpinvval2 19017
Description: A df-neg 11497-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
grpsubcl.b 𝐵 = (Base‘𝐺)
grpsubcl.m = (-g𝐺)
grpinvsub.n 𝑁 = (invg𝐺)
grpinvval2.z 0 = (0g𝐺)
Assertion
Ref Expression
grpinvval2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) = ( 0 𝑋))

Proof of Theorem grpinvval2
StepHypRef Expression
1 grpsubcl.b . . . 4 𝐵 = (Base‘𝐺)
2 grpinvval2.z . . . 4 0 = (0g𝐺)
31, 2grpidcl 18960 . . 3 (𝐺 ∈ Grp → 0𝐵)
4 eqid 2726 . . . 4 (+g𝐺) = (+g𝐺)
5 grpinvsub.n . . . 4 𝑁 = (invg𝐺)
6 grpsubcl.m . . . 4 = (-g𝐺)
71, 4, 5, 6grpsubval 18980 . . 3 (( 0𝐵𝑋𝐵) → ( 0 𝑋) = ( 0 (+g𝐺)(𝑁𝑋)))
83, 7sylan 578 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 𝑋) = ( 0 (+g𝐺)(𝑁𝑋)))
91, 5grpinvcl 18982 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
101, 4, 2grplid 18962 . . 3 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝑁𝑋)) = (𝑁𝑋))
119, 10syldan 589 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)(𝑁𝑋)) = (𝑁𝑋))
128, 11eqtr2d 2767 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) = ( 0 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cfv 6554  (class class class)co 7424  Basecbs 17213  +gcplusg 17266  0gc0g 17454  Grpcgrp 18928  invgcminusg 18929  -gcsg 18930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-0g 17456  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-minusg 18932  df-sbg 18933
This theorem is referenced by:  grpsubadd0sub  19021  odm1inv  19551  matinvgcell  22428  istgp2  24086  nrmmetd  24574  nminv  24621
  Copyright terms: Public domain W3C validator