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Theorem grpinvid1 18876
Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
Hypotheses
Ref Expression
grpinv.b 𝐵 = (Base‘𝐺)
grpinv.p + = (+g𝐺)
grpinv.u 0 = (0g𝐺)
grpinv.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvid1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 ))

Proof of Theorem grpinvid1
StepHypRef Expression
1 oveq2 7417 . . . 4 ((𝑁𝑋) = 𝑌 → (𝑋 + (𝑁𝑋)) = (𝑋 + 𝑌))
21adantl 483 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑋 + (𝑁𝑋)) = (𝑋 + 𝑌))
3 grpinv.b . . . . . 6 𝐵 = (Base‘𝐺)
4 grpinv.p . . . . . 6 + = (+g𝐺)
5 grpinv.u . . . . . 6 0 = (0g𝐺)
6 grpinv.n . . . . . 6 𝑁 = (invg𝐺)
73, 4, 5, 6grprinv 18875 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
873adant3 1133 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
98adantr 482 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑋 + (𝑁𝑋)) = 0 )
102, 9eqtr3d 2775 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑋 + 𝑌) = 0 )
11 oveq2 7417 . . . 4 ((𝑋 + 𝑌) = 0 → ((𝑁𝑋) + (𝑋 + 𝑌)) = ((𝑁𝑋) + 0 ))
1211adantl 483 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + (𝑋 + 𝑌)) = ((𝑁𝑋) + 0 ))
133, 4, 5, 6grplinv 18874 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
1413oveq1d 7424 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ( 0 + 𝑌))
15143adant3 1133 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ( 0 + 𝑌))
163, 6grpinvcl 18872 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
1716adantrr 716 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑁𝑋) ∈ 𝐵)
18 simprl 770 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
19 simprr 772 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
2017, 18, 193jca 1129 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑁𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵))
213, 4grpass 18828 . . . . . . . 8 ((𝐺 ∈ Grp ∧ ((𝑁𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
2220, 21syldan 592 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
23223impb 1116 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
2415, 23eqtr3d 2775 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
253, 4, 5grplid 18852 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ( 0 + 𝑌) = 𝑌)
26253adant2 1132 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + 𝑌) = 𝑌)
2724, 26eqtr3d 2775 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + (𝑋 + 𝑌)) = 𝑌)
2827adantr 482 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + (𝑋 + 𝑌)) = 𝑌)
293, 4, 5grprid 18853 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3016, 29syldan 592 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
31303adant3 1133 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3231adantr 482 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3312, 28, 323eqtr3rd 2782 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → (𝑁𝑋) = 𝑌)
3410, 33impbida 800 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Grpcgrp 18819  invgcminusg 18820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-riota 7365  df-ov 7412  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823
This theorem is referenced by:  grpinvid  18884  grpinvcnv  18891  grpinvadd  18901  subginv  19013  qusinv  19069  ghminv  19099  symginv  19270  frgpinv  19632  cnaddinv  19739  ringnegl  20114  lmodindp1  20625  lmodvsinv2  20648  cnfldneg  20971  zringinvg  21035  mdetunilem6  22119  invrvald  22178  dchrinv  26764  baerlem3lem1  40578  rngmneg1  46666
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