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Theorem grpinvid2 18877
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
grpinvid2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 0 ))

Proof of Theorem grpinvid2
StepHypRef Expression
1 oveq1 7416 . . . 4 ((𝑁𝑋) = 𝑌 → ((𝑁𝑋) + 𝑋) = (𝑌 + 𝑋))
21adantl 483 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → ((𝑁𝑋) + 𝑋) = (𝑌 + 𝑋))
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, 6grplinv 18874 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
873adant3 1133 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
98adantr 482 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → ((𝑁𝑋) + 𝑋) = 0 )
102, 9eqtr3d 2775 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑌 + 𝑋) = 0 )
113, 6grpinvcl 18872 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
123, 4, 5grplid 18852 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
1311, 12syldan 592 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
14133adant3 1133 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
1514eqcomd 2739 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑋) = ( 0 + (𝑁𝑋)))
1615adantr 482 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁𝑋) = ( 0 + (𝑁𝑋)))
17 oveq1 7416 . . . 4 ((𝑌 + 𝑋) = 0 → ((𝑌 + 𝑋) + (𝑁𝑋)) = ( 0 + (𝑁𝑋)))
1817adantl 483 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁𝑋)) = ( 0 + (𝑁𝑋)))
19 simprr 772 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
20 simprl 770 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
2111adantrr 716 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑁𝑋) ∈ 𝐵)
2219, 20, 213jca 1129 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑌𝐵𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵))
233, 4grpass 18828 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵)) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
2422, 23syldan 592 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
25243impb 1116 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
263, 4, 5, 6grprinv 18875 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
2726oveq2d 7425 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑌 + (𝑋 + (𝑁𝑋))) = (𝑌 + 0 ))
28273adant3 1133 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + (𝑋 + (𝑁𝑋))) = (𝑌 + 0 ))
293, 4, 5grprid 18853 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + 0 ) = 𝑌)
30293adant2 1132 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + 0 ) = 𝑌)
3125, 28, 303eqtrd 2777 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + 𝑋) + (𝑁𝑋)) = 𝑌)
3231adantr 482 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁𝑋)) = 𝑌)
3316, 18, 323eqtr2d 2779 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁𝑋) = 𝑌)
3410, 33impbida 800 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 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:  grpinvcnv  18891  grpsubeq0  18909  prdsinvgd  18934  xpsinv  18943  eqg0subg  19073  ringnegr  20115  islindf4  21393  psrneg  21520  pi1inv  24568  fldhmf1  40955  rngmneg2  46667  lindslinindimp2lem4  47142  lincresunit3  47162
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