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Theorem grpinvid1 17391
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 6612 . . . 4 ((𝑁𝑋) = 𝑌 → (𝑋 + (𝑁𝑋)) = (𝑋 + 𝑌))
21adantl 482 . . 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 17390 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
873adant3 1079 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
98adantr 481 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑋 + (𝑁𝑋)) = 0 )
102, 9eqtr3d 2657 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑋 + 𝑌) = 0 )
11 oveq2 6612 . . . 4 ((𝑋 + 𝑌) = 0 → ((𝑁𝑋) + (𝑋 + 𝑌)) = ((𝑁𝑋) + 0 ))
1211adantl 482 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + (𝑋 + 𝑌)) = ((𝑁𝑋) + 0 ))
133, 4, 5, 6grplinv 17389 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
1413oveq1d 6619 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ( 0 + 𝑌))
15143adant3 1079 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ( 0 + 𝑌))
163, 6grpinvcl 17388 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
1716adantrr 752 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑁𝑋) ∈ 𝐵)
18 simprl 793 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
19 simprr 795 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
2017, 18, 193jca 1240 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑁𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵))
213, 4grpass 17352 . . . . . . . 8 ((𝐺 ∈ Grp ∧ ((𝑁𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
2220, 21syldan 487 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
23223impb 1257 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
2415, 23eqtr3d 2657 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
253, 4, 5grplid 17373 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ( 0 + 𝑌) = 𝑌)
26253adant2 1078 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + 𝑌) = 𝑌)
2724, 26eqtr3d 2657 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + (𝑋 + 𝑌)) = 𝑌)
2827adantr 481 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + (𝑋 + 𝑌)) = 𝑌)
293, 4, 5grprid 17374 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3016, 29syldan 487 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
31303adant3 1079 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3231adantr 481 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3312, 28, 323eqtr3rd 2664 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → (𝑁𝑋) = 𝑌)
3410, 33impbida 876 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Grpcgrp 17343  invgcminusg 17344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347
This theorem is referenced by:  grpinvid  17397  grpinvcnv  17404  grpinvadd  17414  subginv  17522  qusinv  17574  ghminv  17588  symginv  17743  frgpinv  18098  cnaddinv  18195  ringnegl  18515  lmodindp1  18933  lmodvsinv2  18956  cnfldneg  19691  zringinvg  19754  mdetunilem6  20342  invrvald  20401  dchrinv  24886  baerlem3lem1  36473
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