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Theorem grpinvid2 17392
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 6611 . . . 4 ((𝑁𝑋) = 𝑌 → ((𝑁𝑋) + 𝑋) = (𝑌 + 𝑋))
21adantl 482 . . 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 17389 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
873adant3 1079 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
98adantr 481 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → ((𝑁𝑋) + 𝑋) = 0 )
102, 9eqtr3d 2657 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑌 + 𝑋) = 0 )
113, 6grpinvcl 17388 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
123, 4, 5grplid 17373 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
1311, 12syldan 487 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
14133adant3 1079 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
1514eqcomd 2627 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑋) = ( 0 + (𝑁𝑋)))
1615adantr 481 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁𝑋) = ( 0 + (𝑁𝑋)))
17 oveq1 6611 . . . 4 ((𝑌 + 𝑋) = 0 → ((𝑌 + 𝑋) + (𝑁𝑋)) = ( 0 + (𝑁𝑋)))
1817adantl 482 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁𝑋)) = ( 0 + (𝑁𝑋)))
19 simprr 795 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
20 simprl 793 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
2111adantrr 752 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑁𝑋) ∈ 𝐵)
2219, 20, 213jca 1240 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑌𝐵𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵))
233, 4grpass 17352 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵)) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
2422, 23syldan 487 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
25243impb 1257 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
263, 4, 5, 6grprinv 17390 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
2726oveq2d 6620 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑌 + (𝑋 + (𝑁𝑋))) = (𝑌 + 0 ))
28273adant3 1079 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + (𝑋 + (𝑁𝑋))) = (𝑌 + 0 ))
293, 4, 5grprid 17374 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + 0 ) = 𝑌)
30293adant2 1078 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + 0 ) = 𝑌)
3125, 28, 303eqtrd 2659 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + 𝑋) + (𝑁𝑋)) = 𝑌)
3231adantr 481 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁𝑋)) = 𝑌)
3316, 18, 323eqtr2d 2661 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁𝑋) = 𝑌)
3410, 33impbida 876 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 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:  grpinvcnv  17404  grpsubeq0  17422  prdsinvgd  17447  rngnegr  18516  psrneg  19319  islindf4  20096  pi1inv  22760  lindslinindimp2lem4  41538  lincresunit3  41558
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