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Theorem grpinvid2 19010
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 7438 . . . 4 ((𝑁𝑋) = 𝑌 → ((𝑁𝑋) + 𝑋) = (𝑌 + 𝑋))
21adantl 481 . . 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 19007 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
873adant3 1133 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
98adantr 480 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → ((𝑁𝑋) + 𝑋) = 0 )
102, 9eqtr3d 2779 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑌 + 𝑋) = 0 )
113, 6grpinvcl 19005 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
123, 4, 5grplid 18985 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
1311, 12syldan 591 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
14133adant3 1133 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
1514eqcomd 2743 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑋) = ( 0 + (𝑁𝑋)))
1615adantr 480 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁𝑋) = ( 0 + (𝑁𝑋)))
17 oveq1 7438 . . . 4 ((𝑌 + 𝑋) = 0 → ((𝑌 + 𝑋) + (𝑁𝑋)) = ( 0 + (𝑁𝑋)))
1817adantl 481 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁𝑋)) = ( 0 + (𝑁𝑋)))
19 simprr 773 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
20 simprl 771 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
2111adantrr 717 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑁𝑋) ∈ 𝐵)
2219, 20, 213jca 1129 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑌𝐵𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵))
233, 4grpass 18960 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵)) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
2422, 23syldan 591 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
25243impb 1115 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
263, 4, 5, 6grprinv 19008 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
2726oveq2d 7447 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑌 + (𝑋 + (𝑁𝑋))) = (𝑌 + 0 ))
28273adant3 1133 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + (𝑋 + (𝑁𝑋))) = (𝑌 + 0 ))
293, 4, 5grprid 18986 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + 0 ) = 𝑌)
30293adant2 1132 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + 0 ) = 𝑌)
3125, 28, 303eqtrd 2781 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + 𝑋) + (𝑁𝑋)) = 𝑌)
3231adantr 480 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁𝑋)) = 𝑌)
3316, 18, 323eqtr2d 2783 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁𝑋) = 𝑌)
3410, 33impbida 801 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  invgcminusg 18952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955
This theorem is referenced by:  grpinvcnv  19024  grpsubeq0  19044  prdsinvgd  19069  xpsinv  19078  eqg0subg  19214  rngmneg2  20165  ringnegr  20300  islindf4  21858  psrneg  21979  pi1inv  25085  fldhmf1  42091  lindslinindimp2lem4  48378  lincresunit3  48398
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