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Theorem grpinvid2 18873
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 7412 . . . 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 18870 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
873adant3 1132 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
98adantr 481 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → ((𝑁𝑋) + 𝑋) = 0 )
102, 9eqtr3d 2774 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑌 + 𝑋) = 0 )
113, 6grpinvcl 18868 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
123, 4, 5grplid 18848 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
1311, 12syldan 591 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
14133adant3 1132 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
1514eqcomd 2738 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑋) = ( 0 + (𝑁𝑋)))
1615adantr 481 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁𝑋) = ( 0 + (𝑁𝑋)))
17 oveq1 7412 . . . 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 771 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
20 simprl 769 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
2111adantrr 715 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑁𝑋) ∈ 𝐵)
2219, 20, 213jca 1128 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑌𝐵𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵))
233, 4grpass 18824 . . . . . . 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 18871 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
2726oveq2d 7421 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑌 + (𝑋 + (𝑁𝑋))) = (𝑌 + 0 ))
28273adant3 1132 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + (𝑋 + (𝑁𝑋))) = (𝑌 + 0 ))
293, 4, 5grprid 18849 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + 0 ) = 𝑌)
30293adant2 1131 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + 0 ) = 𝑌)
3125, 28, 303eqtrd 2776 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + 𝑋) + (𝑁𝑋)) = 𝑌)
3231adantr 481 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁𝑋)) = 𝑌)
3316, 18, 323eqtr2d 2778 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁𝑋) = 𝑌)
3410, 33impbida 799 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  0gc0g 17381  Grpcgrp 18815  invgcminusg 18816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7361  df-ov 7408  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819
This theorem is referenced by:  grpinvcnv  18887  grpsubeq0  18905  prdsinvgd  18930  eqg0subg  19067  ringnegr  20108  islindf4  21384  psrneg  21511  pi1inv  24559  fldhmf1  40943  rngmneg2  46653  lindslinindimp2lem4  47095  lincresunit3  47115
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