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Theorem grpinvid2 17518
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 6697 . . . 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 17515 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
873adant3 1101 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
98adantr 480 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → ((𝑁𝑋) + 𝑋) = 0 )
102, 9eqtr3d 2687 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑌 + 𝑋) = 0 )
113, 6grpinvcl 17514 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
123, 4, 5grplid 17499 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
1311, 12syldan 486 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
14133adant3 1101 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + (𝑁𝑋)) = (𝑁𝑋))
1514eqcomd 2657 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑋) = ( 0 + (𝑁𝑋)))
1615adantr 480 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁𝑋) = ( 0 + (𝑁𝑋)))
17 oveq1 6697 . . . 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 811 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
20 simprl 809 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
2111adantrr 753 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑁𝑋) ∈ 𝐵)
2219, 20, 213jca 1261 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑌𝐵𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵))
233, 4grpass 17478 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵)) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
2422, 23syldan 486 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
25243impb 1279 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + 𝑋) + (𝑁𝑋)) = (𝑌 + (𝑋 + (𝑁𝑋))))
263, 4, 5, 6grprinv 17516 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
2726oveq2d 6706 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑌 + (𝑋 + (𝑁𝑋))) = (𝑌 + 0 ))
28273adant3 1101 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + (𝑋 + (𝑁𝑋))) = (𝑌 + 0 ))
293, 4, 5grprid 17500 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + 0 ) = 𝑌)
30293adant2 1100 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + 0 ) = 𝑌)
3125, 28, 303eqtrd 2689 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + 𝑋) + (𝑁𝑋)) = 𝑌)
3231adantr 480 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → ((𝑌 + 𝑋) + (𝑁𝑋)) = 𝑌)
3316, 18, 323eqtr2d 2691 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑌 + 𝑋) = 0 ) → (𝑁𝑋) = 𝑌)
3410, 33impbida 895 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑌 + 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Grpcgrp 17469  invgcminusg 17470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473
This theorem is referenced by:  grpinvcnv  17530  grpsubeq0  17548  prdsinvgd  17573  rngnegr  18641  psrneg  19448  islindf4  20225  pi1inv  22898  lindslinindimp2lem4  42575  lincresunit3  42595
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