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Theorem grpinvid1 17671
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 6821 . . . 4 ((𝑁𝑋) = 𝑌 → (𝑋 + (𝑁𝑋)) = (𝑋 + 𝑌))
21adantl 473 . . 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 17670 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
873adant3 1127 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
98adantr 472 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑋 + (𝑁𝑋)) = 0 )
102, 9eqtr3d 2796 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑋 + 𝑌) = 0 )
11 oveq2 6821 . . . 4 ((𝑋 + 𝑌) = 0 → ((𝑁𝑋) + (𝑋 + 𝑌)) = ((𝑁𝑋) + 0 ))
1211adantl 473 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + (𝑋 + 𝑌)) = ((𝑁𝑋) + 0 ))
133, 4, 5, 6grplinv 17669 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
1413oveq1d 6828 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ( 0 + 𝑌))
15143adant3 1127 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ( 0 + 𝑌))
163, 6grpinvcl 17668 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
1716adantrr 755 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑁𝑋) ∈ 𝐵)
18 simprl 811 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
19 simprr 813 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
2017, 18, 193jca 1123 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑁𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵))
213, 4grpass 17632 . . . . . . . 8 ((𝐺 ∈ Grp ∧ ((𝑁𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
2220, 21syldan 488 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
23223impb 1108 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
2415, 23eqtr3d 2796 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
253, 4, 5grplid 17653 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ( 0 + 𝑌) = 𝑌)
26253adant2 1126 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + 𝑌) = 𝑌)
2724, 26eqtr3d 2796 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + (𝑋 + 𝑌)) = 𝑌)
2827adantr 472 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + (𝑋 + 𝑌)) = 𝑌)
293, 4, 5grprid 17654 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3016, 29syldan 488 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
31303adant3 1127 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3231adantr 472 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3312, 28, 323eqtr3rd 2803 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → (𝑁𝑋) = 𝑌)
3410, 33impbida 913 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  0gc0g 16302  Grpcgrp 17623  invgcminusg 17624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627
This theorem is referenced by:  grpinvid  17677  grpinvcnv  17684  grpinvadd  17694  subginv  17802  qusinv  17854  ghminv  17868  symginv  18022  frgpinv  18377  cnaddinv  18474  ringnegl  18794  lmodindp1  19216  lmodvsinv2  19239  cnfldneg  19974  zringinvg  20037  mdetunilem6  20625  invrvald  20684  dchrinv  25185  baerlem3lem1  37498
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