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Theorem grpinvid1 12801
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 5876 . . . 4 ((𝑁𝑋) = 𝑌 → (𝑋 + (𝑁𝑋)) = (𝑋 + 𝑌))
21adantl 277 . . 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 12800 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
873adant3 1017 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
98adantr 276 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑋 + (𝑁𝑋)) = 0 )
102, 9eqtr3d 2212 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑁𝑋) = 𝑌) → (𝑋 + 𝑌) = 0 )
11 oveq2 5876 . . . 4 ((𝑋 + 𝑌) = 0 → ((𝑁𝑋) + (𝑋 + 𝑌)) = ((𝑁𝑋) + 0 ))
1211adantl 277 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + (𝑋 + 𝑌)) = ((𝑁𝑋) + 0 ))
133, 4, 5, 6grplinv 12799 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 𝑋) = 0 )
1413oveq1d 5883 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ( 0 + 𝑌))
15143adant3 1017 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ( 0 + 𝑌))
163, 6grpinvcl 12798 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
1716adantrr 479 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (𝑁𝑋) ∈ 𝐵)
18 simprl 529 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
19 simprr 531 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
2017, 18, 193jca 1177 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → ((𝑁𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵))
213, 4grpass 12763 . . . . . . . 8 ((𝐺 ∈ Grp ∧ ((𝑁𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
2220, 21syldan 282 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵)) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
23223impb 1199 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (((𝑁𝑋) + 𝑋) + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
2415, 23eqtr3d 2212 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + 𝑌) = ((𝑁𝑋) + (𝑋 + 𝑌)))
253, 4, 5grplid 12783 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ( 0 + 𝑌) = 𝑌)
26253adant2 1016 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ( 0 + 𝑌) = 𝑌)
2724, 26eqtr3d 2212 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + (𝑋 + 𝑌)) = 𝑌)
2827adantr 276 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + (𝑋 + 𝑌)) = 𝑌)
293, 4, 5grprid 12784 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3016, 29syldan 282 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
31303adant3 1017 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3231adantr 276 . . 3 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → ((𝑁𝑋) + 0 ) = (𝑁𝑋))
3312, 28, 323eqtr3rd 2219 . 2 (((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 + 𝑌) = 0 ) → (𝑁𝑋) = 𝑌)
3410, 33impbida 596 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  cfv 5211  (class class class)co 5868  Basecbs 12432  +gcplusg 12505  0gc0g 12640  Grpcgrp 12754  invgcminusg 12755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757  df-minusg 12758
This theorem is referenced by:  grpinvid  12807  grpinvcnv  12814  grpinvadd  12824  ringnegl  13041
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