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Theorem grpinvadd 18978
Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.)
Hypotheses
Ref Expression
grpinvadd.b 𝐵 = (Base‘𝐺)
grpinvadd.p + = (+g𝐺)
grpinvadd.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvadd ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)))

Proof of Theorem grpinvadd
StepHypRef Expression
1 simp1 1133 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Grp)
2 simp2 1134 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 simp3 1135 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
4 grpinvadd.b . . . . . . 7 𝐵 = (Base‘𝐺)
5 grpinvadd.n . . . . . . 7 𝑁 = (invg𝐺)
64, 5grpinvcl 18948 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
763adant2 1128 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
84, 5grpinvcl 18948 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
983adant3 1129 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑋) ∈ 𝐵)
10 grpinvadd.p . . . . . 6 + = (+g𝐺)
114, 10grpcl 18902 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑁𝑌) ∈ 𝐵 ∧ (𝑁𝑋) ∈ 𝐵) → ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)
121, 7, 9, 11syl3anc 1368 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)
134, 10grpass 18903 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵 ∧ ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)) → ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (𝑋 + (𝑌 + ((𝑁𝑌) + (𝑁𝑋)))))
141, 2, 3, 12, 13syl13anc 1369 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (𝑋 + (𝑌 + ((𝑁𝑌) + (𝑁𝑋)))))
15 eqid 2725 . . . . . . . 8 (0g𝐺) = (0g𝐺)
164, 10, 15, 5grprinv 18951 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (𝑁𝑌)) = (0g𝐺))
17163adant2 1128 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + (𝑁𝑌)) = (0g𝐺))
1817oveq1d 7431 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + (𝑁𝑌)) + (𝑁𝑋)) = ((0g𝐺) + (𝑁𝑋)))
194, 10grpass 18903 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑌𝐵 ∧ (𝑁𝑌) ∈ 𝐵 ∧ (𝑁𝑋) ∈ 𝐵)) → ((𝑌 + (𝑁𝑌)) + (𝑁𝑋)) = (𝑌 + ((𝑁𝑌) + (𝑁𝑋))))
201, 3, 7, 9, 19syl13anc 1369 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + (𝑁𝑌)) + (𝑁𝑋)) = (𝑌 + ((𝑁𝑌) + (𝑁𝑋))))
214, 10, 15grplid 18928 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ((0g𝐺) + (𝑁𝑋)) = (𝑁𝑋))
221, 9, 21syl2anc 582 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + (𝑁𝑋)) = (𝑁𝑋))
2318, 20, 223eqtr3d 2773 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + ((𝑁𝑌) + (𝑁𝑋))) = (𝑁𝑋))
2423oveq2d 7432 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑌 + ((𝑁𝑌) + (𝑁𝑋)))) = (𝑋 + (𝑁𝑋)))
254, 10, 15, 5grprinv 18951 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
26253adant3 1129 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
2714, 24, 263eqtrd 2769 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (0g𝐺))
284, 10grpcl 18902 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
294, 10, 15, 5grpinvid1 18952 . . 3 ((𝐺 ∈ Grp ∧ (𝑋 + 𝑌) ∈ 𝐵 ∧ ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵) → ((𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)) ↔ ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (0g𝐺)))
301, 28, 12, 29syl3anc 1368 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)) ↔ ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (0g𝐺)))
3127, 30mpbird 256 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  cfv 6543  (class class class)co 7416  Basecbs 17179  +gcplusg 17232  0gc0g 17420  Grpcgrp 18894  invgcminusg 18895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7372  df-ov 7419  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18897  df-minusg 18898
This theorem is referenced by:  grpinvsub  18982  mulgaddcomlem  19056  mulginvcom  19058  mulgdir  19065  eqger  19137  eqgcpbl  19141  invoppggim  19318  sylow2blem1  19579  lsmsubg  19613  ablinvadd  19766  ablsub2inv  19767  invghm  19792  rdivmuldivd  20356  dvrcan5  32988
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