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Theorem grpinvadd 12837
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 997 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Grp)
2 simp2 998 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 simp3 999 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
4 grpinvadd.b . . . . . . 7 𝐵 = (Base‘𝐺)
5 grpinvadd.n . . . . . . 7 𝑁 = (invg𝐺)
64, 5grpinvcl 12811 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
763adant2 1016 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
84, 5grpinvcl 12811 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
983adant3 1017 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑋) ∈ 𝐵)
10 grpinvadd.p . . . . . 6 + = (+g𝐺)
114, 10grpcl 12775 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑁𝑌) ∈ 𝐵 ∧ (𝑁𝑋) ∈ 𝐵) → ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)
121, 7, 9, 11syl3anc 1238 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)
134, 10grpass 12776 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵 ∧ ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)) → ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (𝑋 + (𝑌 + ((𝑁𝑌) + (𝑁𝑋)))))
141, 2, 3, 12, 13syl13anc 1240 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (𝑋 + (𝑌 + ((𝑁𝑌) + (𝑁𝑋)))))
15 eqid 2177 . . . . . . . 8 (0g𝐺) = (0g𝐺)
164, 10, 15, 5grprinv 12813 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (𝑁𝑌)) = (0g𝐺))
17163adant2 1016 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + (𝑁𝑌)) = (0g𝐺))
1817oveq1d 5884 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + (𝑁𝑌)) + (𝑁𝑋)) = ((0g𝐺) + (𝑁𝑋)))
194, 10grpass 12776 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑌𝐵 ∧ (𝑁𝑌) ∈ 𝐵 ∧ (𝑁𝑋) ∈ 𝐵)) → ((𝑌 + (𝑁𝑌)) + (𝑁𝑋)) = (𝑌 + ((𝑁𝑌) + (𝑁𝑋))))
201, 3, 7, 9, 19syl13anc 1240 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + (𝑁𝑌)) + (𝑁𝑋)) = (𝑌 + ((𝑁𝑌) + (𝑁𝑋))))
214, 10, 15grplid 12796 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ((0g𝐺) + (𝑁𝑋)) = (𝑁𝑋))
221, 9, 21syl2anc 411 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + (𝑁𝑋)) = (𝑁𝑋))
2318, 20, 223eqtr3d 2218 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + ((𝑁𝑌) + (𝑁𝑋))) = (𝑁𝑋))
2423oveq2d 5885 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑌 + ((𝑁𝑌) + (𝑁𝑋)))) = (𝑋 + (𝑁𝑋)))
254, 10, 15, 5grprinv 12813 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
26253adant3 1017 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
2714, 24, 263eqtrd 2214 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (0g𝐺))
284, 10grpcl 12775 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
294, 10, 15, 5grpinvid1 12814 . . 3 ((𝐺 ∈ Grp ∧ (𝑋 + 𝑌) ∈ 𝐵 ∧ ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵) → ((𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)) ↔ ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (0g𝐺)))
301, 28, 12, 29syl3anc 1238 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)) ↔ ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (0g𝐺)))
3127, 30mpbird 167 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  w3a 978   = wceq 1353  wcel 2148  cfv 5212  (class class class)co 5869  Basecbs 12445  +gcplusg 12518  0gc0g 12653  Grpcgrp 12767  invgcminusg 12768
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 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
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 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-0g 12655  df-mgm 12667  df-sgrp 12700  df-mnd 12710  df-grp 12770  df-minusg 12771
This theorem is referenced by:  grpinvsub  12841  mulgaddcomlem  12894  mulginvcom  12896  mulgdir  12903  ablinvadd  12940  ablsub2inv  12941
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