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Theorem grpinvadd 18179
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 18153 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
763adant2 1128 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
84, 5grpinvcl 18153 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
983adant3 1129 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑋) ∈ 𝐵)
10 grpinvadd.p . . . . . 6 + = (+g𝐺)
114, 10grpcl 18113 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑁𝑌) ∈ 𝐵 ∧ (𝑁𝑋) ∈ 𝐵) → ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)
121, 7, 9, 11syl3anc 1368 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)
134, 10grpass 18114 . . . 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 2824 . . . . . . . 8 (0g𝐺) = (0g𝐺)
164, 10, 15, 5grprinv 18155 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (𝑁𝑌)) = (0g𝐺))
17163adant2 1128 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + (𝑁𝑌)) = (0g𝐺))
1817oveq1d 7166 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + (𝑁𝑌)) + (𝑁𝑋)) = ((0g𝐺) + (𝑁𝑋)))
194, 10grpass 18114 . . . . . 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 18135 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ((0g𝐺) + (𝑁𝑋)) = (𝑁𝑋))
221, 9, 21syl2anc 587 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + (𝑁𝑋)) = (𝑁𝑋))
2318, 20, 223eqtr3d 2867 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + ((𝑁𝑌) + (𝑁𝑋))) = (𝑁𝑋))
2423oveq2d 7167 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑌 + ((𝑁𝑌) + (𝑁𝑋)))) = (𝑋 + (𝑁𝑋)))
254, 10, 15, 5grprinv 18155 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
26253adant3 1129 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
2714, 24, 263eqtrd 2863 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (0g𝐺))
284, 10grpcl 18113 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
294, 10, 15, 5grpinvid1 18156 . . 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 260 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2115  cfv 6345  (class class class)co 7151  Basecbs 16485  +gcplusg 16567  0gc0g 16715  Grpcgrp 18105  invgcminusg 18106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-fv 6353  df-riota 7109  df-ov 7154  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109
This theorem is referenced by:  grpinvsub  18183  mulgaddcomlem  18252  mulginvcom  18254  mulgdir  18261  eqger  18332  eqgcpbl  18336  invoppggim  18490  sylow2blem1  18747  lsmsubg  18781  ablinvadd  18932  ablsub2inv  18933  invghm  18956  rdivmuldivd  30905  dvrcan5  30907
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