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Theorem grpinvadd 17964
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 1116 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Grp)
2 simp2 1117 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 simp3 1118 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
4 grpinvadd.b . . . . . . 7 𝐵 = (Base‘𝐺)
5 grpinvadd.n . . . . . . 7 𝑁 = (invg𝐺)
64, 5grpinvcl 17938 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
763adant2 1111 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
84, 5grpinvcl 17938 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
983adant3 1112 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑋) ∈ 𝐵)
10 grpinvadd.p . . . . . 6 + = (+g𝐺)
114, 10grpcl 17899 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑁𝑌) ∈ 𝐵 ∧ (𝑁𝑋) ∈ 𝐵) → ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)
121, 7, 9, 11syl3anc 1351 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)
134, 10grpass 17900 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵 ∧ ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵)) → ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (𝑋 + (𝑌 + ((𝑁𝑌) + (𝑁𝑋)))))
141, 2, 3, 12, 13syl13anc 1352 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (𝑋 + (𝑌 + ((𝑁𝑌) + (𝑁𝑋)))))
15 eqid 2778 . . . . . . . 8 (0g𝐺) = (0g𝐺)
164, 10, 15, 5grprinv 17940 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (𝑁𝑌)) = (0g𝐺))
17163adant2 1111 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + (𝑁𝑌)) = (0g𝐺))
1817oveq1d 6991 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + (𝑁𝑌)) + (𝑁𝑋)) = ((0g𝐺) + (𝑁𝑋)))
194, 10grpass 17900 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑌𝐵 ∧ (𝑁𝑌) ∈ 𝐵 ∧ (𝑁𝑋) ∈ 𝐵)) → ((𝑌 + (𝑁𝑌)) + (𝑁𝑋)) = (𝑌 + ((𝑁𝑌) + (𝑁𝑋))))
201, 3, 7, 9, 19syl13anc 1352 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑌 + (𝑁𝑌)) + (𝑁𝑋)) = (𝑌 + ((𝑁𝑌) + (𝑁𝑋))))
214, 10, 15grplid 17921 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ((0g𝐺) + (𝑁𝑋)) = (𝑁𝑋))
221, 9, 21syl2anc 576 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + (𝑁𝑋)) = (𝑁𝑋))
2318, 20, 223eqtr3d 2822 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 + ((𝑁𝑌) + (𝑁𝑋))) = (𝑁𝑋))
2423oveq2d 6992 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑌 + ((𝑁𝑌) + (𝑁𝑋)))) = (𝑋 + (𝑁𝑋)))
254, 10, 15, 5grprinv 17940 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
26253adant3 1112 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
2714, 24, 263eqtrd 2818 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (0g𝐺))
284, 10grpcl 17899 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
294, 10, 15, 5grpinvid1 17941 . . 3 ((𝐺 ∈ Grp ∧ (𝑋 + 𝑌) ∈ 𝐵 ∧ ((𝑁𝑌) + (𝑁𝑋)) ∈ 𝐵) → ((𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)) ↔ ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (0g𝐺)))
301, 28, 12, 29syl3anc 1351 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)) ↔ ((𝑋 + 𝑌) + ((𝑁𝑌) + (𝑁𝑋))) = (0g𝐺)))
3127, 30mpbird 249 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1068   = wceq 1507  wcel 2050  cfv 6188  (class class class)co 6976  Basecbs 16339  +gcplusg 16421  0gc0g 16569  Grpcgrp 17891  invgcminusg 17892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-riota 6937  df-ov 6979  df-0g 16571  df-mgm 17710  df-sgrp 17752  df-mnd 17763  df-grp 17894  df-minusg 17895
This theorem is referenced by:  grpinvsub  17968  mulgaddcomlem  18034  mulginvcom  18036  mulgdir  18043  eqger  18113  eqgcpbl  18117  invoppggim  18259  sylow2blem1  18506  lsmsubg  18540  ablinvadd  18688  ablsub2inv  18689  invghm  18712  rdivmuldivd  30547  dvrcan5  30549
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