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Theorem grpinvsub 17705
Description: Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)
Hypotheses
Ref Expression
grpsubcl.b 𝐵 = (Base‘𝐺)
grpsubcl.m = (-g𝐺)
grpinvsub.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvsub ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))

Proof of Theorem grpinvsub
StepHypRef Expression
1 grpsubcl.b . . . . . 6 𝐵 = (Base‘𝐺)
2 grpinvsub.n . . . . . 6 𝑁 = (invg𝐺)
31, 2grpinvcl 17675 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
433adant2 1125 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
5 eqid 2771 . . . . 5 (+g𝐺) = (+g𝐺)
61, 5, 2grpinvadd 17701 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ (𝑁𝑌) ∈ 𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
74, 6syld3an3 1515 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
81, 2grpinvinv 17690 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
983adant2 1125 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
109oveq1d 6811 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)) = (𝑌(+g𝐺)(𝑁𝑋)))
117, 10eqtrd 2805 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = (𝑌(+g𝐺)(𝑁𝑋)))
12 grpsubcl.m . . . . 5 = (-g𝐺)
131, 5, 2, 12grpsubval 17673 . . . 4 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
14133adant1 1124 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
1514fveq2d 6337 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
161, 5, 2, 12grpsubval 17673 . . . 4 ((𝑌𝐵𝑋𝐵) → (𝑌 𝑋) = (𝑌(+g𝐺)(𝑁𝑋)))
1716ancoms 446 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = (𝑌(+g𝐺)(𝑁𝑋)))
18173adant1 1124 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = (𝑌(+g𝐺)(𝑁𝑋)))
1911, 15, 183eqtr4d 2815 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  Grpcgrp 17630  invgcminusg 17631  -gcsg 17632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-sbg 17635
This theorem is referenced by:  grpsubsub  17712  ablsub2inv  18423  lspsnsub  19220  ghmcnp  22138  nrmmetd  22599  nmsub  22647  mapdpglem14  37493
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