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Theorem grpinvsub 17619
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 17589 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
433adant2 1123 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
5 eqid 2724 . . . . 5 (+g𝐺) = (+g𝐺)
61, 5, 2grpinvadd 17615 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ (𝑁𝑌) ∈ 𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
74, 6syld3an3 1484 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)))
81, 2grpinvinv 17604 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
983adant2 1123 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑁𝑌)) = 𝑌)
109oveq1d 6780 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁‘(𝑁𝑌))(+g𝐺)(𝑁𝑋)) = (𝑌(+g𝐺)(𝑁𝑋)))
117, 10eqtrd 2758 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))) = (𝑌(+g𝐺)(𝑁𝑋)))
12 grpsubcl.m . . . . 5 = (-g𝐺)
131, 5, 2, 12grpsubval 17587 . . . 4 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
14133adant1 1122 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋(+g𝐺)(𝑁𝑌)))
1514fveq2d 6308 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑁‘(𝑋(+g𝐺)(𝑁𝑌))))
161, 5, 2, 12grpsubval 17587 . . . 4 ((𝑌𝐵𝑋𝐵) → (𝑌 𝑋) = (𝑌(+g𝐺)(𝑁𝑋)))
1716ancoms 468 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = (𝑌(+g𝐺)(𝑁𝑋)))
18173adant1 1122 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = (𝑌(+g𝐺)(𝑁𝑋)))
1911, 15, 183eqtr4d 2768 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 𝑌)) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1596  wcel 2103  cfv 6001  (class class class)co 6765  Basecbs 15980  +gcplusg 16064  Grpcgrp 17544  invgcminusg 17545  -gcsg 17546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-0g 16225  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-grp 17547  df-minusg 17548  df-sbg 17549
This theorem is referenced by:  grpsubsub  17626  ablsub2inv  18337  lspsnsub  19130  ghmcnp  22040  nrmmetd  22501  nmsub  22549  mapdpglem14  37393
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