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Theorem grpinvf 18095
Description: The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015.)
Hypotheses
Ref Expression
grpinvcl.b 𝐵 = (Base‘𝐺)
grpinvcl.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvf (𝐺 ∈ Grp → 𝑁:𝐵𝐵)

Proof of Theorem grpinvf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpinvcl.b . . . 4 𝐵 = (Base‘𝐺)
2 eqid 2826 . . . 4 (+g𝐺) = (+g𝐺)
3 eqid 2826 . . . 4 (0g𝐺) = (0g𝐺)
41, 2, 3grpinveu 18083 . . 3 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃!𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺))
5 riotacl 7125 . . 3 (∃!𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺) → (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ 𝐵)
64, 5syl 17 . 2 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ 𝐵)
7 grpinvcl.n . . 3 𝑁 = (invg𝐺)
81, 2, 3, 7grpinvfval 18087 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)))
96, 8fmptd 6876 1 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  ∃!wreu 3145  wf 6350  cfv 6354  crio 7107  (class class class)co 7150  Basecbs 16478  +gcplusg 16560  0gc0g 16708  Grpcgrp 18048  invgcminusg 18049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-riota 7108  df-ov 7153  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18051  df-minusg 18052
This theorem is referenced by:  grpinvcl  18096  isgrpinv  18101  grpinvcnv  18112  grpinvf1o  18114  grp1inv  18152  pwsinvg  18157  pwssub  18158  oppginv  18432  invoppggim  18433  symgtrinv  18536  invghm  18890  gsumzinv  19001  dprdfinv  19077  mhpinvcl  20274  grpvlinv  20941  grpvrinv  20942  mdetralt  21152  istgp2  22634  symgtgp  22644  subgtgp  22648  tgpconncomp  22655  prdstgpd  22667  tsmssub  22691  tsmsxplem1  22695  tlmtgp  22738  nrginvrcn  23235
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