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Theorem grpinvf 19004
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 2737 . . . 4 (+g𝐺) = (+g𝐺)
3 eqid 2737 . . . 4 (0g𝐺) = (0g𝐺)
41, 2, 3grpinveu 18992 . . 3 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃!𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺))
5 riotacl 7405 . . 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 18996 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)))
96, 8fmptd 7134 1 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  ∃!wreu 3378  wf 6557  cfv 6561  crio 7387  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  invgcminusg 18952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955
This theorem is referenced by:  grpinvcl  19005  isgrpinv  19011  grpinvcnv  19024  grpinvf1o  19027  grp1inv  19066  pwsinvg  19071  pwssub  19072  oppginv  19378  invoppggim  19379  symgtrinv  19490  invghm  19851  gsumzinv  19963  dprdfinv  20039  grpvlinv  22402  grpvrinv  22403  mdetralt  22614  istgp2  24099  subgtgp  24113  symgtgp  24114  tgpconncomp  24121  prdstgpd  24133  tsmssub  24157  tsmsxplem1  24161  tlmtgp  24204  nrginvrcn  24713
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