MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoinvfval Structured version   Visualization version   GIF version

Theorem grpoinvfval 27949
Description: The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvfval.1 𝑋 = ran 𝐺
grpinvfval.2 𝑈 = (GId‘𝐺)
grpinvfval.3 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinvfval (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑋,𝑦   𝑥,𝑈
Allowed substitution hints:   𝑈(𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem grpoinvfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpinvfval.3 . 2 𝑁 = (inv‘𝐺)
2 grpinvfval.1 . . . . 5 𝑋 = ran 𝐺
3 rnexg 7376 . . . . 5 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
42, 3syl5eqel 2862 . . . 4 (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5 mptexg 6756 . . . 4 (𝑋 ∈ V → (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
64, 5syl 17 . . 3 (𝐺 ∈ GrpOp → (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V)
7 rneq 5596 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
87, 2syl6eqr 2831 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9 oveq 6928 . . . . . . 7 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
10 fveq2 6446 . . . . . . . 8 (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
11 grpinvfval.2 . . . . . . . 8 𝑈 = (GId‘𝐺)
1210, 11syl6eqr 2831 . . . . . . 7 (𝑔 = 𝐺 → (GId‘𝑔) = 𝑈)
139, 12eqeq12d 2792 . . . . . 6 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = (GId‘𝑔) ↔ (𝑦𝐺𝑥) = 𝑈))
148, 13riotaeqbidv 6886 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔)) = (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
158, 14mpteq12dv 4969 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔 ↦ (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
16 df-ginv 27922 . . . 4 inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (𝑦 ∈ ran 𝑔(𝑦𝑔𝑥) = (GId‘𝑔))))
1715, 16fvmptg 6540 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) ∈ V) → (inv‘𝐺) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
186, 17mpdan 677 . 2 (𝐺 ∈ GrpOp → (inv‘𝐺) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
191, 18syl5eq 2825 1 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2106  Vcvv 3397  cmpt 4965  ran crn 5356  cfv 6135  crio 6882  (class class class)co 6922  GrpOpcgr 27916  GIdcgi 27917  invcgn 27918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-ginv 27922
This theorem is referenced by:  grpoinvval  27950  grpoinvf  27959
  Copyright terms: Public domain W3C validator