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Theorem invrfval 18719
Description: Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
Hypotheses
Ref Expression
invrfval.u 𝑈 = (Unit‘𝑅)
invrfval.g 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
invrfval.i 𝐼 = (invr𝑅)
Assertion
Ref Expression
invrfval 𝐼 = (invg𝐺)

Proof of Theorem invrfval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 invrfval.i . 2 𝐼 = (invr𝑅)
2 fveq2 6229 . . . . . . 7 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
3 fveq2 6229 . . . . . . . 8 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
4 invrfval.u . . . . . . . 8 𝑈 = (Unit‘𝑅)
53, 4syl6eqr 2703 . . . . . . 7 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
62, 5oveq12d 6708 . . . . . 6 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = ((mulGrp‘𝑅) ↾s 𝑈))
7 invrfval.g . . . . . 6 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
86, 7syl6eqr 2703 . . . . 5 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = 𝐺)
98fveq2d 6233 . . . 4 (𝑟 = 𝑅 → (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))) = (invg𝐺))
10 df-invr 18718 . . . 4 invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))
11 fvex 6239 . . . 4 (invg𝐺) ∈ V
129, 10, 11fvmpt 6321 . . 3 (𝑅 ∈ V → (invr𝑅) = (invg𝐺))
13 fvprc 6223 . . . . 5 𝑅 ∈ V → (invr𝑅) = ∅)
14 base0 15959 . . . . . . 7 ∅ = (Base‘∅)
15 eqid 2651 . . . . . . 7 (invg‘∅) = (invg‘∅)
1614, 15grpinvfn 17509 . . . . . 6 (invg‘∅) Fn ∅
17 fn0 6049 . . . . . 6 ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅)
1816, 17mpbi 220 . . . . 5 (invg‘∅) = ∅
1913, 18syl6eqr 2703 . . . 4 𝑅 ∈ V → (invr𝑅) = (invg‘∅))
20 fvprc 6223 . . . . . . . 8 𝑅 ∈ V → (mulGrp‘𝑅) = ∅)
2120oveq1d 6705 . . . . . . 7 𝑅 ∈ V → ((mulGrp‘𝑅) ↾s 𝑈) = (∅ ↾s 𝑈))
227, 21syl5eq 2697 . . . . . 6 𝑅 ∈ V → 𝐺 = (∅ ↾s 𝑈))
23 ress0 15981 . . . . . 6 (∅ ↾s 𝑈) = ∅
2422, 23syl6eq 2701 . . . . 5 𝑅 ∈ V → 𝐺 = ∅)
2524fveq2d 6233 . . . 4 𝑅 ∈ V → (invg𝐺) = (invg‘∅))
2619, 25eqtr4d 2688 . . 3 𝑅 ∈ V → (invr𝑅) = (invg𝐺))
2712, 26pm2.61i 176 . 2 (invr𝑅) = (invg𝐺)
281, 27eqtri 2673 1 𝐼 = (invg𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948   Fn wfn 5921  cfv 5926  (class class class)co 6690  s cress 15905  invgcminusg 17470  mulGrpcmgp 18535  Unitcui 18685  invrcinvr 18717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-slot 15908  df-base 15910  df-ress 15912  df-minusg 17473  df-invr 18718
This theorem is referenced by:  unitinvcl  18720  unitinvinv  18721  unitlinv  18723  unitrinv  18724  invrpropd  18744  subrgugrp  18847  cnmsubglem  19857  psgninv  19976  invrvald  20530  invrcn2  22030  nrginvrcn  22543  nrgtdrg  22544  sum2dchr  25044  rdivmuldivd  29919  ringinvval  29920  dvrcan5  29921  cntzsdrg  38089
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