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Theorem invrfval 18871
 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 6350 . . . . . . 7 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
3 fveq2 6350 . . . . . . . 8 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
4 invrfval.u . . . . . . . 8 𝑈 = (Unit‘𝑅)
53, 4syl6eqr 2810 . . . . . . 7 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
62, 5oveq12d 6829 . . . . . 6 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = ((mulGrp‘𝑅) ↾s 𝑈))
7 invrfval.g . . . . . 6 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
86, 7syl6eqr 2810 . . . . 5 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = 𝐺)
98fveq2d 6354 . . . 4 (𝑟 = 𝑅 → (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))) = (invg𝐺))
10 df-invr 18870 . . . 4 invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))
11 fvex 6360 . . . 4 (invg𝐺) ∈ V
129, 10, 11fvmpt 6442 . . 3 (𝑅 ∈ V → (invr𝑅) = (invg𝐺))
13 fvprc 6344 . . . . 5 𝑅 ∈ V → (invr𝑅) = ∅)
14 base0 16112 . . . . . . 7 ∅ = (Base‘∅)
15 eqid 2758 . . . . . . 7 (invg‘∅) = (invg‘∅)
1614, 15grpinvfn 17661 . . . . . 6 (invg‘∅) Fn ∅
17 fn0 6170 . . . . . 6 ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅)
1816, 17mpbi 220 . . . . 5 (invg‘∅) = ∅
1913, 18syl6eqr 2810 . . . 4 𝑅 ∈ V → (invr𝑅) = (invg‘∅))
20 fvprc 6344 . . . . . . . 8 𝑅 ∈ V → (mulGrp‘𝑅) = ∅)
2120oveq1d 6826 . . . . . . 7 𝑅 ∈ V → ((mulGrp‘𝑅) ↾s 𝑈) = (∅ ↾s 𝑈))
227, 21syl5eq 2804 . . . . . 6 𝑅 ∈ V → 𝐺 = (∅ ↾s 𝑈))
23 ress0 16134 . . . . . 6 (∅ ↾s 𝑈) = ∅
2422, 23syl6eq 2808 . . . . 5 𝑅 ∈ V → 𝐺 = ∅)
2524fveq2d 6354 . . . 4 𝑅 ∈ V → (invg𝐺) = (invg‘∅))
2619, 25eqtr4d 2795 . . 3 𝑅 ∈ V → (invr𝑅) = (invg𝐺))
2712, 26pm2.61i 176 . 2 (invr𝑅) = (invg𝐺)
281, 27eqtri 2780 1 𝐼 = (invg𝐺)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1630   ∈ wcel 2137  Vcvv 3338  ∅c0 4056   Fn wfn 6042  ‘cfv 6047  (class class class)co 6811   ↾s cress 16058  invgcminusg 17622  mulGrpcmgp 18687  Unitcui 18837  invrcinvr 18869 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-slot 16061  df-base 16063  df-ress 16065  df-minusg 17625  df-invr 18870 This theorem is referenced by:  unitinvcl  18872  unitinvinv  18873  unitlinv  18875  unitrinv  18876  invrpropd  18896  subrgugrp  18999  cnmsubglem  20009  psgninv  20128  invrvald  20682  invrcn2  22182  nrginvrcn  22695  nrgtdrg  22696  sum2dchr  25196  rdivmuldivd  30098  ringinvval  30099  dvrcan5  30100  cntzsdrg  38272
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