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Theorem invrfval 19422
 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 6649 . . . . . . 7 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
3 fveq2 6649 . . . . . . . 8 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
4 invrfval.u . . . . . . . 8 𝑈 = (Unit‘𝑅)
53, 4eqtr4di 2854 . . . . . . 7 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
62, 5oveq12d 7157 . . . . . 6 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = ((mulGrp‘𝑅) ↾s 𝑈))
7 invrfval.g . . . . . 6 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
86, 7eqtr4di 2854 . . . . 5 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = 𝐺)
98fveq2d 6653 . . . 4 (𝑟 = 𝑅 → (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))) = (invg𝐺))
10 df-invr 19421 . . . 4 invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))
11 fvex 6662 . . . 4 (invg𝐺) ∈ V
129, 10, 11fvmpt 6749 . . 3 (𝑅 ∈ V → (invr𝑅) = (invg𝐺))
13 fvprc 6642 . . . . 5 𝑅 ∈ V → (invr𝑅) = ∅)
14 base0 16531 . . . . . . 7 ∅ = (Base‘∅)
15 eqid 2801 . . . . . . 7 (invg‘∅) = (invg‘∅)
1614, 15grpinvfn 18140 . . . . . 6 (invg‘∅) Fn ∅
17 fn0 6455 . . . . . 6 ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅)
1816, 17mpbi 233 . . . . 5 (invg‘∅) = ∅
1913, 18eqtr4di 2854 . . . 4 𝑅 ∈ V → (invr𝑅) = (invg‘∅))
20 fvprc 6642 . . . . . . . 8 𝑅 ∈ V → (mulGrp‘𝑅) = ∅)
2120oveq1d 7154 . . . . . . 7 𝑅 ∈ V → ((mulGrp‘𝑅) ↾s 𝑈) = (∅ ↾s 𝑈))
227, 21syl5eq 2848 . . . . . 6 𝑅 ∈ V → 𝐺 = (∅ ↾s 𝑈))
23 ress0 16553 . . . . . 6 (∅ ↾s 𝑈) = ∅
2422, 23eqtrdi 2852 . . . . 5 𝑅 ∈ V → 𝐺 = ∅)
2524fveq2d 6653 . . . 4 𝑅 ∈ V → (invg𝐺) = (invg‘∅))
2619, 25eqtr4d 2839 . . 3 𝑅 ∈ V → (invr𝑅) = (invg𝐺))
2712, 26pm2.61i 185 . 2 (invr𝑅) = (invg𝐺)
281, 27eqtri 2824 1 𝐼 = (invg𝐺)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ∅c0 4246   Fn wfn 6323  ‘cfv 6328  (class class class)co 7139   ↾s cress 16479  invgcminusg 18099  mulGrpcmgp 19235  Unitcui 19388  invrcinvr 19420 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-slot 16482  df-base 16484  df-ress 16486  df-minusg 18102  df-invr 19421 This theorem is referenced by:  unitinvcl  19423  unitinvinv  19424  unitlinv  19426  unitrinv  19427  invrpropd  19447  subrgugrp  19550  cntzsdrg  19577  cnmsubglem  20157  psgninv  20274  invrvald  21284  invrcn2  22788  nrginvrcn  23301  nrgtdrg  23302  sum2dchr  25861  rdivmuldivd  30916  ringinvval  30917  dvrcan5  30918
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