![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > invrfval | Structured version Visualization version GIF version |
Description: Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.) |
Ref | Expression |
---|---|
invrfval.u | β’ π = (Unitβπ ) |
invrfval.g | β’ πΊ = ((mulGrpβπ ) βΎs π) |
invrfval.i | β’ πΌ = (invrβπ ) |
Ref | Expression |
---|---|
invrfval | β’ πΌ = (invgβπΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invrfval.i | . 2 β’ πΌ = (invrβπ ) | |
2 | fveq2 6885 | . . . . . . 7 β’ (π = π β (mulGrpβπ) = (mulGrpβπ )) | |
3 | fveq2 6885 | . . . . . . . 8 β’ (π = π β (Unitβπ) = (Unitβπ )) | |
4 | invrfval.u | . . . . . . . 8 β’ π = (Unitβπ ) | |
5 | 3, 4 | eqtr4di 2784 | . . . . . . 7 β’ (π = π β (Unitβπ) = π) |
6 | 2, 5 | oveq12d 7423 | . . . . . 6 β’ (π = π β ((mulGrpβπ) βΎs (Unitβπ)) = ((mulGrpβπ ) βΎs π)) |
7 | invrfval.g | . . . . . 6 β’ πΊ = ((mulGrpβπ ) βΎs π) | |
8 | 6, 7 | eqtr4di 2784 | . . . . 5 β’ (π = π β ((mulGrpβπ) βΎs (Unitβπ)) = πΊ) |
9 | 8 | fveq2d 6889 | . . . 4 β’ (π = π β (invgβ((mulGrpβπ) βΎs (Unitβπ))) = (invgβπΊ)) |
10 | df-invr 20290 | . . . 4 β’ invr = (π β V β¦ (invgβ((mulGrpβπ) βΎs (Unitβπ)))) | |
11 | fvex 6898 | . . . 4 β’ (invgβπΊ) β V | |
12 | 9, 10, 11 | fvmpt 6992 | . . 3 β’ (π β V β (invrβπ ) = (invgβπΊ)) |
13 | fvprc 6877 | . . . . 5 β’ (Β¬ π β V β (invrβπ ) = β ) | |
14 | base0 17158 | . . . . . . 7 β’ β = (Baseββ ) | |
15 | eqid 2726 | . . . . . . 7 β’ (invgββ ) = (invgββ ) | |
16 | 14, 15 | grpinvfn 18911 | . . . . . 6 β’ (invgββ ) Fn β |
17 | fn0 6675 | . . . . . 6 β’ ((invgββ ) Fn β β (invgββ ) = β ) | |
18 | 16, 17 | mpbi 229 | . . . . 5 β’ (invgββ ) = β |
19 | 13, 18 | eqtr4di 2784 | . . . 4 β’ (Β¬ π β V β (invrβπ ) = (invgββ )) |
20 | fvprc 6877 | . . . . . . . 8 β’ (Β¬ π β V β (mulGrpβπ ) = β ) | |
21 | 20 | oveq1d 7420 | . . . . . . 7 β’ (Β¬ π β V β ((mulGrpβπ ) βΎs π) = (β βΎs π)) |
22 | 7, 21 | eqtrid 2778 | . . . . . 6 β’ (Β¬ π β V β πΊ = (β βΎs π)) |
23 | ress0 17197 | . . . . . 6 β’ (β βΎs π) = β | |
24 | 22, 23 | eqtrdi 2782 | . . . . 5 β’ (Β¬ π β V β πΊ = β ) |
25 | 24 | fveq2d 6889 | . . . 4 β’ (Β¬ π β V β (invgβπΊ) = (invgββ )) |
26 | 19, 25 | eqtr4d 2769 | . . 3 β’ (Β¬ π β V β (invrβπ ) = (invgβπΊ)) |
27 | 12, 26 | pm2.61i 182 | . 2 β’ (invrβπ ) = (invgβπΊ) |
28 | 1, 27 | eqtri 2754 | 1 β’ πΌ = (invgβπΊ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1533 β wcel 2098 Vcvv 3468 β c0 4317 Fn wfn 6532 βcfv 6537 (class class class)co 7405 βΎs cress 17182 invgcminusg 18864 mulGrpcmgp 20039 Unitcui 20257 invrcinvr 20289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-1cn 11170 ax-addcl 11172 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-nn 12217 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-minusg 18867 df-invr 20290 |
This theorem is referenced by: unitinvcl 20292 unitinvinv 20293 unitlinv 20295 unitrinv 20296 rdivmuldivd 20315 invrpropd 20320 subrgugrp 20493 cntzsdrg 20653 cnmsubglem 21324 psgninv 21475 invrvald 22533 invrcn2 24039 nrginvrcn 24564 nrgtdrg 24565 sum2dchr 27162 ringinvval 32886 dvrcan5 32887 |
Copyright terms: Public domain | W3C validator |