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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 6843 | . . . . . . 7 β’ (π = π β (mulGrpβπ) = (mulGrpβπ )) | |
3 | fveq2 6843 | . . . . . . . 8 β’ (π = π β (Unitβπ) = (Unitβπ )) | |
4 | invrfval.u | . . . . . . . 8 β’ π = (Unitβπ ) | |
5 | 3, 4 | eqtr4di 2795 | . . . . . . 7 β’ (π = π β (Unitβπ) = π) |
6 | 2, 5 | oveq12d 7376 | . . . . . 6 β’ (π = π β ((mulGrpβπ) βΎs (Unitβπ)) = ((mulGrpβπ ) βΎs π)) |
7 | invrfval.g | . . . . . 6 β’ πΊ = ((mulGrpβπ ) βΎs π) | |
8 | 6, 7 | eqtr4di 2795 | . . . . 5 β’ (π = π β ((mulGrpβπ) βΎs (Unitβπ)) = πΊ) |
9 | 8 | fveq2d 6847 | . . . 4 β’ (π = π β (invgβ((mulGrpβπ) βΎs (Unitβπ))) = (invgβπΊ)) |
10 | df-invr 20102 | . . . 4 β’ invr = (π β V β¦ (invgβ((mulGrpβπ) βΎs (Unitβπ)))) | |
11 | fvex 6856 | . . . 4 β’ (invgβπΊ) β V | |
12 | 9, 10, 11 | fvmpt 6949 | . . 3 β’ (π β V β (invrβπ ) = (invgβπΊ)) |
13 | fvprc 6835 | . . . . 5 β’ (Β¬ π β V β (invrβπ ) = β ) | |
14 | base0 17089 | . . . . . . 7 β’ β = (Baseββ ) | |
15 | eqid 2737 | . . . . . . 7 β’ (invgββ ) = (invgββ ) | |
16 | 14, 15 | grpinvfn 18793 | . . . . . 6 β’ (invgββ ) Fn β |
17 | fn0 6633 | . . . . . 6 β’ ((invgββ ) Fn β β (invgββ ) = β ) | |
18 | 16, 17 | mpbi 229 | . . . . 5 β’ (invgββ ) = β |
19 | 13, 18 | eqtr4di 2795 | . . . 4 β’ (Β¬ π β V β (invrβπ ) = (invgββ )) |
20 | fvprc 6835 | . . . . . . . 8 β’ (Β¬ π β V β (mulGrpβπ ) = β ) | |
21 | 20 | oveq1d 7373 | . . . . . . 7 β’ (Β¬ π β V β ((mulGrpβπ ) βΎs π) = (β βΎs π)) |
22 | 7, 21 | eqtrid 2789 | . . . . . 6 β’ (Β¬ π β V β πΊ = (β βΎs π)) |
23 | ress0 17125 | . . . . . 6 β’ (β βΎs π) = β | |
24 | 22, 23 | eqtrdi 2793 | . . . . 5 β’ (Β¬ π β V β πΊ = β ) |
25 | 24 | fveq2d 6847 | . . . 4 β’ (Β¬ π β V β (invgβπΊ) = (invgββ )) |
26 | 19, 25 | eqtr4d 2780 | . . 3 β’ (Β¬ π β V β (invrβπ ) = (invgβπΊ)) |
27 | 12, 26 | pm2.61i 182 | . 2 β’ (invrβπ ) = (invgβπΊ) |
28 | 1, 27 | eqtri 2765 | 1 β’ πΌ = (invgβπΊ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1542 β wcel 2107 Vcvv 3446 β c0 4283 Fn wfn 6492 βcfv 6497 (class class class)co 7358 βΎs cress 17113 invgcminusg 18750 mulGrpcmgp 19897 Unitcui 20069 invrcinvr 20101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-1cn 11110 ax-addcl 11112 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-nn 12155 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-minusg 18753 df-invr 20102 |
This theorem is referenced by: unitinvcl 20104 unitinvinv 20105 unitlinv 20107 unitrinv 20108 invrpropd 20128 subrgugrp 20244 cntzsdrg 20272 cnmsubglem 20863 psgninv 20989 invrvald 22028 invrcn2 23534 nrginvrcn 24059 nrgtdrg 24060 sum2dchr 26625 rdivmuldivd 32074 ringinvval 32075 dvrcan5 32076 |
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