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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 | ⊢ 𝑈 = (Unit‘𝑅) |
invrfval.g | ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) |
invrfval.i | ⊢ 𝐼 = (invr‘𝑅) |
Ref | Expression |
---|---|
invrfval | ⊢ 𝐼 = (invg‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invrfval.i | . 2 ⊢ 𝐼 = (invr‘𝑅) | |
2 | fveq2 6901 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
3 | fveq2 6901 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
4 | invrfval.u | . . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅) | |
5 | 3, 4 | eqtr4di 2784 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
6 | 2, 5 | oveq12d 7442 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = ((mulGrp‘𝑅) ↾s 𝑈)) |
7 | invrfval.g | . . . . . 6 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) | |
8 | 6, 7 | eqtr4di 2784 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = 𝐺) |
9 | 8 | fveq2d 6905 | . . . 4 ⊢ (𝑟 = 𝑅 → (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))) = (invg‘𝐺)) |
10 | df-invr 20370 | . . . 4 ⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟)))) | |
11 | fvex 6914 | . . . 4 ⊢ (invg‘𝐺) ∈ V | |
12 | 9, 10, 11 | fvmpt 7009 | . . 3 ⊢ (𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺)) |
13 | fvprc 6893 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = ∅) | |
14 | base0 17218 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
15 | eqid 2726 | . . . . . . 7 ⊢ (invg‘∅) = (invg‘∅) | |
16 | 14, 15 | grpinvfn 18976 | . . . . . 6 ⊢ (invg‘∅) Fn ∅ |
17 | fn0 6692 | . . . . . 6 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅) | |
18 | 16, 17 | mpbi 229 | . . . . 5 ⊢ (invg‘∅) = ∅ |
19 | 13, 18 | eqtr4di 2784 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘∅)) |
20 | fvprc 6893 | . . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
21 | 20 | oveq1d 7439 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → ((mulGrp‘𝑅) ↾s 𝑈) = (∅ ↾s 𝑈)) |
22 | 7, 21 | eqtrid 2778 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐺 = (∅ ↾s 𝑈)) |
23 | ress0 17257 | . . . . . 6 ⊢ (∅ ↾s 𝑈) = ∅ | |
24 | 22, 23 | eqtrdi 2782 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐺 = ∅) |
25 | 24 | fveq2d 6905 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (invg‘𝐺) = (invg‘∅)) |
26 | 19, 25 | eqtr4d 2769 | . . 3 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺)) |
27 | 12, 26 | pm2.61i 182 | . 2 ⊢ (invr‘𝑅) = (invg‘𝐺) |
28 | 1, 27 | eqtri 2754 | 1 ⊢ 𝐼 = (invg‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 Fn wfn 6549 ‘cfv 6554 (class class class)co 7424 ↾s cress 17242 invgcminusg 18929 mulGrpcmgp 20117 Unitcui 20337 invrcinvr 20369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-1cn 11216 ax-addcl 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-nn 12265 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-minusg 18932 df-invr 20370 |
This theorem is referenced by: unitinvcl 20372 unitinvinv 20373 unitlinv 20375 unitrinv 20376 rdivmuldivd 20395 invrpropd 20400 subrgugrp 20575 cntzsdrg 20781 cnmsubglem 21427 psgninv 21578 invrvald 22669 invrcn2 24175 nrginvrcn 24700 nrgtdrg 24701 sum2dchr 27303 ringinvval 33100 dvrcan5 33101 |
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