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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 6695 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
3 | fveq2 6695 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
4 | invrfval.u | . . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅) | |
5 | 3, 4 | eqtr4di 2789 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
6 | 2, 5 | oveq12d 7209 | . . . . . 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 2789 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = 𝐺) |
9 | 8 | fveq2d 6699 | . . . 4 ⊢ (𝑟 = 𝑅 → (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))) = (invg‘𝐺)) |
10 | df-invr 19644 | . . . 4 ⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟)))) | |
11 | fvex 6708 | . . . 4 ⊢ (invg‘𝐺) ∈ V | |
12 | 9, 10, 11 | fvmpt 6796 | . . 3 ⊢ (𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺)) |
13 | fvprc 6687 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = ∅) | |
14 | base0 16726 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
15 | eqid 2736 | . . . . . . 7 ⊢ (invg‘∅) = (invg‘∅) | |
16 | 14, 15 | grpinvfn 18363 | . . . . . 6 ⊢ (invg‘∅) Fn ∅ |
17 | fn0 6487 | . . . . . 6 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅) | |
18 | 16, 17 | mpbi 233 | . . . . 5 ⊢ (invg‘∅) = ∅ |
19 | 13, 18 | eqtr4di 2789 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘∅)) |
20 | fvprc 6687 | . . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
21 | 20 | oveq1d 7206 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → ((mulGrp‘𝑅) ↾s 𝑈) = (∅ ↾s 𝑈)) |
22 | 7, 21 | syl5eq 2783 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐺 = (∅ ↾s 𝑈)) |
23 | ress0 16742 | . . . . . 6 ⊢ (∅ ↾s 𝑈) = ∅ | |
24 | 22, 23 | eqtrdi 2787 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐺 = ∅) |
25 | 24 | fveq2d 6699 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (invg‘𝐺) = (invg‘∅)) |
26 | 19, 25 | eqtr4d 2774 | . . 3 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺)) |
27 | 12, 26 | pm2.61i 185 | . 2 ⊢ (invr‘𝑅) = (invg‘𝐺) |
28 | 1, 27 | eqtri 2759 | 1 ⊢ 𝐼 = (invg‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∅c0 4223 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 ↾s cress 16667 invgcminusg 18320 mulGrpcmgp 19458 Unitcui 19611 invrcinvr 19643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-1cn 10752 ax-addcl 10754 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-nn 11796 df-ndx 16669 df-slot 16670 df-base 16672 df-ress 16674 df-minusg 18323 df-invr 19644 |
This theorem is referenced by: unitinvcl 19646 unitinvinv 19647 unitlinv 19649 unitrinv 19650 invrpropd 19670 subrgugrp 19773 cntzsdrg 19800 cnmsubglem 20380 psgninv 20498 invrvald 21527 invrcn2 23031 nrginvrcn 23544 nrgtdrg 23545 sum2dchr 26109 rdivmuldivd 31161 ringinvval 31162 dvrcan5 31163 |
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