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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 6664 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
3 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
4 | invrfval.u | . . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅) | |
5 | 3, 4 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
6 | 2, 5 | oveq12d 7168 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = ((mulGrp‘𝑅) ↾s 𝑈)) |
7 | invrfval.g | . . . . . 6 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) | |
8 | 6, 7 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = 𝐺) |
9 | 8 | fveq2d 6668 | . . . 4 ⊢ (𝑟 = 𝑅 → (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))) = (invg‘𝐺)) |
10 | df-invr 19416 | . . . 4 ⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟)))) | |
11 | fvex 6677 | . . . 4 ⊢ (invg‘𝐺) ∈ V | |
12 | 9, 10, 11 | fvmpt 6762 | . . 3 ⊢ (𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺)) |
13 | fvprc 6657 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = ∅) | |
14 | base0 16530 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
15 | eqid 2821 | . . . . . . 7 ⊢ (invg‘∅) = (invg‘∅) | |
16 | 14, 15 | grpinvfn 18139 | . . . . . 6 ⊢ (invg‘∅) Fn ∅ |
17 | fn0 6473 | . . . . . 6 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅) | |
18 | 16, 17 | mpbi 232 | . . . . 5 ⊢ (invg‘∅) = ∅ |
19 | 13, 18 | syl6eqr 2874 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘∅)) |
20 | fvprc 6657 | . . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
21 | 20 | oveq1d 7165 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → ((mulGrp‘𝑅) ↾s 𝑈) = (∅ ↾s 𝑈)) |
22 | 7, 21 | syl5eq 2868 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐺 = (∅ ↾s 𝑈)) |
23 | ress0 16552 | . . . . . 6 ⊢ (∅ ↾s 𝑈) = ∅ | |
24 | 22, 23 | syl6eq 2872 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐺 = ∅) |
25 | 24 | fveq2d 6668 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (invg‘𝐺) = (invg‘∅)) |
26 | 19, 25 | eqtr4d 2859 | . . 3 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺)) |
27 | 12, 26 | pm2.61i 184 | . 2 ⊢ (invr‘𝑅) = (invg‘𝐺) |
28 | 1, 27 | eqtri 2844 | 1 ⊢ 𝐼 = (invg‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 Fn wfn 6344 ‘cfv 6349 (class class class)co 7150 ↾s cress 16478 invgcminusg 18098 mulGrpcmgp 19233 Unitcui 19383 invrcinvr 19415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-slot 16481 df-base 16483 df-ress 16485 df-minusg 18101 df-invr 19416 |
This theorem is referenced by: unitinvcl 19418 unitinvinv 19419 unitlinv 19421 unitrinv 19422 invrpropd 19442 subrgugrp 19548 cntzsdrg 19575 cnmsubglem 20602 psgninv 20720 invrvald 21279 invrcn2 22782 nrginvrcn 23295 nrgtdrg 23296 sum2dchr 25844 rdivmuldivd 30857 ringinvval 30858 dvrcan5 30859 |
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