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Mirrors > Home > ILE Home > Th. List > invrfvald | GIF version |
Description: Multiplicative inverse function for a ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.) |
Ref | Expression |
---|---|
invrfvald.u | ⊢ (𝜑 → 𝑈 = (Unit‘𝑅)) |
invrfvald.g | ⊢ (𝜑 → 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)) |
invrfvald.i | ⊢ (𝜑 → 𝐼 = (invr‘𝑅)) |
invrfvald.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
invrfvald | ⊢ (𝜑 → 𝐼 = (invg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invrfvald.u | . . . 4 ⊢ (𝜑 → 𝑈 = (Unit‘𝑅)) | |
2 | 1 | oveq2d 5888 | . . 3 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s 𝑈) = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))) |
3 | 2 | fveq2d 5518 | . 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝑅) ↾s 𝑈)) = (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅)))) |
4 | invrfvald.g | . . 3 ⊢ (𝜑 → 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)) | |
5 | 4 | fveq2d 5518 | . 2 ⊢ (𝜑 → (invg‘𝐺) = (invg‘((mulGrp‘𝑅) ↾s 𝑈))) |
6 | invrfvald.i | . . 3 ⊢ (𝜑 → 𝐼 = (invr‘𝑅)) | |
7 | df-invr 13221 | . . . 4 ⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟)))) | |
8 | fveq2 5514 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
9 | fveq2 5514 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
10 | 8, 9 | oveq12d 5890 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))) |
11 | 10 | fveq2d 5518 | . . . 4 ⊢ (𝑟 = 𝑅 → (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))) = (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅)))) |
12 | invrfvald.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
13 | 12 | elexd 2750 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ V) |
14 | eqid 2177 | . . . . . . . 8 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
15 | eqid 2177 | . . . . . . . 8 ⊢ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) = ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) | |
16 | 14, 15 | unitgrp 13216 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ Grp) |
17 | 12, 16 | syl 14 | . . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ Grp) |
18 | eqid 2177 | . . . . . . 7 ⊢ (Base‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) = (Base‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) | |
19 | eqid 2177 | . . . . . . 7 ⊢ (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) = (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) | |
20 | 18, 19 | grpinvfng 12849 | . . . . . 6 ⊢ (((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ Grp → (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) Fn (Base‘((mulGrp‘𝑅) ↾s (Unit‘𝑅)))) |
21 | 17, 20 | syl 14 | . . . . 5 ⊢ (𝜑 → (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) Fn (Base‘((mulGrp‘𝑅) ↾s (Unit‘𝑅)))) |
22 | basfn 12512 | . . . . . 6 ⊢ Base Fn V | |
23 | 17 | elexd 2750 | . . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ V) |
24 | funfvex 5531 | . . . . . . 7 ⊢ ((Fun Base ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ dom Base) → (Base‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) ∈ V) | |
25 | 24 | funfni 5315 | . . . . . 6 ⊢ ((Base Fn V ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ V) → (Base‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) ∈ V) |
26 | 22, 23, 25 | sylancr 414 | . . . . 5 ⊢ (𝜑 → (Base‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) ∈ V) |
27 | fnex 5737 | . . . . 5 ⊢ (((invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) Fn (Base‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) ∧ (Base‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) ∈ V) → (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) ∈ V) | |
28 | 21, 26, 27 | syl2anc 411 | . . . 4 ⊢ (𝜑 → (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅))) ∈ V) |
29 | 7, 11, 13, 28 | fvmptd3 5608 | . . 3 ⊢ (𝜑 → (invr‘𝑅) = (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅)))) |
30 | 6, 29 | eqtrd 2210 | . 2 ⊢ (𝜑 → 𝐼 = (invg‘((mulGrp‘𝑅) ↾s (Unit‘𝑅)))) |
31 | 3, 5, 30 | 3eqtr4rd 2221 | 1 ⊢ (𝜑 → 𝐼 = (invg‘𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 Fn wfn 5210 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 ↾s cress 12455 Grpcgrp 12809 invgcminusg 12810 mulGrpcmgp 13061 Ringcrg 13110 Unitcui 13187 invrcinvr 13220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-tpos 6243 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-inn 8916 df-2 8974 df-3 8975 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-iress 12462 df-plusg 12541 df-mulr 12542 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-minusg 12813 df-cmn 13021 df-abl 13022 df-mgp 13062 df-ur 13074 df-srg 13078 df-ring 13112 df-oppr 13171 df-dvdsr 13189 df-unit 13190 df-invr 13221 |
This theorem is referenced by: unitinvcl 13223 unitinvinv 13224 unitlinv 13226 unitrinv 13227 rdivmuldivd 13244 invrpropdg 13249 subrgugrp 13299 |
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