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Mirrors > Home > MPE Home > Th. List > unitsubm | Structured version Visualization version GIF version |
Description: The group of units is a submonoid of the multiplicative monoid of the ring. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
unitsubm.1 | ⊢ 𝑈 = (Unit‘𝑅) |
unitsubm.2 | ⊢ 𝑀 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
unitsubm | ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubMnd‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | unitsubm.1 | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | 1, 2 | unitss 18864 | . . 3 ⊢ 𝑈 ⊆ (Base‘𝑅) |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅)) |
5 | eqid 2771 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | 2, 5 | 1unit 18862 | . 2 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈) |
7 | unitsubm.2 | . . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅) | |
8 | 7 | oveq1i 6802 | . . . 4 ⊢ (𝑀 ↾s 𝑈) = ((mulGrp‘𝑅) ↾s 𝑈) |
9 | 2, 8 | unitgrp 18871 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑀 ↾s 𝑈) ∈ Grp) |
10 | grpmnd 17633 | . . 3 ⊢ ((𝑀 ↾s 𝑈) ∈ Grp → (𝑀 ↾s 𝑈) ∈ Mnd) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → (𝑀 ↾s 𝑈) ∈ Mnd) |
12 | 7 | ringmgp 18757 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
13 | 7, 1 | mgpbas 18699 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑀) |
14 | 7, 5 | ringidval 18707 | . . . 4 ⊢ (1r‘𝑅) = (0g‘𝑀) |
15 | eqid 2771 | . . . 4 ⊢ (𝑀 ↾s 𝑈) = (𝑀 ↾s 𝑈) | |
16 | 13, 14, 15 | issubm2 17552 | . . 3 ⊢ (𝑀 ∈ Mnd → (𝑈 ∈ (SubMnd‘𝑀) ↔ (𝑈 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝑈 ∧ (𝑀 ↾s 𝑈) ∈ Mnd))) |
17 | 12, 16 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → (𝑈 ∈ (SubMnd‘𝑀) ↔ (𝑈 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝑈 ∧ (𝑀 ↾s 𝑈) ∈ Mnd))) |
18 | 4, 6, 11, 17 | mpbir3and 1427 | 1 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubMnd‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ‘cfv 6029 (class class class)co 6792 Basecbs 16060 ↾s cress 16061 Mndcmnd 17498 SubMndcsubmnd 17538 Grpcgrp 17626 mulGrpcmgp 18693 1rcur 18705 Ringcrg 18751 Unitcui 18843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7213 df-tpos 7504 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-ndx 16063 df-slot 16064 df-base 16066 df-sets 16067 df-ress 16068 df-plusg 16158 df-mulr 16159 df-0g 16306 df-mgm 17446 df-sgrp 17488 df-mnd 17499 df-submnd 17540 df-grp 17629 df-mgp 18694 df-ur 18706 df-ring 18753 df-oppr 18827 df-dvdsr 18845 df-unit 18846 |
This theorem is referenced by: zrhpsgnmhm 20141 nrgtdrg 22713 amgmlem 24933 dchrfi 25197 dchrghm 25198 dchrabs 25202 lgseisenlem3 25319 lgseisenlem4 25320 idomodle 38297 proot1ex 38302 amgmwlem 43076 amgmlemALT 43077 |
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