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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 | β’ π = (Unitβπ ) |
unitsubm.2 | β’ π = (mulGrpβπ ) |
Ref | Expression |
---|---|
unitsubm | β’ (π β Ring β π β (SubMndβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
2 | unitsubm.1 | . . . 4 β’ π = (Unitβπ ) | |
3 | 1, 2 | unitss 20317 | . . 3 β’ π β (Baseβπ ) |
4 | 3 | a1i 11 | . 2 β’ (π β Ring β π β (Baseβπ )) |
5 | eqid 2725 | . . 3 β’ (1rβπ ) = (1rβπ ) | |
6 | 2, 5 | 1unit 20315 | . 2 β’ (π β Ring β (1rβπ ) β π) |
7 | unitsubm.2 | . . . . 5 β’ π = (mulGrpβπ ) | |
8 | 7 | oveq1i 7424 | . . . 4 β’ (π βΎs π) = ((mulGrpβπ ) βΎs π) |
9 | 2, 8 | unitgrp 20324 | . . 3 β’ (π β Ring β (π βΎs π) β Grp) |
10 | 9 | grpmndd 18905 | . 2 β’ (π β Ring β (π βΎs π) β Mnd) |
11 | 7 | ringmgp 20181 | . . 3 β’ (π β Ring β π β Mnd) |
12 | 7, 1 | mgpbas 20082 | . . . 4 β’ (Baseβπ ) = (Baseβπ) |
13 | 7, 5 | ringidval 20125 | . . . 4 β’ (1rβπ ) = (0gβπ) |
14 | eqid 2725 | . . . 4 β’ (π βΎs π) = (π βΎs π) | |
15 | 12, 13, 14 | issubm2 18758 | . . 3 β’ (π β Mnd β (π β (SubMndβπ) β (π β (Baseβπ ) β§ (1rβπ ) β π β§ (π βΎs π) β Mnd))) |
16 | 11, 15 | syl 17 | . 2 β’ (π β Ring β (π β (SubMndβπ) β (π β (Baseβπ ) β§ (1rβπ ) β π β§ (π βΎs π) β Mnd))) |
17 | 4, 6, 10, 16 | mpbir3and 1339 | 1 β’ (π β Ring β π β (SubMndβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3939 βcfv 6541 (class class class)co 7414 Basecbs 17177 βΎs cress 17206 Mndcmnd 18691 SubMndcsubmnd 18736 mulGrpcmgp 20076 1rcur 20123 Ringcrg 20175 Unitcui 20296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-grp 18895 df-minusg 18896 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-dvdsr 20298 df-unit 20299 |
This theorem is referenced by: zrhpsgnmhm 21518 nrgtdrg 24626 amgmlem 26938 dchrfi 27204 dchrghm 27205 dchrabs 27209 lgseisenlem3 27326 lgseisenlem4 27327 idomodle 42656 proot1ex 42661 amgmwlem 48319 amgmlemALT 48320 |
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