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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 2821 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | unitsubm.1 | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | 1, 2 | unitss 19404 | . . 3 ⊢ 𝑈 ⊆ (Base‘𝑅) |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅)) |
5 | eqid 2821 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | 2, 5 | 1unit 19402 | . 2 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈) |
7 | unitsubm.2 | . . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅) | |
8 | 7 | oveq1i 7160 | . . . 4 ⊢ (𝑀 ↾s 𝑈) = ((mulGrp‘𝑅) ↾s 𝑈) |
9 | 2, 8 | unitgrp 19411 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑀 ↾s 𝑈) ∈ Grp) |
10 | grpmnd 18104 | . . 3 ⊢ ((𝑀 ↾s 𝑈) ∈ Grp → (𝑀 ↾s 𝑈) ∈ Mnd) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → (𝑀 ↾s 𝑈) ∈ Mnd) |
12 | 7 | ringmgp 19297 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
13 | 7, 1 | mgpbas 19239 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑀) |
14 | 7, 5 | ringidval 19247 | . . . 4 ⊢ (1r‘𝑅) = (0g‘𝑀) |
15 | eqid 2821 | . . . 4 ⊢ (𝑀 ↾s 𝑈) = (𝑀 ↾s 𝑈) | |
16 | 13, 14, 15 | issubm2 17963 | . . 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 1338 | 1 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubMnd‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 ↾s cress 16478 Mndcmnd 17905 SubMndcsubmnd 17949 Grpcgrp 18097 mulGrpcmgp 19233 1rcur 19245 Ringcrg 19291 Unitcui 19383 |
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-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 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-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 |
This theorem is referenced by: zrhpsgnmhm 20722 nrgtdrg 23296 amgmlem 25561 dchrfi 25825 dchrghm 25826 dchrabs 25830 lgseisenlem3 25947 lgseisenlem4 25948 idomodle 39789 proot1ex 39794 amgmwlem 44897 amgmlemALT 44898 |
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