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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 2727 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | unitsubm.1 | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | 1, 2 | unitss 20297 | . . 3 ⊢ 𝑈 ⊆ (Base‘𝑅) |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅)) |
5 | eqid 2727 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
6 | 2, 5 | 1unit 20295 | . 2 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈) |
7 | unitsubm.2 | . . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅) | |
8 | 7 | oveq1i 7424 | . . . 4 ⊢ (𝑀 ↾s 𝑈) = ((mulGrp‘𝑅) ↾s 𝑈) |
9 | 2, 8 | unitgrp 20304 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑀 ↾s 𝑈) ∈ Grp) |
10 | 9 | grpmndd 18888 | . 2 ⊢ (𝑅 ∈ Ring → (𝑀 ↾s 𝑈) ∈ Mnd) |
11 | 7 | ringmgp 20163 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
12 | 7, 1 | mgpbas 20064 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑀) |
13 | 7, 5 | ringidval 20107 | . . . 4 ⊢ (1r‘𝑅) = (0g‘𝑀) |
14 | eqid 2727 | . . . 4 ⊢ (𝑀 ↾s 𝑈) = (𝑀 ↾s 𝑈) | |
15 | 12, 13, 14 | issubm2 18741 | . . 3 ⊢ (𝑀 ∈ Mnd → (𝑈 ∈ (SubMnd‘𝑀) ↔ (𝑈 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝑈 ∧ (𝑀 ↾s 𝑈) ∈ Mnd))) |
16 | 11, 15 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → (𝑈 ∈ (SubMnd‘𝑀) ↔ (𝑈 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝑈 ∧ (𝑀 ↾s 𝑈) ∈ Mnd))) |
17 | 4, 6, 10, 16 | mpbir3and 1340 | 1 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubMnd‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 ‘cfv 6542 (class class class)co 7414 Basecbs 17165 ↾s cress 17194 Mndcmnd 18679 SubMndcsubmnd 18724 mulGrpcmgp 20058 1rcur 20105 Ringcrg 20157 Unitcui 20276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-0g 17408 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-grp 18878 df-minusg 18879 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-oppr 20255 df-dvdsr 20278 df-unit 20279 |
This theorem is referenced by: zrhpsgnmhm 21496 nrgtdrg 24584 amgmlem 26896 dchrfi 27162 dchrghm 27163 dchrabs 27167 lgseisenlem3 27284 lgseisenlem4 27285 idomodle 42531 proot1ex 42536 amgmwlem 48148 amgmlemALT 48149 |
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