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Mirrors > Home > ILE Home > Th. List > unitsubm | 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 | eqidd 2178 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅)) | |
2 | unitsubm.1 | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑈 = (Unit‘𝑅)) |
4 | ringsrg 13177 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | |
5 | 1, 3, 4 | unitssd 13231 | . 2 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅)) |
6 | eqid 2177 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 2, 6 | 1unit 13229 | . 2 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈) |
8 | unitsubm.2 | . . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅) | |
9 | 8 | oveq1i 5884 | . . . 4 ⊢ (𝑀 ↾s 𝑈) = ((mulGrp‘𝑅) ↾s 𝑈) |
10 | 2, 9 | unitgrp 13238 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑀 ↾s 𝑈) ∈ Grp) |
11 | 10 | grpmndd 12843 | . 2 ⊢ (𝑅 ∈ Ring → (𝑀 ↾s 𝑈) ∈ Mnd) |
12 | 8 | ringmgp 13138 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
13 | eqid 2177 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
14 | eqid 2177 | . . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
15 | eqid 2177 | . . . . 5 ⊢ (𝑀 ↾s 𝑈) = (𝑀 ↾s 𝑈) | |
16 | 13, 14, 15 | issubm2 12818 | . . . 4 ⊢ (𝑀 ∈ Mnd → (𝑈 ∈ (SubMnd‘𝑀) ↔ (𝑈 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑈 ∧ (𝑀 ↾s 𝑈) ∈ Mnd))) |
17 | 12, 16 | syl 14 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑈 ∈ (SubMnd‘𝑀) ↔ (𝑈 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑈 ∧ (𝑀 ↾s 𝑈) ∈ Mnd))) |
18 | eqid 2177 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
19 | 8, 18 | mgpbasg 13089 | . . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑀)) |
20 | 19 | sseq2d 3185 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑈 ⊆ (Base‘𝑅) ↔ 𝑈 ⊆ (Base‘𝑀))) |
21 | 8, 6 | ringidvalg 13097 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (0g‘𝑀)) |
22 | 21 | eleq1d 2246 | . . . 4 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ∈ 𝑈 ↔ (0g‘𝑀) ∈ 𝑈)) |
23 | 20, 22 | 3anbi12d 1313 | . . 3 ⊢ (𝑅 ∈ Ring → ((𝑈 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝑈 ∧ (𝑀 ↾s 𝑈) ∈ Mnd) ↔ (𝑈 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑈 ∧ (𝑀 ↾s 𝑈) ∈ Mnd))) |
24 | 17, 23 | bitr4d 191 | . 2 ⊢ (𝑅 ∈ Ring → (𝑈 ∈ (SubMnd‘𝑀) ↔ (𝑈 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝑈 ∧ (𝑀 ↾s 𝑈) ∈ Mnd))) |
25 | 5, 7, 11, 24 | mpbir3and 1180 | 1 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubMnd‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ⊆ wss 3129 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 ↾s cress 12457 0gc0g 12695 Mndcmnd 12771 SubMndcsubmnd 12804 mulGrpcmgp 13083 1rcur 13095 Ringcrg 13132 Unitcui 13209 |
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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
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 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-tpos 6245 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-iress 12464 df-plusg 12543 df-mulr 12544 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-submnd 12806 df-grp 12834 df-minusg 12835 df-cmn 13043 df-abl 13044 df-mgp 13084 df-ur 13096 df-srg 13100 df-ring 13134 df-oppr 13193 df-dvdsr 13211 df-unit 13212 |
This theorem is referenced by: (None) |
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