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Mirrors > Home > MPE Home > Th. List > subrgugrp | Structured version Visualization version GIF version |
Description: The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
subrgugrp.1 | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
subrgugrp.2 | ⊢ 𝑈 = (Unit‘𝑅) |
subrgugrp.3 | ⊢ 𝑉 = (Unit‘𝑆) |
subrgugrp.4 | ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) |
Ref | Expression |
---|---|
subrgugrp | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgugrp.1 | . . 3 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
2 | subrgugrp.2 | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | subrgugrp.3 | . . 3 ⊢ 𝑉 = (Unit‘𝑆) | |
4 | 1, 2, 3 | subrguss 19286 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈) |
5 | 1 | subrgring 19274 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) |
6 | eqid 2773 | . . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
7 | 3, 6 | 1unit 19144 | . . 3 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ 𝑉) |
8 | ne0i 4181 | . . 3 ⊢ ((1r‘𝑆) ∈ 𝑉 → 𝑉 ≠ ∅) | |
9 | 5, 7, 8 | 3syl 18 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ≠ ∅) |
10 | eqid 2773 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 1, 10 | ressmulr 16480 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
12 | 11 | 3ad2ant1 1114 | . . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆)) |
13 | 12 | oveqd 6992 | . . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑆)𝑦)) |
14 | eqid 2773 | . . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
15 | 3, 14 | unitmulcl 19150 | . . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉) |
16 | 5, 15 | syl3an1 1144 | . . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉) |
17 | 13, 16 | eqeltrd 2861 | . . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
18 | 17 | 3expa 1099 | . . . . 5 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
19 | 18 | ralrimiva 3127 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
20 | eqid 2773 | . . . . . 6 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
21 | eqid 2773 | . . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆) | |
22 | 1, 20, 3, 21 | subrginv 19287 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) = ((invr‘𝑆)‘𝑥)) |
23 | 3, 21 | unitinvcl 19160 | . . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉) |
24 | 5, 23 | sylan 572 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉) |
25 | 22, 24 | eqeltrd 2861 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) ∈ 𝑉) |
26 | 19, 25 | jca 504 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)) |
27 | 26 | ralrimiva 3127 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)) |
28 | subrgrcl 19276 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
29 | subrgugrp.4 | . . . 4 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) | |
30 | 2, 29 | unitgrp 19153 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp) |
31 | 2, 29 | unitgrpbas 19152 | . . . 4 ⊢ 𝑈 = (Base‘𝐺) |
32 | 2 | fvexi 6511 | . . . . 5 ⊢ 𝑈 ∈ V |
33 | eqid 2773 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
34 | 33, 10 | mgpplusg 18979 | . . . . . 6 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
35 | 29, 34 | ressplusg 16467 | . . . . 5 ⊢ (𝑈 ∈ V → (.r‘𝑅) = (+g‘𝐺)) |
36 | 32, 35 | ax-mp 5 | . . . 4 ⊢ (.r‘𝑅) = (+g‘𝐺) |
37 | 2, 29, 20 | invrfval 19159 | . . . 4 ⊢ (invr‘𝑅) = (invg‘𝐺) |
38 | 31, 36, 37 | issubg2 18091 | . . 3 ⊢ (𝐺 ∈ Grp → (𝑉 ∈ (SubGrp‘𝐺) ↔ (𝑉 ⊆ 𝑈 ∧ 𝑉 ≠ ∅ ∧ ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)))) |
39 | 28, 30, 38 | 3syl 18 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑉 ∈ (SubGrp‘𝐺) ↔ (𝑉 ⊆ 𝑈 ∧ 𝑉 ≠ ∅ ∧ ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)))) |
40 | 4, 9, 27, 39 | mpbir3and 1323 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 ∀wral 3083 Vcvv 3410 ⊆ wss 3824 ∅c0 4173 ‘cfv 6186 (class class class)co 6975 ↾s cress 16339 +gcplusg 16420 .rcmulr 16421 Grpcgrp 17904 SubGrpcsubg 18070 mulGrpcmgp 18975 1rcur 18987 Ringcrg 19033 Unitcui 19125 invrcinvr 19157 SubRingcsubrg 19267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-tpos 7694 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-3 11503 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-ress 16346 df-plusg 16433 df-mulr 16434 df-0g 16570 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-grp 17907 df-minusg 17908 df-subg 18073 df-mgp 18976 df-ur 18988 df-ring 19035 df-oppr 19109 df-dvdsr 19127 df-unit 19128 df-invr 19158 df-subrg 19269 |
This theorem is referenced by: (None) |
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