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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 19815 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈) |
5 | 1 | subrgring 19803 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) |
6 | eqid 2737 | . . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
7 | 3, 6 | 1unit 19676 | . . 3 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ 𝑉) |
8 | ne0i 4249 | . . 3 ⊢ ((1r‘𝑆) ∈ 𝑉 → 𝑉 ≠ ∅) | |
9 | 5, 7, 8 | 3syl 18 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ≠ ∅) |
10 | eqid 2737 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 1, 10 | ressmulr 16848 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
12 | 11 | 3ad2ant1 1135 | . . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆)) |
13 | 12 | oveqd 7230 | . . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑆)𝑦)) |
14 | eqid 2737 | . . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
15 | 3, 14 | unitmulcl 19682 | . . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉) |
16 | 5, 15 | syl3an1 1165 | . . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉) |
17 | 13, 16 | eqeltrd 2838 | . . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
18 | 17 | 3expa 1120 | . . . . 5 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
19 | 18 | ralrimiva 3105 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
20 | eqid 2737 | . . . . . 6 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
21 | eqid 2737 | . . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆) | |
22 | 1, 20, 3, 21 | subrginv 19816 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) = ((invr‘𝑆)‘𝑥)) |
23 | 3, 21 | unitinvcl 19692 | . . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉) |
24 | 5, 23 | sylan 583 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉) |
25 | 22, 24 | eqeltrd 2838 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) ∈ 𝑉) |
26 | 19, 25 | jca 515 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)) |
27 | 26 | ralrimiva 3105 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)) |
28 | subrgrcl 19805 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
29 | subrgugrp.4 | . . . 4 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) | |
30 | 2, 29 | unitgrp 19685 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp) |
31 | 2, 29 | unitgrpbas 19684 | . . . 4 ⊢ 𝑈 = (Base‘𝐺) |
32 | 2 | fvexi 6731 | . . . . 5 ⊢ 𝑈 ∈ V |
33 | eqid 2737 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
34 | 33, 10 | mgpplusg 19508 | . . . . . 6 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
35 | 29, 34 | ressplusg 16834 | . . . . 5 ⊢ (𝑈 ∈ V → (.r‘𝑅) = (+g‘𝐺)) |
36 | 32, 35 | ax-mp 5 | . . . 4 ⊢ (.r‘𝑅) = (+g‘𝐺) |
37 | 2, 29, 20 | invrfval 19691 | . . . 4 ⊢ (invr‘𝑅) = (invg‘𝐺) |
38 | 31, 36, 37 | issubg2 18558 | . . 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 1344 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 Vcvv 3408 ⊆ wss 3866 ∅c0 4237 ‘cfv 6380 (class class class)co 7213 ↾s cress 16784 +gcplusg 16802 .rcmulr 16803 Grpcgrp 18365 SubGrpcsubg 18537 mulGrpcmgp 19504 1rcur 19516 Ringcrg 19562 Unitcui 19657 invrcinvr 19689 SubRingcsubrg 19796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-minusg 18369 df-subg 18540 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-subrg 19798 |
This theorem is referenced by: (None) |
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