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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 20355 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈) |
5 | 1 | subrgring 20343 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) |
6 | eqid 2733 | . . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
7 | 3, 6 | 1unit 20166 | . . 3 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ 𝑉) |
8 | ne0i 4332 | . . 3 ⊢ ((1r‘𝑆) ∈ 𝑉 → 𝑉 ≠ ∅) | |
9 | 5, 7, 8 | 3syl 18 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ≠ ∅) |
10 | eqid 2733 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 1, 10 | ressmulr 17239 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
12 | 11 | 3ad2ant1 1134 | . . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆)) |
13 | 12 | oveqd 7413 | . . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑆)𝑦)) |
14 | eqid 2733 | . . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
15 | 3, 14 | unitmulcl 20172 | . . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉) |
16 | 5, 15 | syl3an1 1164 | . . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉) |
17 | 13, 16 | eqeltrd 2834 | . . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
18 | 17 | 3expa 1119 | . . . . 5 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
19 | 18 | ralrimiva 3147 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
20 | eqid 2733 | . . . . . 6 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
21 | eqid 2733 | . . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆) | |
22 | 1, 20, 3, 21 | subrginv 20356 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) = ((invr‘𝑆)‘𝑥)) |
23 | 3, 21 | unitinvcl 20182 | . . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉) |
24 | 5, 23 | sylan 581 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉) |
25 | 22, 24 | eqeltrd 2834 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) ∈ 𝑉) |
26 | 19, 25 | jca 513 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)) |
27 | 26 | ralrimiva 3147 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)) |
28 | subrgrcl 20345 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
29 | subrgugrp.4 | . . . 4 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) | |
30 | 2, 29 | unitgrp 20175 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp) |
31 | 2, 29 | unitgrpbas 20174 | . . . 4 ⊢ 𝑈 = (Base‘𝐺) |
32 | 2 | fvexi 6895 | . . . . 5 ⊢ 𝑈 ∈ V |
33 | eqid 2733 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
34 | 33, 10 | mgpplusg 19974 | . . . . . 6 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
35 | 29, 34 | ressplusg 17222 | . . . . 5 ⊢ (𝑈 ∈ V → (.r‘𝑅) = (+g‘𝐺)) |
36 | 32, 35 | ax-mp 5 | . . . 4 ⊢ (.r‘𝑅) = (+g‘𝐺) |
37 | 2, 29, 20 | invrfval 20181 | . . . 4 ⊢ (invr‘𝑅) = (invg‘𝐺) |
38 | 31, 36, 37 | issubg2 19006 | . . 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 1343 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ⊆ wss 3946 ∅c0 4320 ‘cfv 6535 (class class class)co 7396 ↾s cress 17160 +gcplusg 17184 .rcmulr 17185 Grpcgrp 18806 SubGrpcsubg 18985 mulGrpcmgp 19970 1rcur 19987 Ringcrg 20038 Unitcui 20147 invrcinvr 20179 SubRingcsubrg 20336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 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 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-tpos 8198 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-0g 17374 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-grp 18809 df-minusg 18810 df-subg 18988 df-mgp 19971 df-ur 19988 df-ring 20040 df-oppr 20128 df-dvdsr 20149 df-unit 20150 df-invr 20180 df-subrg 20338 |
This theorem is referenced by: (None) |
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