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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 19550 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈) |
5 | 1 | subrgring 19538 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) |
6 | eqid 2821 | . . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
7 | 3, 6 | 1unit 19408 | . . 3 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ 𝑉) |
8 | ne0i 4300 | . . 3 ⊢ ((1r‘𝑆) ∈ 𝑉 → 𝑉 ≠ ∅) | |
9 | 5, 7, 8 | 3syl 18 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ≠ ∅) |
10 | eqid 2821 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 1, 10 | ressmulr 16625 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
12 | 11 | 3ad2ant1 1129 | . . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆)) |
13 | 12 | oveqd 7173 | . . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑆)𝑦)) |
14 | eqid 2821 | . . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
15 | 3, 14 | unitmulcl 19414 | . . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉) |
16 | 5, 15 | syl3an1 1159 | . . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉) |
17 | 13, 16 | eqeltrd 2913 | . . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
18 | 17 | 3expa 1114 | . . . . 5 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
19 | 18 | ralrimiva 3182 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉) |
20 | eqid 2821 | . . . . . 6 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
21 | eqid 2821 | . . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆) | |
22 | 1, 20, 3, 21 | subrginv 19551 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) = ((invr‘𝑆)‘𝑥)) |
23 | 3, 21 | unitinvcl 19424 | . . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉) |
24 | 5, 23 | sylan 582 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉) |
25 | 22, 24 | eqeltrd 2913 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) ∈ 𝑉) |
26 | 19, 25 | jca 514 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)) |
27 | 26 | ralrimiva 3182 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)) |
28 | subrgrcl 19540 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
29 | subrgugrp.4 | . . . 4 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) | |
30 | 2, 29 | unitgrp 19417 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp) |
31 | 2, 29 | unitgrpbas 19416 | . . . 4 ⊢ 𝑈 = (Base‘𝐺) |
32 | 2 | fvexi 6684 | . . . . 5 ⊢ 𝑈 ∈ V |
33 | eqid 2821 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
34 | 33, 10 | mgpplusg 19243 | . . . . . 6 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
35 | 29, 34 | ressplusg 16612 | . . . . 5 ⊢ (𝑈 ∈ V → (.r‘𝑅) = (+g‘𝐺)) |
36 | 32, 35 | ax-mp 5 | . . . 4 ⊢ (.r‘𝑅) = (+g‘𝐺) |
37 | 2, 29, 20 | invrfval 19423 | . . . 4 ⊢ (invr‘𝑅) = (invg‘𝐺) |
38 | 31, 36, 37 | issubg2 18294 | . . 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 1338 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 ‘cfv 6355 (class class class)co 7156 ↾s cress 16484 +gcplusg 16565 .rcmulr 16566 Grpcgrp 18103 SubGrpcsubg 18273 mulGrpcmgp 19239 1rcur 19251 Ringcrg 19297 Unitcui 19389 invrcinvr 19421 SubRingcsubrg 19531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-subg 18276 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-subrg 19533 |
This theorem is referenced by: (None) |
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