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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 20587 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈) |
| 5 | 1 | subrgring 20574 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) |
| 6 | eqid 2737 | . . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
| 7 | 3, 6 | 1unit 20374 | . . 3 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ 𝑉) |
| 8 | ne0i 4341 | . . 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 17351 | . . . . . . . . 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 7448 | . . . . . . 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 20380 | . . . . . . . 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 2841 | . . . . . 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 3146 | . . . 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 20588 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) = ((invr‘𝑆)‘𝑥)) |
| 23 | 3, 21 | unitinvcl 20390 | . . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉) |
| 24 | 5, 23 | sylan 580 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉) |
| 25 | 22, 24 | eqeltrd 2841 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) ∈ 𝑉) |
| 26 | 19, 25 | jca 511 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)) |
| 27 | 26 | ralrimiva 3146 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉)) |
| 28 | subrgrcl 20576 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
| 29 | subrgugrp.4 | . . . 4 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) | |
| 30 | 2, 29 | unitgrp 20383 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp) |
| 31 | 2, 29 | unitgrpbas 20382 | . . . 4 ⊢ 𝑈 = (Base‘𝐺) |
| 32 | 2 | fvexi 6920 | . . . . 5 ⊢ 𝑈 ∈ V |
| 33 | eqid 2737 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
| 34 | 33, 10 | mgpplusg 20141 | . . . . . 6 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
| 35 | 29, 34 | ressplusg 17334 | . . . . 5 ⊢ (𝑈 ∈ V → (.r‘𝑅) = (+g‘𝐺)) |
| 36 | 32, 35 | ax-mp 5 | . . . 4 ⊢ (.r‘𝑅) = (+g‘𝐺) |
| 37 | 2, 29, 20 | invrfval 20389 | . . . 4 ⊢ (invr‘𝑅) = (invg‘𝐺) |
| 38 | 31, 36, 37 | issubg2 19159 | . . 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 ↾s cress 17274 +gcplusg 17297 .rcmulr 17298 Grpcgrp 18951 SubGrpcsubg 19138 mulGrpcmgp 20137 1rcur 20178 Ringcrg 20230 Unitcui 20355 invrcinvr 20387 SubRingcsubrg 20569 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-subrg 20570 |
| This theorem is referenced by: (None) |
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