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Mirrors > Home > MPE Home > Th. List > issubrg3 | Structured version Visualization version GIF version |
Description: A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
issubrg3.m | ⊢ 𝑀 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
issubrg3 | ⊢ (𝑅 ∈ Ring → (𝑆 ∈ (SubRing‘𝑅) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ 𝑆 ∈ (SubMnd‘𝑀)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2725 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | eqid 2725 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | 1, 2, 3 | issubrg2 20543 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑆 ∈ (SubRing‘𝑅) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ (1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆))) |
5 | 3anass 1092 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝑅) ∧ (1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ ((1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆))) | |
6 | 4, 5 | bitrdi 286 | . 2 ⊢ (𝑅 ∈ Ring → (𝑆 ∈ (SubRing‘𝑅) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ ((1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆)))) |
7 | issubrg3.m | . . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅) | |
8 | 7 | ringmgp 20191 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
9 | 1 | subgss 19090 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → 𝑆 ⊆ (Base‘𝑅)) |
10 | 7, 1 | mgpbas 20092 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑀) |
11 | 7, 2 | ringidval 20135 | . . . . . . 7 ⊢ (1r‘𝑅) = (0g‘𝑀) |
12 | 7, 3 | mgpplusg 20090 | . . . . . . 7 ⊢ (.r‘𝑅) = (+g‘𝑀) |
13 | 10, 11, 12 | issubm 18763 | . . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆))) |
14 | 3anass 1092 | . . . . . 6 ⊢ ((𝑆 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ (Base‘𝑅) ∧ ((1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆))) | |
15 | 13, 14 | bitrdi 286 | . . . . 5 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑅) ∧ ((1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆)))) |
16 | 15 | baibd 538 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ (Base‘𝑅)) → (𝑆 ∈ (SubMnd‘𝑀) ↔ ((1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆))) |
17 | 8, 9, 16 | syl2an 594 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (SubGrp‘𝑅)) → (𝑆 ∈ (SubMnd‘𝑀) ↔ ((1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆))) |
18 | 17 | pm5.32da 577 | . 2 ⊢ (𝑅 ∈ Ring → ((𝑆 ∈ (SubGrp‘𝑅) ∧ 𝑆 ∈ (SubMnd‘𝑀)) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ ((1r‘𝑅) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑆)))) |
19 | 6, 18 | bitr4d 281 | 1 ⊢ (𝑅 ∈ Ring → (𝑆 ∈ (SubRing‘𝑅) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ 𝑆 ∈ (SubMnd‘𝑀)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ⊆ wss 3944 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 .rcmulr 17237 Mndcmnd 18697 SubMndcsubmnd 18742 SubGrpcsubg 19083 mulGrpcmgp 20086 1rcur 20133 Ringcrg 20185 SubRingcsubrg 20518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-subg 19086 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-subrng 20495 df-subrg 20520 |
This theorem is referenced by: rhmeql 20554 rhmima 20555 cntzsubr 20557 subrgacs 20700 |
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