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Mirrors > Home > MPE Home > Th. List > issubrngd2 | Structured version Visualization version GIF version |
Description: Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.) |
Ref | Expression |
---|---|
issubrngd.s | ⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) |
issubrngd.z | ⊢ (𝜑 → 0 = (0g‘𝐼)) |
issubrngd.p | ⊢ (𝜑 → + = (+g‘𝐼)) |
issubrngd.ss | ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) |
issubrngd.zcl | ⊢ (𝜑 → 0 ∈ 𝐷) |
issubrngd.acl | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) |
issubrngd.ncl | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) |
issubrngd.o | ⊢ (𝜑 → 1 = (1r‘𝐼)) |
issubrngd.t | ⊢ (𝜑 → · = (.r‘𝐼)) |
issubrngd.ocl | ⊢ (𝜑 → 1 ∈ 𝐷) |
issubrngd.tcl | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 · 𝑦) ∈ 𝐷) |
issubrngd.g | ⊢ (𝜑 → 𝐼 ∈ Ring) |
Ref | Expression |
---|---|
issubrngd2 | ⊢ (𝜑 → 𝐷 ∈ (SubRing‘𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubrngd.s | . . 3 ⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) | |
2 | issubrngd.z | . . 3 ⊢ (𝜑 → 0 = (0g‘𝐼)) | |
3 | issubrngd.p | . . 3 ⊢ (𝜑 → + = (+g‘𝐼)) | |
4 | issubrngd.ss | . . 3 ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) | |
5 | issubrngd.zcl | . . 3 ⊢ (𝜑 → 0 ∈ 𝐷) | |
6 | issubrngd.acl | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) | |
7 | issubrngd.ncl | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) | |
8 | issubrngd.g | . . . 4 ⊢ (𝜑 → 𝐼 ∈ Ring) | |
9 | ringgrp 18552 | . . . 4 ⊢ (𝐼 ∈ Ring → 𝐼 ∈ Grp) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 ∈ Grp) |
11 | 1, 2, 3, 4, 5, 6, 7, 10 | issubgrpd2 17610 | . 2 ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼)) |
12 | issubrngd.o | . . 3 ⊢ (𝜑 → 1 = (1r‘𝐼)) | |
13 | issubrngd.ocl | . . 3 ⊢ (𝜑 → 1 ∈ 𝐷) | |
14 | 12, 13 | eqeltrrd 2702 | . 2 ⊢ (𝜑 → (1r‘𝐼) ∈ 𝐷) |
15 | issubrngd.t | . . . . 5 ⊢ (𝜑 → · = (.r‘𝐼)) | |
16 | 15 | oveqdr 6674 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 · 𝑦) = (𝑥(.r‘𝐼)𝑦)) |
17 | issubrngd.tcl | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 · 𝑦) ∈ 𝐷) | |
18 | 17 | 3expb 1266 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 · 𝑦) ∈ 𝐷) |
19 | 16, 18 | eqeltrrd 2702 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥(.r‘𝐼)𝑦) ∈ 𝐷) |
20 | 19 | ralrimivva 2971 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥(.r‘𝐼)𝑦) ∈ 𝐷) |
21 | eqid 2622 | . . . 4 ⊢ (Base‘𝐼) = (Base‘𝐼) | |
22 | eqid 2622 | . . . 4 ⊢ (1r‘𝐼) = (1r‘𝐼) | |
23 | eqid 2622 | . . . 4 ⊢ (.r‘𝐼) = (.r‘𝐼) | |
24 | 21, 22, 23 | issubrg2 18800 | . . 3 ⊢ (𝐼 ∈ Ring → (𝐷 ∈ (SubRing‘𝐼) ↔ (𝐷 ∈ (SubGrp‘𝐼) ∧ (1r‘𝐼) ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥(.r‘𝐼)𝑦) ∈ 𝐷))) |
25 | 8, 24 | syl 17 | . 2 ⊢ (𝜑 → (𝐷 ∈ (SubRing‘𝐼) ↔ (𝐷 ∈ (SubGrp‘𝐼) ∧ (1r‘𝐼) ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥(.r‘𝐼)𝑦) ∈ 𝐷))) |
26 | 11, 14, 20, 25 | mpbir3and 1245 | 1 ⊢ (𝜑 → 𝐷 ∈ (SubRing‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ↾s cress 15858 +gcplusg 15941 .rcmulr 15942 0gc0g 16100 Grpcgrp 17422 invgcminusg 17423 SubGrpcsubg 17588 1rcur 18501 Ringcrg 18547 SubRingcsubrg 18776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 |
This theorem is referenced by: rngunsnply 37743 |
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