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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 19539 | . . . 4 ⊢ (𝐼 ∈ Ring → 𝐼 ∈ Grp) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 ∈ Grp) |
11 | 1, 2, 3, 4, 5, 6, 7, 10 | issubgrpd2 18531 | . 2 ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼)) |
12 | issubrngd.o | . . 3 ⊢ (𝜑 → 1 = (1r‘𝐼)) | |
13 | issubrngd.ocl | . . 3 ⊢ (𝜑 → 1 ∈ 𝐷) | |
14 | 12, 13 | eqeltrrd 2835 | . 2 ⊢ (𝜑 → (1r‘𝐼) ∈ 𝐷) |
15 | issubrngd.t | . . . . 5 ⊢ (𝜑 → · = (.r‘𝐼)) | |
16 | 15 | oveqdr 7230 | . . . 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 1122 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 · 𝑦) ∈ 𝐷) |
19 | 16, 18 | eqeltrrd 2835 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥(.r‘𝐼)𝑦) ∈ 𝐷) |
20 | 19 | ralrimivva 3105 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥(.r‘𝐼)𝑦) ∈ 𝐷) |
21 | eqid 2734 | . . . 4 ⊢ (Base‘𝐼) = (Base‘𝐼) | |
22 | eqid 2734 | . . . 4 ⊢ (1r‘𝐼) = (1r‘𝐼) | |
23 | eqid 2734 | . . . 4 ⊢ (.r‘𝐼) = (.r‘𝐼) | |
24 | 21, 22, 23 | issubrg2 19792 | . . 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 1344 | 1 ⊢ (𝜑 → 𝐷 ∈ (SubRing‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ⊆ wss 3857 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 ↾s cress 16685 +gcplusg 16767 .rcmulr 16768 0gc0g 16916 Grpcgrp 18337 invgcminusg 18338 SubGrpcsubg 18509 1rcur 19488 Ringcrg 19534 SubRingcsubrg 19768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-subg 18512 df-mgp 19477 df-ur 19489 df-ring 19536 df-subrg 19770 |
This theorem is referenced by: rngunsnply 40653 |
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