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| Mirrors > Home > MPE Home > Th. List > issubrgd | Structured version Visualization version GIF version | ||
| Description: Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| issubrgd.s | ⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) | 
| issubrgd.z | ⊢ (𝜑 → 0 = (0g‘𝐼)) | 
| issubrgd.p | ⊢ (𝜑 → + = (+g‘𝐼)) | 
| issubrgd.ss | ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) | 
| issubrgd.zcl | ⊢ (𝜑 → 0 ∈ 𝐷) | 
| issubrgd.acl | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) | 
| issubrgd.ncl | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) | 
| issubrgd.o | ⊢ (𝜑 → 1 = (1r‘𝐼)) | 
| issubrgd.t | ⊢ (𝜑 → · = (.r‘𝐼)) | 
| issubrgd.ocl | ⊢ (𝜑 → 1 ∈ 𝐷) | 
| issubrgd.tcl | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 · 𝑦) ∈ 𝐷) | 
| issubrgd.g | ⊢ (𝜑 → 𝐼 ∈ Ring) | 
| Ref | Expression | 
|---|---|
| issubrgd | ⊢ (𝜑 → 𝐷 ∈ (SubRing‘𝐼)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | issubrgd.s | . . 3 ⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) | |
| 2 | issubrgd.z | . . 3 ⊢ (𝜑 → 0 = (0g‘𝐼)) | |
| 3 | issubrgd.p | . . 3 ⊢ (𝜑 → + = (+g‘𝐼)) | |
| 4 | issubrgd.ss | . . 3 ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) | |
| 5 | issubrgd.zcl | . . 3 ⊢ (𝜑 → 0 ∈ 𝐷) | |
| 6 | issubrgd.acl | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) | |
| 7 | issubrgd.ncl | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) | |
| 8 | issubrgd.g | . . . 4 ⊢ (𝜑 → 𝐼 ∈ Ring) | |
| 9 | ringgrp 20236 | . . . 4 ⊢ (𝐼 ∈ Ring → 𝐼 ∈ Grp) | |
| 10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 ∈ Grp) | 
| 11 | 1, 2, 3, 4, 5, 6, 7, 10 | issubgrpd2 19161 | . 2 ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼)) | 
| 12 | issubrgd.o | . . 3 ⊢ (𝜑 → 1 = (1r‘𝐼)) | |
| 13 | issubrgd.ocl | . . 3 ⊢ (𝜑 → 1 ∈ 𝐷) | |
| 14 | 12, 13 | eqeltrrd 2841 | . 2 ⊢ (𝜑 → (1r‘𝐼) ∈ 𝐷) | 
| 15 | issubrgd.t | . . . . 5 ⊢ (𝜑 → · = (.r‘𝐼)) | |
| 16 | 15 | oveqdr 7460 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 · 𝑦) = (𝑥(.r‘𝐼)𝑦)) | 
| 17 | issubrgd.tcl | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 · 𝑦) ∈ 𝐷) | |
| 18 | 17 | 3expb 1120 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 · 𝑦) ∈ 𝐷) | 
| 19 | 16, 18 | eqeltrrd 2841 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥(.r‘𝐼)𝑦) ∈ 𝐷) | 
| 20 | 19 | ralrimivva 3201 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥(.r‘𝐼)𝑦) ∈ 𝐷) | 
| 21 | eqid 2736 | . . . 4 ⊢ (Base‘𝐼) = (Base‘𝐼) | |
| 22 | eqid 2736 | . . . 4 ⊢ (1r‘𝐼) = (1r‘𝐼) | |
| 23 | eqid 2736 | . . . 4 ⊢ (.r‘𝐼) = (.r‘𝐼) | |
| 24 | 21, 22, 23 | issubrg2 20593 | . . 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 1342 | 1 ⊢ (𝜑 → 𝐷 ∈ (SubRing‘𝐼)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ⊆ wss 3950 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 ↾s cress 17275 +gcplusg 17298 .rcmulr 17299 0gc0g 17485 Grpcgrp 18952 invgcminusg 18953 SubGrpcsubg 19139 1rcur 20179 Ringcrg 20231 SubRingcsubrg 20570 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-subg 19142 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-subrng 20547 df-subrg 20571 | 
| This theorem is referenced by: rngunsnply 43186 | 
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