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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 19302 | . . . 4 ⊢ (𝐼 ∈ Ring → 𝐼 ∈ Grp) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 ∈ Grp) |
11 | 1, 2, 3, 4, 5, 6, 7, 10 | issubgrpd2 18295 | . 2 ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼)) |
12 | issubrngd.o | . . 3 ⊢ (𝜑 → 1 = (1r‘𝐼)) | |
13 | issubrngd.ocl | . . 3 ⊢ (𝜑 → 1 ∈ 𝐷) | |
14 | 12, 13 | eqeltrrd 2914 | . 2 ⊢ (𝜑 → (1r‘𝐼) ∈ 𝐷) |
15 | issubrngd.t | . . . . 5 ⊢ (𝜑 → · = (.r‘𝐼)) | |
16 | 15 | oveqdr 7184 | . . . 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 1116 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 · 𝑦) ∈ 𝐷) |
19 | 16, 18 | eqeltrrd 2914 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥(.r‘𝐼)𝑦) ∈ 𝐷) |
20 | 19 | ralrimivva 3191 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥(.r‘𝐼)𝑦) ∈ 𝐷) |
21 | eqid 2821 | . . . 4 ⊢ (Base‘𝐼) = (Base‘𝐼) | |
22 | eqid 2821 | . . . 4 ⊢ (1r‘𝐼) = (1r‘𝐼) | |
23 | eqid 2821 | . . . 4 ⊢ (.r‘𝐼) = (.r‘𝐼) | |
24 | 21, 22, 23 | issubrg2 19555 | . . 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 1338 | 1 ⊢ (𝜑 → 𝐷 ∈ (SubRing‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 +gcplusg 16565 .rcmulr 16566 0gc0g 16713 Grpcgrp 18103 invgcminusg 18104 SubGrpcsubg 18273 1rcur 19251 Ringcrg 19297 SubRingcsubrg 19531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-subg 18276 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 |
This theorem is referenced by: rngunsnply 39793 |
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