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Mirrors > Home > MPE Home > Th. List > issdrg2 | Structured version Visualization version GIF version |
Description: Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.) |
Ref | Expression |
---|---|
issdrg2.i | ⊢ 𝐼 = (invr‘𝑅) |
issdrg2.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
issdrg2 | ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issdrg 19839 | . 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) | |
2 | eqid 2737 | . . . . 5 ⊢ (𝑅 ↾s 𝑆) = (𝑅 ↾s 𝑆) | |
3 | issdrg2.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
4 | issdrg2.i | . . . . 5 ⊢ 𝐼 = (invr‘𝑅) | |
5 | 2, 3, 4 | issubdrg 19825 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → ((𝑅 ↾s 𝑆) ∈ DivRing ↔ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆)) |
6 | 5 | pm5.32i 578 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅)) ∧ (𝑅 ↾s 𝑆) ∈ DivRing) ↔ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆)) |
7 | df-3an 1091 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing) ↔ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅)) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) | |
8 | df-3an 1091 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆) ↔ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆)) | |
9 | 6, 7, 8 | 3bitr4i 306 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆)) |
10 | 1, 9 | bitri 278 | 1 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∖ cdif 3863 {csn 4541 ‘cfv 6380 (class class class)co 7213 ↾s cress 16784 0gc0g 16944 invrcinvr 19689 DivRingcdr 19767 SubRingcsubrg 19796 SubDRingcsdrg 19837 |
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 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-minusg 18369 df-subg 18540 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-drng 19769 df-subrg 19798 df-sdrg 19838 |
This theorem is referenced by: sdrgacs 19845 cntzsdrg 19846 |
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