Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rgspncl | Structured version Visualization version GIF version |
Description: The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
Ref | Expression |
---|---|
rgspnval.r | β’ (π β π β Ring) |
rgspnval.b | β’ (π β π΅ = (Baseβπ )) |
rgspnval.ss | β’ (π β π΄ β π΅) |
rgspnval.n | β’ (π β π = (RingSpanβπ )) |
rgspnval.sp | β’ (π β π = (πβπ΄)) |
Ref | Expression |
---|---|
rgspncl | β’ (π β π β (SubRingβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rgspnval.r | . . 3 β’ (π β π β Ring) | |
2 | rgspnval.b | . . 3 β’ (π β π΅ = (Baseβπ )) | |
3 | rgspnval.ss | . . 3 β’ (π β π΄ β π΅) | |
4 | rgspnval.n | . . 3 β’ (π β π = (RingSpanβπ )) | |
5 | rgspnval.sp | . . 3 β’ (π β π = (πβπ΄)) | |
6 | 1, 2, 3, 4, 5 | rgspnval 41030 | . 2 β’ (π β π = β© {π‘ β (SubRingβπ ) β£ π΄ β π‘}) |
7 | ssrab2 4019 | . . 3 β’ {π‘ β (SubRingβπ ) β£ π΄ β π‘} β (SubRingβπ ) | |
8 | eqid 2736 | . . . . . . . 8 β’ (Baseβπ ) = (Baseβπ ) | |
9 | 8 | subrgid 20067 | . . . . . . 7 β’ (π β Ring β (Baseβπ ) β (SubRingβπ )) |
10 | 1, 9 | syl 17 | . . . . . 6 β’ (π β (Baseβπ ) β (SubRingβπ )) |
11 | 2, 10 | eqeltrd 2837 | . . . . 5 β’ (π β π΅ β (SubRingβπ )) |
12 | sseq2 3952 | . . . . . 6 β’ (π‘ = π΅ β (π΄ β π‘ β π΄ β π΅)) | |
13 | 12 | rspcev 3566 | . . . . 5 β’ ((π΅ β (SubRingβπ ) β§ π΄ β π΅) β βπ‘ β (SubRingβπ )π΄ β π‘) |
14 | 11, 3, 13 | syl2anc 585 | . . . 4 β’ (π β βπ‘ β (SubRingβπ )π΄ β π‘) |
15 | rabn0 4325 | . . . 4 β’ ({π‘ β (SubRingβπ ) β£ π΄ β π‘} β β β βπ‘ β (SubRingβπ )π΄ β π‘) | |
16 | 14, 15 | sylibr 234 | . . 3 β’ (π β {π‘ β (SubRingβπ ) β£ π΄ β π‘} β β ) |
17 | subrgint 20087 | . . 3 β’ (({π‘ β (SubRingβπ ) β£ π΄ β π‘} β (SubRingβπ ) β§ {π‘ β (SubRingβπ ) β£ π΄ β π‘} β β ) β β© {π‘ β (SubRingβπ ) β£ π΄ β π‘} β (SubRingβπ )) | |
18 | 7, 16, 17 | sylancr 588 | . 2 β’ (π β β© {π‘ β (SubRingβπ ) β£ π΄ β π‘} β (SubRingβπ )) |
19 | 6, 18 | eqeltrd 2837 | 1 β’ (π β π β (SubRingβπ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 β wne 2941 βwrex 3071 {crab 3284 β wss 3892 β c0 4262 β© cint 4886 βcfv 6454 Basecbs 16953 Ringcrg 19824 SubRingcsubrg 20061 RingSpancrgspn 20062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-nn 12016 df-2 12078 df-3 12079 df-sets 16906 df-slot 16924 df-ndx 16936 df-base 16954 df-ress 16983 df-plusg 17016 df-mulr 17017 df-0g 17193 df-mgm 18367 df-sgrp 18416 df-mnd 18427 df-grp 18621 df-minusg 18622 df-subg 18793 df-mgp 19762 df-ur 19779 df-ring 19826 df-subrg 20063 df-rgspn 20064 |
This theorem is referenced by: rngunsnply 41035 |
Copyright terms: Public domain | W3C validator |