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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rgspnssid | Structured version Visualization version GIF version |
Description: The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-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 |
---|---|
rgspnssid | β’ (π β π΄ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintub 4962 | . 2 β’ π΄ β β© {π‘ β (SubRingβπ ) β£ π΄ β π‘} | |
2 | rgspnval.r | . . 3 β’ (π β π β Ring) | |
3 | rgspnval.b | . . 3 β’ (π β π΅ = (Baseβπ )) | |
4 | rgspnval.ss | . . 3 β’ (π β π΄ β π΅) | |
5 | rgspnval.n | . . 3 β’ (π β π = (RingSpanβπ )) | |
6 | rgspnval.sp | . . 3 β’ (π β π = (πβπ΄)) | |
7 | 2, 3, 4, 5, 6 | rgspnval 42629 | . 2 β’ (π β π = β© {π‘ β (SubRingβπ ) β£ π΄ β π‘}) |
8 | 1, 7 | sseqtrrid 4025 | 1 β’ (π β π΄ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {crab 3419 β wss 3939 β© cint 4942 βcfv 6541 Basecbs 17177 Ringcrg 20175 SubRingcsubrg 20508 RingSpancrgspn 20509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mgp 20077 df-ur 20124 df-ring 20177 df-subrg 20510 df-rgspn 20511 |
This theorem is referenced by: rgspnid 42633 rngunsnply 42634 |
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