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Theorem rgspncl 27353
Description: The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
Hypotheses
Ref Expression
rgspnval.r  |-  ( ph  ->  R  e.  Ring )
rgspnval.b  |-  ( ph  ->  B  =  ( Base `  R ) )
rgspnval.ss  |-  ( ph  ->  A  C_  B )
rgspnval.n  |-  ( ph  ->  N  =  (RingSpan `  R
) )
rgspnval.sp  |-  ( ph  ->  U  =  ( N `
 A ) )
Assertion
Ref Expression
rgspncl  |-  ( ph  ->  U  e.  (SubRing `  R
) )

Proof of Theorem rgspncl
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 rgspnval.r . . 3  |-  ( ph  ->  R  e.  Ring )
2 rgspnval.b . . 3  |-  ( ph  ->  B  =  ( Base `  R ) )
3 rgspnval.ss . . 3  |-  ( ph  ->  A  C_  B )
4 rgspnval.n . . 3  |-  ( ph  ->  N  =  (RingSpan `  R
) )
5 rgspnval.sp . . 3  |-  ( ph  ->  U  =  ( N `
 A ) )
61, 2, 3, 4, 5rgspnval 27352 . 2  |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
7 ssrab2 3430 . . 3  |-  { t  e.  (SubRing `  R
)  |  A  C_  t }  C_  (SubRing `  R
)
8 eqid 2438 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
98subrgid 15872 . . . . . . 7  |-  ( R  e.  Ring  ->  ( Base `  R )  e.  (SubRing `  R ) )
101, 9syl 16 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  e.  (SubRing `  R
) )
112, 10eqeltrd 2512 . . . . 5  |-  ( ph  ->  B  e.  (SubRing `  R
) )
12 sseq2 3372 . . . . . 6  |-  ( t  =  B  ->  ( A  C_  t  <->  A  C_  B
) )
1312rspcev 3054 . . . . 5  |-  ( ( B  e.  (SubRing `  R
)  /\  A  C_  B
)  ->  E. t  e.  (SubRing `  R ) A  C_  t )
1411, 3, 13syl2anc 644 . . . 4  |-  ( ph  ->  E. t  e.  (SubRing `  R ) A  C_  t )
15 rabn0 3649 . . . 4  |-  ( { t  e.  (SubRing `  R
)  |  A  C_  t }  =/=  (/)  <->  E. t  e.  (SubRing `  R ) A  C_  t )
1614, 15sylibr 205 . . 3  |-  ( ph  ->  { t  e.  (SubRing `  R )  |  A  C_  t }  =/=  (/) )
17 subrgint 15892 . . 3  |-  ( ( { t  e.  (SubRing `  R )  |  A  C_  t }  C_  (SubRing `  R )  /\  {
t  e.  (SubRing `  R
)  |  A  C_  t }  =/=  (/) )  ->  |^| { t  e.  (SubRing `  R )  |  A  C_  t }  e.  (SubRing `  R ) )
187, 16, 17sylancr 646 . 2  |-  ( ph  ->  |^| { t  e.  (SubRing `  R )  |  A  C_  t }  e.  (SubRing `  R
) )
196, 18eqeltrd 2512 1  |-  ( ph  ->  U  e.  (SubRing `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711    C_ wss 3322   (/)c0 3630   |^|cint 4052   ` cfv 5456   Basecbs 13471   Ringcrg 15662  SubRingcsubrg 15866  RingSpancrgspn 15867
This theorem is referenced by:  rngunsnply  27357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-subg 14943  df-mgp 15651  df-rng 15665  df-ur 15667  df-subrg 15868  df-rgspn 15869
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