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Theorem rgspncl 29673
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 29672 . 2  |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
7 ssrab2 3544 . . 3  |-  { t  e.  (SubRing `  R
)  |  A  C_  t }  C_  (SubRing `  R
)
8 eqid 2454 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
98subrgid 16989 . . . . . . 7  |-  ( R  e.  Ring  ->  ( Base `  R )  e.  (SubRing `  R ) )
101, 9syl 16 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  e.  (SubRing `  R
) )
112, 10eqeltrd 2542 . . . . 5  |-  ( ph  ->  B  e.  (SubRing `  R
) )
12 sseq2 3485 . . . . . 6  |-  ( t  =  B  ->  ( A  C_  t  <->  A  C_  B
) )
1312rspcev 3177 . . . . 5  |-  ( ( B  e.  (SubRing `  R
)  /\  A  C_  B
)  ->  E. t  e.  (SubRing `  R ) A  C_  t )
1411, 3, 13syl2anc 661 . . . 4  |-  ( ph  ->  E. t  e.  (SubRing `  R ) A  C_  t )
15 rabn0 3764 . . . 4  |-  ( { t  e.  (SubRing `  R
)  |  A  C_  t }  =/=  (/)  <->  E. t  e.  (SubRing `  R ) A  C_  t )
1614, 15sylibr 212 . . 3  |-  ( ph  ->  { t  e.  (SubRing `  R )  |  A  C_  t }  =/=  (/) )
17 subrgint 17009 . . 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 663 . 2  |-  ( ph  ->  |^| { t  e.  (SubRing `  R )  |  A  C_  t }  e.  (SubRing `  R
) )
196, 18eqeltrd 2542 1  |-  ( ph  ->  U  e.  (SubRing `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799   {crab 2802    C_ wss 3435   (/)c0 3744   |^|cint 4235   ` cfv 5525   Basecbs 14291   Ringcrg 16767  SubRingcsubrg 16983  RingSpancrgspn 16984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-0g 14498  df-mnd 15533  df-grp 15663  df-minusg 15664  df-subg 15796  df-mgp 16713  df-ur 16725  df-rng 16769  df-subrg 16985  df-rgspn 16986
This theorem is referenced by:  rngunsnply  29677
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