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Theorem rgspncl 31286
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 31285 . 2  |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
7 ssrab2 3499 . . 3  |-  { t  e.  (SubRing `  R
)  |  A  C_  t }  C_  (SubRing `  R
)
8 eqid 2382 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
98subrgid 17544 . . . . . . 7  |-  ( R  e.  Ring  ->  ( Base `  R )  e.  (SubRing `  R ) )
101, 9syl 16 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  e.  (SubRing `  R
) )
112, 10eqeltrd 2470 . . . . 5  |-  ( ph  ->  B  e.  (SubRing `  R
) )
12 sseq2 3439 . . . . . 6  |-  ( t  =  B  ->  ( A  C_  t  <->  A  C_  B
) )
1312rspcev 3135 . . . . 5  |-  ( ( B  e.  (SubRing `  R
)  /\  A  C_  B
)  ->  E. t  e.  (SubRing `  R ) A  C_  t )
1411, 3, 13syl2anc 659 . . . 4  |-  ( ph  ->  E. t  e.  (SubRing `  R ) A  C_  t )
15 rabn0 3732 . . . 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 17564 . . 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 661 . 2  |-  ( ph  ->  |^| { t  e.  (SubRing `  R )  |  A  C_  t }  e.  (SubRing `  R
) )
196, 18eqeltrd 2470 1  |-  ( ph  ->  U  e.  (SubRing `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826    =/= wne 2577   E.wrex 2733   {crab 2736    C_ wss 3389   (/)c0 3711   |^|cint 4199   ` cfv 5496   Basecbs 14634   Ringcrg 17311  SubRingcsubrg 17538  RingSpancrgspn 17539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-subg 16315  df-mgp 17255  df-ur 17267  df-ring 17313  df-subrg 17540  df-rgspn 17541
This theorem is referenced by:  rngunsnply  31290
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