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Theorem rgspnmin 27344
Description: The ring-span is contained in all subspaces which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-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 ) )
rgspnmin.sr  |-  ( ph  ->  S  e.  (SubRing `  R
) )
rgspnmin.ss  |-  ( ph  ->  A  C_  S )
Assertion
Ref Expression
rgspnmin  |-  ( ph  ->  U  C_  S )

Proof of Theorem rgspnmin
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 27341 . 2  |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
7 rgspnmin.sr . . . 4  |-  ( ph  ->  S  e.  (SubRing `  R
) )
8 rgspnmin.ss . . . 4  |-  ( ph  ->  A  C_  S )
9 sseq2 3362 . . . . 5  |-  ( t  =  S  ->  ( A  C_  t  <->  A  C_  S
) )
109elrab 3084 . . . 4  |-  ( S  e.  { t  e.  (SubRing `  R )  |  A  C_  t }  <-> 
( S  e.  (SubRing `  R )  /\  A  C_  S ) )
117, 8, 10sylanbrc 646 . . 3  |-  ( ph  ->  S  e.  { t  e.  (SubRing `  R
)  |  A  C_  t } )
12 intss1 4057 . . 3  |-  ( S  e.  { t  e.  (SubRing `  R )  |  A  C_  t }  ->  |^| { t  e.  (SubRing `  R )  |  A  C_  t } 
C_  S )
1311, 12syl 16 . 2  |-  ( ph  ->  |^| { t  e.  (SubRing `  R )  |  A  C_  t } 
C_  S )
146, 13eqsstrd 3374 1  |-  ( ph  ->  U  C_  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2701    C_ wss 3312   |^|cint 4042   ` cfv 5446   Basecbs 13461   Ringcrg 15652  SubRingcsubrg 15856  RingSpancrgspn 15857
This theorem is referenced by:  rgspnid  27345  rngunsnply  27346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-mgp 15641  df-rng 15655  df-ur 15657  df-subrg 15858  df-rgspn 15859
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