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Theorem rgspnval 27341
Description: Value of the ring-span of a set of elements in a ring. (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
rgspnval  |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
Distinct variable groups:    ph, t    t, R    t, B    t, A
Allowed substitution hints:    U( t)    N( t)

Proof of Theorem rgspnval
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rgspnval.sp . 2  |-  ( ph  ->  U  =  ( N `
 A ) )
2 rgspnval.n . . 3  |-  ( ph  ->  N  =  (RingSpan `  R
) )
32fveq1d 5722 . 2  |-  ( ph  ->  ( N `  A
)  =  ( (RingSpan `  R ) `  A
) )
4 rgspnval.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
5 elex 2956 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
_V )
6 fveq2 5720 . . . . . . . 8  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
76pweqd 3796 . . . . . . 7  |-  ( a  =  R  ->  ~P ( Base `  a )  =  ~P ( Base `  R
) )
8 fveq2 5720 . . . . . . . . 9  |-  ( a  =  R  ->  (SubRing `  a )  =  (SubRing `  R ) )
9 rabeq 2942 . . . . . . . . 9  |-  ( (SubRing `  a )  =  (SubRing `  R )  ->  { t  e.  (SubRing `  a
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  b  C_  t } )
108, 9syl 16 . . . . . . . 8  |-  ( a  =  R  ->  { t  e.  (SubRing `  a
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  b  C_  t } )
1110inteqd 4047 . . . . . . 7  |-  ( a  =  R  ->  |^| { t  e.  (SubRing `  a
)  |  b  C_  t }  =  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } )
127, 11mpteq12dv 4279 . . . . . 6  |-  ( a  =  R  ->  (
b  e.  ~P ( Base `  a )  |->  |^|
{ t  e.  (SubRing `  a )  |  b 
C_  t } )  =  ( b  e. 
~P ( Base `  R
)  |->  |^| { t  e.  (SubRing `  R )  |  b  C_  t } ) )
13 df-rgspn 15859 . . . . . 6  |- RingSpan  =  ( a  e.  _V  |->  ( b  e.  ~P ( Base `  a )  |->  |^|
{ t  e.  (SubRing `  a )  |  b 
C_  t } ) )
14 fvex 5734 . . . . . . . 8  |-  ( Base `  R )  e.  _V
1514pwex 4374 . . . . . . 7  |-  ~P ( Base `  R )  e. 
_V
1615mptex 5958 . . . . . 6  |-  ( b  e.  ~P ( Base `  R )  |->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } )  e.  _V
1712, 13, 16fvmpt 5798 . . . . 5  |-  ( R  e.  _V  ->  (RingSpan `  R )  =  ( b  e.  ~P ( Base `  R )  |->  |^|
{ t  e.  (SubRing `  R )  |  b 
C_  t } ) )
184, 5, 173syl 19 . . . 4  |-  ( ph  ->  (RingSpan `  R )  =  ( b  e. 
~P ( Base `  R
)  |->  |^| { t  e.  (SubRing `  R )  |  b  C_  t } ) )
1918fveq1d 5722 . . 3  |-  ( ph  ->  ( (RingSpan `  R
) `  A )  =  ( ( b  e.  ~P ( Base `  R )  |->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } ) `  A
) )
20 rgspnval.ss . . . . . 6  |-  ( ph  ->  A  C_  B )
21 rgspnval.b . . . . . 6  |-  ( ph  ->  B  =  ( Base `  R ) )
2220, 21sseqtrd 3376 . . . . 5  |-  ( ph  ->  A  C_  ( Base `  R ) )
2314elpw2 4356 . . . . 5  |-  ( A  e.  ~P ( Base `  R )  <->  A  C_  ( Base `  R ) )
2422, 23sylibr 204 . . . 4  |-  ( ph  ->  A  e.  ~P ( Base `  R ) )
25 eqid 2435 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
2625subrgid 15862 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  R )  e.  (SubRing `  R ) )
274, 26syl 16 . . . . . . 7  |-  ( ph  ->  ( Base `  R
)  e.  (SubRing `  R
) )
2821, 27eqeltrd 2509 . . . . . 6  |-  ( ph  ->  B  e.  (SubRing `  R
) )
29 sseq2 3362 . . . . . . 7  |-  ( t  =  B  ->  ( A  C_  t  <->  A  C_  B
) )
3029rspcev 3044 . . . . . 6  |-  ( ( B  e.  (SubRing `  R
)  /\  A  C_  B
)  ->  E. t  e.  (SubRing `  R ) A  C_  t )
3128, 20, 30syl2anc 643 . . . . 5  |-  ( ph  ->  E. t  e.  (SubRing `  R ) A  C_  t )
32 intexrab 4351 . . . . 5  |-  ( E. t  e.  (SubRing `  R
) A  C_  t  <->  |^|
{ t  e.  (SubRing `  R )  |  A  C_  t }  e.  _V )
3331, 32sylib 189 . . . 4  |-  ( ph  ->  |^| { t  e.  (SubRing `  R )  |  A  C_  t }  e.  _V )
34 sseq1 3361 . . . . . . 7  |-  ( b  =  A  ->  (
b  C_  t  <->  A  C_  t
) )
3534rabbidv 2940 . . . . . 6  |-  ( b  =  A  ->  { t  e.  (SubRing `  R
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  A  C_  t } )
3635inteqd 4047 . . . . 5  |-  ( b  =  A  ->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t }  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
37 eqid 2435 . . . . 5  |-  ( b  e.  ~P ( Base `  R )  |->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } )  =  ( b  e.  ~P ( Base `  R )  |->  |^|
{ t  e.  (SubRing `  R )  |  b 
C_  t } )
3836, 37fvmptg 5796 . . . 4  |-  ( ( A  e.  ~P ( Base `  R )  /\  |^|
{ t  e.  (SubRing `  R )  |  A  C_  t }  e.  _V )  ->  ( ( b  e.  ~P ( Base `  R )  |->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } ) `  A
)  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
3924, 33, 38syl2anc 643 . . 3  |-  ( ph  ->  ( ( b  e. 
~P ( Base `  R
)  |->  |^| { t  e.  (SubRing `  R )  |  b  C_  t } ) `  A )  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
4019, 39eqtrd 2467 . 2  |-  ( ph  ->  ( (RingSpan `  R
) `  A )  =  |^| { t  e.  (SubRing `  R )  |  A  C_  t } )
411, 3, 403eqtrd 2471 1  |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   |^|cint 4042    e. cmpt 4258   ` cfv 5446   Basecbs 13461   Ringcrg 15652  SubRingcsubrg 15856  RingSpancrgspn 15857
This theorem is referenced by:  rgspncl  27342  rgspnssid  27343  rgspnmin  27344
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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