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Theorem rgspnval 27373
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 5527 . 2  |-  ( ph  ->  ( N `  A
)  =  ( (RingSpan `  R ) `  A
) )
4 rgspnval.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
5 elex 2796 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
_V )
6 fveq2 5525 . . . . . . . 8  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
76pweqd 3630 . . . . . . 7  |-  ( a  =  R  ->  ~P ( Base `  a )  =  ~P ( Base `  R
) )
8 fveq2 5525 . . . . . . . . 9  |-  ( a  =  R  ->  (SubRing `  a )  =  (SubRing `  R ) )
9 rabeq 2782 . . . . . . . . 9  |-  ( (SubRing `  a )  =  (SubRing `  R )  ->  { t  e.  (SubRing `  a
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  b  C_  t } )
108, 9syl 15 . . . . . . . 8  |-  ( a  =  R  ->  { t  e.  (SubRing `  a
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  b  C_  t } )
1110inteqd 3867 . . . . . . 7  |-  ( a  =  R  ->  |^| { t  e.  (SubRing `  a
)  |  b  C_  t }  =  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } )
127, 11mpteq12dv 4098 . . . . . 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 15544 . . . . . 6  |- RingSpan  =  ( a  e.  _V  |->  ( b  e.  ~P ( Base `  a )  |->  |^|
{ t  e.  (SubRing `  a )  |  b 
C_  t } ) )
14 fvex 5539 . . . . . . . 8  |-  ( Base `  R )  e.  _V
1514pwex 4193 . . . . . . 7  |-  ~P ( Base `  R )  e. 
_V
1615mptex 5746 . . . . . 6  |-  ( b  e.  ~P ( Base `  R )  |->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } )  e.  _V
1712, 13, 16fvmpt 5602 . . . . 5  |-  ( R  e.  _V  ->  (RingSpan `  R )  =  ( b  e.  ~P ( Base `  R )  |->  |^|
{ t  e.  (SubRing `  R )  |  b 
C_  t } ) )
184, 5, 173syl 18 . . . 4  |-  ( ph  ->  (RingSpan `  R )  =  ( b  e. 
~P ( Base `  R
)  |->  |^| { t  e.  (SubRing `  R )  |  b  C_  t } ) )
1918fveq1d 5527 . . 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 3214 . . . . 5  |-  ( ph  ->  A  C_  ( Base `  R ) )
2314elpw2 4175 . . . . 5  |-  ( A  e.  ~P ( Base `  R )  <->  A  C_  ( Base `  R ) )
2422, 23sylibr 203 . . . 4  |-  ( ph  ->  A  e.  ~P ( Base `  R ) )
25 eqid 2283 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
2625subrgid 15547 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  R )  e.  (SubRing `  R ) )
274, 26syl 15 . . . . . . 7  |-  ( ph  ->  ( Base `  R
)  e.  (SubRing `  R
) )
2821, 27eqeltrd 2357 . . . . . 6  |-  ( ph  ->  B  e.  (SubRing `  R
) )
29 sseq2 3200 . . . . . . 7  |-  ( t  =  B  ->  ( A  C_  t  <->  A  C_  B
) )
3029rspcev 2884 . . . . . 6  |-  ( ( B  e.  (SubRing `  R
)  /\  A  C_  B
)  ->  E. t  e.  (SubRing `  R ) A  C_  t )
3128, 20, 30syl2anc 642 . . . . 5  |-  ( ph  ->  E. t  e.  (SubRing `  R ) A  C_  t )
32 intexrab 4170 . . . . 5  |-  ( E. t  e.  (SubRing `  R
) A  C_  t  <->  |^|
{ t  e.  (SubRing `  R )  |  A  C_  t }  e.  _V )
3331, 32sylib 188 . . . 4  |-  ( ph  ->  |^| { t  e.  (SubRing `  R )  |  A  C_  t }  e.  _V )
34 sseq1 3199 . . . . . . 7  |-  ( b  =  A  ->  (
b  C_  t  <->  A  C_  t
) )
3534rabbidv 2780 . . . . . 6  |-  ( b  =  A  ->  { t  e.  (SubRing `  R
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  A  C_  t } )
3635inteqd 3867 . . . . 5  |-  ( b  =  A  ->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t }  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
37 eqid 2283 . . . . 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 5600 . . . 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 642 . . 3  |-  ( ph  ->  ( ( b  e. 
~P ( Base `  R
)  |->  |^| { t  e.  (SubRing `  R )  |  b  C_  t } ) `  A )  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
4019, 39eqtrd 2315 . 2  |-  ( ph  ->  ( (RingSpan `  R
) `  A )  =  |^| { t  e.  (SubRing `  R )  |  A  C_  t } )
411, 3, 403eqtrd 2319 1  |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   |^|cint 3862    e. cmpt 4077   ` cfv 5255   Basecbs 13148   Ringcrg 15337  SubRingcsubrg 15541  RingSpancrgspn 15542
This theorem is referenced by:  rgspncl  27374  rgspnssid  27375  rgspnmin  27376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-rgspn 15544
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