MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opsrval2 Unicode version

Theorem opsrval2 16234
Description: Self-referential expression for the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
opsrval2.s  |-  S  =  ( I mPwSer  R )
opsrval2.o  |-  O  =  ( ( I ordPwSer  R
) `  T )
opsrval2.l  |-  .<_  =  ( le `  O )
opsrval2.i  |-  ( ph  ->  I  e.  V )
opsrval2.r  |-  ( ph  ->  R  e.  W )
opsrval2.t  |-  ( ph  ->  T  C_  ( I  X.  I ) )
Assertion
Ref Expression
opsrval2  |-  ( ph  ->  O  =  ( S sSet  <. ( le `  ndx ) ,  .<_  >. )
)

Proof of Theorem opsrval2
Dummy variables  w  h  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opsrval2.s . . 3  |-  S  =  ( I mPwSer  R )
2 opsrval2.o . . 3  |-  O  =  ( ( I ordPwSer  R
) `  T )
3 eqid 2296 . . 3  |-  ( Base `  S )  =  (
Base `  S )
4 eqid 2296 . . 3  |-  ( lt
`  R )  =  ( lt `  R
)
5 eqid 2296 . . 3  |-  ( T  <bag  I )  =  ( T  <bag  I )
6 eqid 2296 . . 3  |-  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
7 eqid 2296 . . 3  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  S )  /\  ( E. z  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
( x `  z
) ( lt `  R ) ( y `
 z )  /\  A. w  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
w ( T  <bag  I ) z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( Base `  S
)  /\  ( E. z  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
( x `  z
) ( lt `  R ) ( y `
 z )  /\  A. w  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
w ( T  <bag  I ) z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }
8 opsrval2.i . . 3  |-  ( ph  ->  I  e.  V )
9 opsrval2.r . . 3  |-  ( ph  ->  R  e.  W )
10 opsrval2.t . . 3  |-  ( ph  ->  T  C_  ( I  X.  I ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10opsrval 16232 . 2  |-  ( ph  ->  O  =  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  S )  /\  ( E. z  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
( x `  z
) ( lt `  R ) ( y `
 z )  /\  A. w  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
w ( T  <bag  I ) z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) )
12 opsrval2.l . . . . 5  |-  .<_  =  ( le `  O )
131, 2, 3, 4, 5, 6, 12, 10opsrle 16233 . . . 4  |-  ( ph  -> 
.<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  S )  /\  ( E. z  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
( x `  z
) ( lt `  R ) ( y `
 z )  /\  A. w  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
w ( T  <bag  I ) z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } )
1413opeq2d 3819 . . 3  |-  ( ph  -> 
<. ( le `  ndx ) ,  .<_  >.  =  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( Base `  S
)  /\  ( E. z  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
( x `  z
) ( lt `  R ) ( y `
 z )  /\  A. w  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
w ( T  <bag  I ) z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. )
1514oveq2d 5890 . 2  |-  ( ph  ->  ( S sSet  <. ( le `  ndx ) , 
.<_  >. )  =  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  S )  /\  ( E. z  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
( x `  z
) ( lt `  R ) ( y `
 z )  /\  A. w  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  (
w ( T  <bag  I ) z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) )
1611, 15eqtr4d 2331 1  |-  ( ph  ->  O  =  ( S sSet  <. ( le `  ndx ) ,  .<_  >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   {cpr 3654   <.cop 3656   class class class wbr 4039   {copab 4092    X. cxp 4703   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   NNcn 9762   NN0cn0 9981   ndxcnx 13161   sSet csts 13162   Basecbs 13164   lecple 13231   ltcplt 14091   mPwSer cmps 16103    <bag cltb 16110   ordPwSer copws 16111
This theorem is referenced by:  opsrbaslem  16235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ple 13244  df-psr 16114  df-opsr 16122
  Copyright terms: Public domain W3C validator