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Theorem opsrval2 16218
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 2283 . . 3  |-  ( Base `  S )  =  (
Base `  S )
4 eqid 2283 . . 3  |-  ( lt
`  R )  =  ( lt `  R
)
5 eqid 2283 . . 3  |-  ( T  <bag  I )  =  ( T  <bag  I )
6 eqid 2283 . . 3  |-  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
7 eqid 2283 . . 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 16216 . 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 16217 . . . 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 3803 . . 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 5874 . 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 2318 1  |-  ( ph  ->  O  =  ( S sSet  <. ( le `  ndx ) ,  .<_  >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152   {cpr 3641   <.cop 3643   class class class wbr 4023   {copab 4076    X. cxp 4687   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   NNcn 9746   NN0cn0 9965   ndxcnx 13145   sSet csts 13146   Basecbs 13148   lecple 13215   ltcplt 14075   mPwSer cmps 16087    <bag cltb 16094   ordPwSer copws 16095
This theorem is referenced by:  opsrbaslem  16219
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-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
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-ral 2548  df-rex 2549  df-reu 2550  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-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-recs 6388  df-rdg 6423  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ple 13228  df-psr 16098  df-opsr 16106
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