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Theorem opsrle 18747
Description: An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
opsrle.s  |-  S  =  ( I mPwSer  R )
opsrle.o  |-  O  =  ( ( I ordPwSer  R
) `  T )
opsrle.b  |-  B  =  ( Base `  S
)
opsrle.q  |-  .<  =  ( lt `  R )
opsrle.c  |-  C  =  ( T  <bag  I )
opsrle.d  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
opsrle.l  |-  .<_  =  ( le `  O )
opsrle.t  |-  ( ph  ->  T  C_  ( I  X.  I ) )
Assertion
Ref Expression
opsrle  |-  ( ph  -> 
.<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } )
Distinct variable groups:    x, y, B    z, w, D    w, h, x, y, z, I   
w, R, x, y, z    ph, w, x, y, z    w, T, x, y, z
Allowed substitution hints:    ph( h)    B( z, w, h)    C( x, y, z, w, h)    D( x, y, h)    R( h)    S( x, y, z, w, h)    .< ( x, y, z, w, h)    T( h)    .<_ ( x, y, z, w, h)    O( x, y, z, w, h)

Proof of Theorem opsrle
StepHypRef Expression
1 opsrle.s . . . . 5  |-  S  =  ( I mPwSer  R )
2 opsrle.o . . . . 5  |-  O  =  ( ( I ordPwSer  R
) `  T )
3 opsrle.b . . . . 5  |-  B  =  ( Base `  S
)
4 opsrle.q . . . . 5  |-  .<  =  ( lt `  R )
5 opsrle.c . . . . 5  |-  C  =  ( T  <bag  I )
6 opsrle.d . . . . 5  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
7 eqid 2461 . . . . 5  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }
8 simprl 769 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  I  e.  _V )
9 simprr 771 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  R  e.  _V )
10 opsrle.t . . . . . 6  |-  ( ph  ->  T  C_  ( I  X.  I ) )
1110adantr 471 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  T  C_  ( I  X.  I ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 11opsrval 18746 . . . 4  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  O  =  ( S sSet  <.
( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) )
1312fveq2d 5891 . . 3  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  -> 
( le `  O
)  =  ( le
`  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) ) )
14 opsrle.l . . 3  |-  .<_  =  ( le `  O )
15 ovex 6342 . . . . 5  |-  ( I mPwSer  R )  e.  _V
161, 15eqeltri 2535 . . . 4  |-  S  e. 
_V
17 fvex 5897 . . . . . . 7  |-  ( Base `  S )  e.  _V
183, 17eqeltri 2535 . . . . . 6  |-  B  e. 
_V
1918, 18xpex 6621 . . . . 5  |-  ( B  X.  B )  e. 
_V
20 vex 3059 . . . . . . . . 9  |-  x  e. 
_V
21 vex 3059 . . . . . . . . 9  |-  y  e. 
_V
2220, 21prss 4138 . . . . . . . 8  |-  ( ( x  e.  B  /\  y  e.  B )  <->  { x ,  y } 
C_  B )
2322anbi1i 706 . . . . . . 7  |-  ( ( ( x  e.  B  /\  y  e.  B
)  /\  ( E. z  e.  D  (
( x `  z
)  .<  ( y `  z )  /\  A. w  e.  D  (
w C z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) )  <->  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) )
2423opabbii 4480 . . . . . 6  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( E. z  e.  D  (
( x `  z
)  .<  ( y `  z )  /\  A. w  e.  D  (
w C z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }
25 opabssxp 4927 . . . . . 6  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( E. z  e.  D  (
( x `  z
)  .<  ( y `  z )  /\  A. w  e.  D  (
w C z  -> 
( x `  w
)  =  ( y `
 w ) ) )  \/  x  =  y ) ) } 
C_  ( B  X.  B )
2624, 25eqsstr3i 3474 . . . . 5  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  C_  ( B  X.  B )
2719, 26ssexi 4561 . . . 4  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  e.  _V
28 pleid 15340 . . . . 5  |-  le  = Slot  ( le `  ndx )
2928setsid 15212 . . . 4  |-  ( ( S  e.  _V  /\  {
<. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( (
x `  z )  .<  ( y `  z
)  /\  A. w  e.  D  ( w C z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  e.  _V )  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  =  ( le
`  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) ) )
3016, 27, 29mp2an 683 . . 3  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  =  ( le
`  ( S sSet  <. ( le `  ndx ) ,  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) )
3113, 14, 303eqtr4g 2520 . 2  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } )
32 reldmopsr 18745 . . . . . . . . . 10  |-  Rel  dom ordPwSer
3332ovprc 6344 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I ordPwSer  R )  =  (/) )
3433adantl 472 . . . . . . . 8  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( I ordPwSer  R )  =  (/) )
3534fveq1d 5889 . . . . . . 7  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( (
I ordPwSer  R ) `  T
)  =  ( (/) `  T ) )
362, 35syl5eq 2507 . . . . . 6  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  O  =  ( (/) `  T ) )
37 0fv 5920 . . . . . 6  |-  ( (/) `  T )  =  (/)
3836, 37syl6eq 2511 . . . . 5  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  O  =  (/) )
3938fveq2d 5891 . . . 4  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( le `  O )  =  ( le `  (/) ) )
4028str0 15209 . . . 4  |-  (/)  =  ( le `  (/) )
4139, 14, 403eqtr4g 2520 . . 3  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  .<_  =  (/) )
42 reldmpsr 18633 . . . . . . . . . . 11  |-  Rel  dom mPwSer
4342ovprc 6344 . . . . . . . . . 10  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
4443adantl 472 . . . . . . . . 9  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( I mPwSer  R )  =  (/) )
451, 44syl5eq 2507 . . . . . . . 8  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  S  =  (/) )
4645fveq2d 5891 . . . . . . 7  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( Base `  S )  =  (
Base `  (/) ) )
47 base0 15210 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
4846, 3, 473eqtr4g 2520 . . . . . 6  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  B  =  (/) )
4948xpeq2d 4876 . . . . 5  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( B  X.  B )  =  ( B  X.  (/) ) )
50 xp0 5273 . . . . 5  |-  ( B  X.  (/) )  =  (/)
5149, 50syl6eq 2511 . . . 4  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( B  X.  B )  =  (/) )
52 sseq0 3777 . . . 4  |-  ( ( { <. x ,  y
>.  |  ( {
x ,  y } 
C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  C_  ( B  X.  B )  /\  ( B  X.  B )  =  (/) )  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  =  (/) )
5326, 51, 52sylancr 674 . . 3  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) }  =  (/) )
5441, 53eqtr4d 2498 . 2  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( (
x `  z )  .<  ( y `  z
)  /\  A. w  e.  D  ( w C z  ->  (
x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } )
5531, 54pm2.61dan 805 1  |-  ( ph  -> 
.<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  B  /\  ( E. z  e.  D  ( ( x `  z )  .<  (
y `  z )  /\  A. w  e.  D  ( w C z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 374    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748   E.wrex 2749   {crab 2752   _Vcvv 3056    C_ wss 3415   (/)c0 3742   {cpr 3981   <.cop 3985   class class class wbr 4415   {copab 4473    X. cxp 4850   `'ccnv 4851   "cima 4855   ` cfv 5600  (class class class)co 6314    ^m cmap 7497   Fincfn 7594   NNcn 10636   NN0cn0 10897   ndxcnx 15166   sSet csts 15167   Basecbs 15169   lecple 15245   ltcplt 16234   mPwSer cmps 18623    <bag cltb 18626   ordPwSer copws 18627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-i2m1 9632  ax-1ne0 9633  ax-rrecex 9636  ax-cnre 9637
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ple 15258  df-psr 18628  df-opsr 18632
This theorem is referenced by:  opsrval2  18748  opsrtoslem1  18755
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