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Theorem opsrle 18011
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 2441 . . . . 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 755 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  I  e.  _V )
9 simprr 756 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  R  e.  _V )
10 opsrle.t . . . . . 6  |-  ( ph  ->  T  C_  ( I  X.  I ) )
1110adantr 465 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  T  C_  ( I  X.  I ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 11opsrval 18010 . . . 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 5857 . . 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 6306 . . . . 5  |-  ( I mPwSer  R )  e.  _V
161, 15eqeltri 2525 . . . 4  |-  S  e. 
_V
17 fvex 5863 . . . . . . 7  |-  ( Base `  S )  e.  _V
183, 17eqeltri 2525 . . . . . 6  |-  B  e. 
_V
1918, 18xpex 6586 . . . . 5  |-  ( B  X.  B )  e. 
_V
20 vex 3096 . . . . . . . . 9  |-  x  e. 
_V
21 vex 3096 . . . . . . . . 9  |-  y  e. 
_V
2220, 21prss 4166 . . . . . . . 8  |-  ( ( x  e.  B  /\  y  e.  B )  <->  { x ,  y } 
C_  B )
2322anbi1i 695 . . . . . . 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 4498 . . . . . 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 5061 . . . . . 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 3518 . . . . 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 4579 . . . 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 14666 . . . . 5  |-  le  = Slot  ( le `  ndx )
2928setsid 14547 . . . 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 672 . . 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 2507 . 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 18009 . . . . . . . . . 10  |-  Rel  dom ordPwSer
3332ovprc 6308 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I ordPwSer  R )  =  (/) )
3433adantl 466 . . . . . . . 8  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( I ordPwSer  R )  =  (/) )
3534fveq1d 5855 . . . . . . 7  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( (
I ordPwSer  R ) `  T
)  =  ( (/) `  T ) )
362, 35syl5eq 2494 . . . . . 6  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  O  =  ( (/) `  T ) )
37 0fv 5886 . . . . . 6  |-  ( (/) `  T )  =  (/)
3836, 37syl6eq 2498 . . . . 5  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  O  =  (/) )
3938fveq2d 5857 . . . 4  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( le `  O )  =  ( le `  (/) ) )
4028str0 14544 . . . 4  |-  (/)  =  ( le `  (/) )
4139, 14, 403eqtr4g 2507 . . 3  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  .<_  =  (/) )
42 reldmpsr 17881 . . . . . . . . . . 11  |-  Rel  dom mPwSer
4342ovprc 6308 . . . . . . . . . 10  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
4443adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( I mPwSer  R )  =  (/) )
451, 44syl5eq 2494 . . . . . . . 8  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  S  =  (/) )
4645fveq2d 5857 . . . . . . 7  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( Base `  S )  =  (
Base `  (/) ) )
47 base0 14545 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
4846, 3, 473eqtr4g 2507 . . . . . 6  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  B  =  (/) )
4948xpeq2d 5010 . . . . 5  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( B  X.  B )  =  ( B  X.  (/) ) )
50 xp0 5412 . . . . 5  |-  ( B  X.  (/) )  =  (/)
5149, 50syl6eq 2498 . . . 4  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( B  X.  B )  =  (/) )
52 sseq0 3800 . . . 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 663 . . 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 2485 . 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 789 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 368    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792   {crab 2795   _Vcvv 3093    C_ wss 3459   (/)c0 3768   {cpr 4013   <.cop 4017   class class class wbr 4434   {copab 4491    X. cxp 4984   `'ccnv 4985   "cima 4989   ` cfv 5575  (class class class)co 6278    ^m cmap 7419   Fincfn 7515   NNcn 10539   NN0cn0 10798   ndxcnx 14503   sSet csts 14504   Basecbs 14506   lecple 14578   ltcplt 15441   mPwSer cmps 17871    <bag cltb 17874   ordPwSer copws 17875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-i2m1 9560  ax-1ne0 9561  ax-rrecex 9564  ax-cnre 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-recs 7041  df-rdg 7075  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-7 10602  df-8 10603  df-9 10604  df-10 10605  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ple 14591  df-psr 17876  df-opsr 17880
This theorem is referenced by:  opsrval2  18012  opsrtoslem1  18019
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