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Theorem opsrle 16233
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 2296 . . . . 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 732 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  I  e.  _V )
9 simprr 733 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  R  e.  _V )
10 opsrle.t . . . . . 6  |-  ( ph  ->  T  C_  ( I  X.  I ) )
1110adantr 451 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  T  C_  ( I  X.  I ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 11opsrval 16232 . . . 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 5545 . . 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 5899 . . . . 5  |-  ( I mPwSer  R )  e.  _V
161, 15eqeltri 2366 . . . 4  |-  S  e. 
_V
17 fvex 5555 . . . . . . 7  |-  ( Base `  S )  e.  _V
183, 17eqeltri 2366 . . . . . 6  |-  B  e. 
_V
1918, 18xpex 4817 . . . . 5  |-  ( B  X.  B )  e. 
_V
20 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
21 vex 2804 . . . . . . . . 9  |-  y  e. 
_V
2220, 21prss 3785 . . . . . . . 8  |-  ( ( x  e.  B  /\  y  e.  B )  <->  { x ,  y } 
C_  B )
2322anbi1i 676 . . . . . . 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 4099 . . . . . 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 4778 . . . . . 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 3222 . . . . 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 4175 . . . 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 13317 . . . . 5  |-  le  = Slot  ( le `  ndx )
2928setsid 13203 . . . 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 653 . . 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 2353 . 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 16231 . . . . . . . . . 10  |-  Rel  dom ordPwSer
3332ovprc 5901 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I ordPwSer  R )  =  (/) )
3433adantl 452 . . . . . . . 8  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( I ordPwSer  R )  =  (/) )
3534fveq1d 5543 . . . . . . 7  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( (
I ordPwSer  R ) `  T
)  =  ( (/) `  T ) )
362, 35syl5eq 2340 . . . . . 6  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  O  =  ( (/) `  T ) )
37 fv01 5575 . . . . . 6  |-  ( (/) `  T )  =  (/)
3836, 37syl6eq 2344 . . . . 5  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  O  =  (/) )
3938fveq2d 5545 . . . 4  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( le `  O )  =  ( le `  (/) ) )
4028str0 13200 . . . 4  |-  (/)  =  ( le `  (/) )
4139, 14, 403eqtr4g 2353 . . 3  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  .<_  =  (/) )
42 reldmpsr 16125 . . . . . . . . . . 11  |-  Rel  dom mPwSer
4342ovprc 5901 . . . . . . . . . 10  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
4443adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( I mPwSer  R )  =  (/) )
451, 44syl5eq 2340 . . . . . . . 8  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  S  =  (/) )
4645fveq2d 5545 . . . . . . 7  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( Base `  S )  =  (
Base `  (/) ) )
47 base0 13201 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
4846, 3, 473eqtr4g 2353 . . . . . 6  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  B  =  (/) )
4948xpeq2d 4729 . . . . 5  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( B  X.  B )  =  ( B  X.  (/) ) )
50 xp0 5114 . . . . 5  |-  ( B  X.  (/) )  =  (/)
5149, 50syl6eq 2344 . . . 4  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( B  X.  B )  =  (/) )
52 sseq0 3499 . . . 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 644 . . 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 2331 . 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 766 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 set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   {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:  opsrval2  16234  opsrtoslem1  16241
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
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