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Theorem opsrle 16536
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 2436 . . . . 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 733 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  I  e.  _V )
9 simprr 734 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  R  e.  _V )
10 opsrle.t . . . . . 6  |-  ( ph  ->  T  C_  ( I  X.  I ) )
1110adantr 452 . . . . 5  |-  ( (
ph  /\  ( I  e.  _V  /\  R  e. 
_V ) )  ->  T  C_  ( I  X.  I ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 11opsrval 16535 . . . 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 5732 . . 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 6106 . . . . 5  |-  ( I mPwSer  R )  e.  _V
161, 15eqeltri 2506 . . . 4  |-  S  e. 
_V
17 fvex 5742 . . . . . . 7  |-  ( Base `  S )  e.  _V
183, 17eqeltri 2506 . . . . . 6  |-  B  e. 
_V
1918, 18xpex 4990 . . . . 5  |-  ( B  X.  B )  e. 
_V
20 vex 2959 . . . . . . . . 9  |-  x  e. 
_V
21 vex 2959 . . . . . . . . 9  |-  y  e. 
_V
2220, 21prss 3952 . . . . . . . 8  |-  ( ( x  e.  B  /\  y  e.  B )  <->  { x ,  y } 
C_  B )
2322anbi1i 677 . . . . . . 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 4272 . . . . . 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 4950 . . . . . 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 3379 . . . . 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 4348 . . . 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 13622 . . . . 5  |-  le  = Slot  ( le `  ndx )
2928setsid 13508 . . . 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 654 . . 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 2493 . 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 16534 . . . . . . . . . 10  |-  Rel  dom ordPwSer
3332ovprc 6108 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I ordPwSer  R )  =  (/) )
3433adantl 453 . . . . . . . 8  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( I ordPwSer  R )  =  (/) )
3534fveq1d 5730 . . . . . . 7  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( (
I ordPwSer  R ) `  T
)  =  ( (/) `  T ) )
362, 35syl5eq 2480 . . . . . 6  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  O  =  ( (/) `  T ) )
37 fv01 5763 . . . . . 6  |-  ( (/) `  T )  =  (/)
3836, 37syl6eq 2484 . . . . 5  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  O  =  (/) )
3938fveq2d 5732 . . . 4  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( le `  O )  =  ( le `  (/) ) )
4028str0 13505 . . . 4  |-  (/)  =  ( le `  (/) )
4139, 14, 403eqtr4g 2493 . . 3  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  .<_  =  (/) )
42 reldmpsr 16428 . . . . . . . . . . 11  |-  Rel  dom mPwSer
4342ovprc 6108 . . . . . . . . . 10  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
4443adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( I mPwSer  R )  =  (/) )
451, 44syl5eq 2480 . . . . . . . 8  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  S  =  (/) )
4645fveq2d 5732 . . . . . . 7  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( Base `  S )  =  (
Base `  (/) ) )
47 base0 13506 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
4846, 3, 473eqtr4g 2493 . . . . . 6  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  B  =  (/) )
4948xpeq2d 4902 . . . . 5  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( B  X.  B )  =  ( B  X.  (/) ) )
50 xp0 5291 . . . . 5  |-  ( B  X.  (/) )  =  (/)
5149, 50syl6eq 2484 . . . 4  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( B  X.  B )  =  (/) )
52 sseq0 3659 . . . 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 645 . . 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 2471 . 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 767 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 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    C_ wss 3320   (/)c0 3628   {cpr 3815   <.cop 3817   class class class wbr 4212   {copab 4265    X. cxp 4876   `'ccnv 4877   "cima 4881   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   Fincfn 7109   NNcn 10000   NN0cn0 10221   ndxcnx 13466   sSet csts 13467   Basecbs 13469   lecple 13536   ltcplt 14398   mPwSer cmps 16406    <bag cltb 16413   ordPwSer copws 16414
This theorem is referenced by:  opsrval2  16537  opsrtoslem1  16544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-i2m1 9058  ax-1ne0 9059  ax-rrecex 9062  ax-cnre 9063
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ple 13549  df-psr 16417  df-opsr 16425
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