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Theorem opsrle 17911
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 2467 . . . . 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 17910 . . . 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 5868 . . 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 6307 . . . . 5  |-  ( I mPwSer  R )  e.  _V
161, 15eqeltri 2551 . . . 4  |-  S  e. 
_V
17 fvex 5874 . . . . . . 7  |-  ( Base `  S )  e.  _V
183, 17eqeltri 2551 . . . . . 6  |-  B  e. 
_V
1918, 18xpex 6711 . . . . 5  |-  ( B  X.  B )  e. 
_V
20 vex 3116 . . . . . . . . 9  |-  x  e. 
_V
21 vex 3116 . . . . . . . . 9  |-  y  e. 
_V
2220, 21prss 4181 . . . . . . . 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 4511 . . . . . 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 5072 . . . . . 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 3535 . . . . 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 4592 . . . 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 14646 . . . . 5  |-  le  = Slot  ( le `  ndx )
2928setsid 14527 . . . 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 2533 . 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 17909 . . . . . . . . . 10  |-  Rel  dom ordPwSer
3332ovprc 6309 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I ordPwSer  R )  =  (/) )
3433adantl 466 . . . . . . . 8  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( I ordPwSer  R )  =  (/) )
3534fveq1d 5866 . . . . . . 7  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( (
I ordPwSer  R ) `  T
)  =  ( (/) `  T ) )
362, 35syl5eq 2520 . . . . . 6  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  O  =  ( (/) `  T ) )
37 0fv 5897 . . . . . 6  |-  ( (/) `  T )  =  (/)
3836, 37syl6eq 2524 . . . . 5  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  O  =  (/) )
3938fveq2d 5868 . . . 4  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( le `  O )  =  ( le `  (/) ) )
4028str0 14524 . . . 4  |-  (/)  =  ( le `  (/) )
4139, 14, 403eqtr4g 2533 . . 3  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  .<_  =  (/) )
42 reldmpsr 17781 . . . . . . . . . . 11  |-  Rel  dom mPwSer
4342ovprc 6309 . . . . . . . . . 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 2520 . . . . . . . 8  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  S  =  (/) )
4645fveq2d 5868 . . . . . . 7  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( Base `  S )  =  (
Base `  (/) ) )
47 base0 14525 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
4846, 3, 473eqtr4g 2533 . . . . . 6  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  B  =  (/) )
4948xpeq2d 5023 . . . . 5  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( B  X.  B )  =  ( B  X.  (/) ) )
50 xp0 5423 . . . . 5  |-  ( B  X.  (/) )  =  (/)
5149, 50syl6eq 2524 . . . 4  |-  ( (
ph  /\  -.  (
I  e.  _V  /\  R  e.  _V )
)  ->  ( B  X.  B )  =  (/) )
52 sseq0 3817 . . . 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 2511 . 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 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {cpr 4029   <.cop 4033   class class class wbr 4447   {copab 4504    X. cxp 4997   `'ccnv 4998   "cima 5002   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   Fincfn 7513   NNcn 10532   NN0cn0 10791   ndxcnx 14483   sSet csts 14484   Basecbs 14486   lecple 14558   ltcplt 15424   mPwSer cmps 17771    <bag cltb 17774   ordPwSer copws 17775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-i2m1 9556  ax-1ne0 9557  ax-rrecex 9560  ax-cnre 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ple 14571  df-psr 17776  df-opsr 17780
This theorem is referenced by:  opsrval2  17912  opsrtoslem1  17919
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