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Theorem pwsle 13641
Description: Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.)
Hypotheses
Ref Expression
pwsle.y  |-  Y  =  ( R  ^s  I )
pwsle.v  |-  B  =  ( Base `  Y
)
pwsle.o  |-  O  =  ( le `  R
)
pwsle.l  |-  .<_  =  ( le `  Y )
Assertion
Ref Expression
pwsle  |-  ( ( R  e.  V  /\  I  e.  W )  -> 
.<_  =  (  o R O  i^i  ( B  X.  B ) ) )

Proof of Theorem pwsle
Dummy variables  f 
g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2902 . . . . . . 7  |-  f  e. 
_V
2 vex 2902 . . . . . . 7  |-  g  e. 
_V
31, 2prss 3895 . . . . . 6  |-  ( ( f  e.  B  /\  g  e.  B )  <->  { f ,  g } 
C_  B )
4 pwsle.v . . . . . . . 8  |-  B  =  ( Base `  Y
)
5 pwsle.y . . . . . . . . . 10  |-  Y  =  ( R  ^s  I )
6 eqid 2387 . . . . . . . . . 10  |-  (Scalar `  R )  =  (Scalar `  R )
75, 6pwsval 13635 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
87fveq2d 5672 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  Y
)  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
94, 8syl5eq 2431 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
109sseq2d 3319 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( { f ,  g }  C_  B  <->  { f ,  g } 
C_  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
113, 10syl5bb 249 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( ( f  e.  B  /\  g  e.  B )  <->  { f ,  g }  C_  ( Base `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
1211anbi1d 686 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( ( ( f  e.  B  /\  g  e.  B )  /\  A. x  e.  I  (
f `  x )
( le `  (
( I  X.  { R } ) `  x
) ) ( g `
 x ) )  <-> 
( { f ,  g }  C_  ( Base `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  /\  A. x  e.  I  ( f `  x ) ( le
`  ( ( I  X.  { R }
) `  x )
) ( g `  x ) ) ) )
13 simpll 731 . . . . . . . . . . . 12  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  ->  R  e.  V )
14 fvconst2g 5884 . . . . . . . . . . . 12  |-  ( ( R  e.  V  /\  x  e.  I )  ->  ( ( I  X.  { R } ) `  x )  =  R )
1513, 14sylan 458 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  (
( I  X.  { R } ) `  x
)  =  R )
1615fveq2d 5672 . . . . . . . . . 10  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  ( le `  ( ( I  X.  { R }
) `  x )
)  =  ( le
`  R ) )
17 pwsle.o . . . . . . . . . 10  |-  O  =  ( le `  R
)
1816, 17syl6eqr 2437 . . . . . . . . 9  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  ( le `  ( ( I  X.  { R }
) `  x )
)  =  O )
1918breqd 4164 . . . . . . . 8  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  (
( f `  x
) ( le `  ( ( I  X.  { R } ) `  x ) ) ( g `  x )  <-> 
( f `  x
) O ( g `
 x ) ) )
2019ralbidva 2665 . . . . . . 7  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
( A. x  e.  I  ( f `  x ) ( le
`  ( ( I  X.  { R }
) `  x )
) ( g `  x )  <->  A. x  e.  I  ( f `  x ) O ( g `  x ) ) )
21 eqid 2387 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
22 simplr 732 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  ->  I  e.  W )
23 simprl 733 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f  e.  B )
245, 21, 4, 13, 22, 23pwselbas 13638 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f : I --> ( Base `  R ) )
25 ffn 5531 . . . . . . . . 9  |-  ( f : I --> ( Base `  R )  ->  f  Fn  I )
2624, 25syl 16 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f  Fn  I )
27 simprr 734 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g  e.  B )
285, 21, 4, 13, 22, 27pwselbas 13638 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g : I --> ( Base `  R ) )
29 ffn 5531 . . . . . . . . 9  |-  ( g : I --> ( Base `  R )  ->  g  Fn  I )
3028, 29syl 16 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g  Fn  I )
31 inidm 3493 . . . . . . . 8  |-  ( I  i^i  I )  =  I
32 eqidd 2388 . . . . . . . 8  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
33 eqidd 2388 . . . . . . . 8  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  (
g `  x )  =  ( g `  x ) )
3426, 30, 22, 22, 31, 32, 33ofrfval 6252 . . . . . . 7  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
( f  o R O g  <->  A. x  e.  I  ( f `  x ) O ( g `  x ) ) )
3520, 34bitr4d 248 . . . . . 6  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
( A. x  e.  I  ( f `  x ) ( le
`  ( ( I  X.  { R }
) `  x )
) ( g `  x )  <->  f  o R O g ) )
3635pm5.32da 623 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( ( ( f  e.  B  /\  g  e.  B )  /\  A. x  e.  I  (
f `  x )
( le `  (
( I  X.  { R } ) `  x
) ) ( g `
 x ) )  <-> 
( ( f  e.  B  /\  g  e.  B )  /\  f  o R O g ) ) )
37 brinxp2 4879 . . . . . 6  |-  ( f (  o R O  i^i  ( B  X.  B ) ) g  <-> 
( f  e.  B  /\  g  e.  B  /\  f  o R O g ) )
38 df-3an 938 . . . . . 6  |-  ( ( f  e.  B  /\  g  e.  B  /\  f  o R O g )  <->  ( ( f  e.  B  /\  g  e.  B )  /\  f  o R O g ) )
3937, 38bitri 241 . . . . 5  |-  ( f (  o R O  i^i  ( B  X.  B ) ) g  <-> 
( ( f  e.  B  /\  g  e.  B )  /\  f  o R O g ) )
4036, 39syl6bbr 255 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( ( ( f  e.  B  /\  g  e.  B )  /\  A. x  e.  I  (
f `  x )
( le `  (
( I  X.  { R } ) `  x
) ) ( g `
 x ) )  <-> 
f (  o R O  i^i  ( B  X.  B ) ) g ) )
4112, 40bitr3d 247 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( ( { f ,  g }  C_  ( Base `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  /\  A. x  e.  I  ( f `  x ) ( le
`  ( ( I  X.  { R }
) `  x )
) ( g `  x ) )  <->  f (  o R O  i^i  ( B  X.  B ) ) g ) )
4241opabbidv 4212 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )  /\  A. x  e.  I  ( f `  x ) ( le
`  ( ( I  X.  { R }
) `  x )
) ( g `  x ) ) }  =  { <. f ,  g >.  |  f (  o R O  i^i  ( B  X.  B ) ) g } )
43 pwsle.l . . . 4  |-  .<_  =  ( le `  Y )
447fveq2d 5672 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( le `  Y
)  =  ( le
`  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
4543, 44syl5eq 2431 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  -> 
.<_  =  ( le `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
46 eqid 2387 . . . 4  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
47 fvex 5682 . . . . 5  |-  (Scalar `  R )  e.  _V
4847a1i 11 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  R )  e.  _V )
49 simpr 448 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  I  e.  W )
50 snex 4346 . . . . 5  |-  { R }  e.  _V
51 xpexg 4929 . . . . 5  |-  ( ( I  e.  W  /\  { R }  e.  _V )  ->  ( I  X.  { R } )  e. 
_V )
5249, 50, 51sylancl 644 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { R } )  e.  _V )
53 eqid 2387 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
54 snnzg 3864 . . . . . 6  |-  ( R  e.  V  ->  { R }  =/=  (/) )
5554adantr 452 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { R }  =/=  (/) )
56 dmxp 5028 . . . . 5  |-  ( { R }  =/=  (/)  ->  dom  ( I  X.  { R } )  =  I )
5755, 56syl 16 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  dom  ( I  X.  { R } )  =  I )
58 eqid 2387 . . . 4  |-  ( le
`  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  ( le `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
5946, 48, 52, 53, 57, 58prdsle 13611 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( le `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  { <. f ,  g >.  |  ( { f ,  g }  C_  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  /\  A. x  e.  I  ( f `  x ) ( le
`  ( ( I  X.  { R }
) `  x )
) ( g `  x ) ) } )
6045, 59eqtrd 2419 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  -> 
.<_  =  { <. f ,  g >.  |  ( { f ,  g }  C_  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  /\  A. x  e.  I  ( f `  x ) ( le
`  ( ( I  X.  { R }
) `  x )
) ( g `  x ) ) } )
61 inss2 3505 . . . . 5  |-  (  o R O  i^i  ( B  X.  B ) ) 
C_  ( B  X.  B )
62 relxp 4923 . . . . 5  |-  Rel  ( B  X.  B )
63 relss 4903 . . . . 5  |-  ( (  o R O  i^i  ( B  X.  B
) )  C_  ( B  X.  B )  -> 
( Rel  ( B  X.  B )  ->  Rel  (  o R O  i^i  ( B  X.  B
) ) ) )
6461, 62, 63mp2 9 . . . 4  |-  Rel  (  o R O  i^i  ( B  X.  B ) )
6564a1i 11 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Rel  (  o R O  i^i  ( B  X.  B ) ) )
66 dfrel4v 5262 . . 3  |-  ( Rel  (  o R O  i^i  ( B  X.  B ) )  <->  (  o R O  i^i  ( B  X.  B ) )  =  { <. f ,  g >.  |  f (  o R O  i^i  ( B  X.  B ) ) g } )
6765, 66sylib 189 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (  o R O  i^i  ( B  X.  B ) )  =  { <. f ,  g
>.  |  f (  o R O  i^i  ( B  X.  B ) ) g } )
6842, 60, 673eqtr4d 2429 1  |-  ( ( R  e.  V  /\  I  e.  W )  -> 
.<_  =  (  o R O  i^i  ( B  X.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   _Vcvv 2899    i^i cin 3262    C_ wss 3263   (/)c0 3571   {csn 3757   {cpr 3758   class class class wbr 4153   {copab 4206    X. cxp 4816   dom cdm 4818   Rel wrel 4823    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020    o Rcofr 6243   Basecbs 13396  Scalarcsca 13459   lecple 13463   X_scprds 13596    ^s cpws 13597
This theorem is referenced by:  pwsleval  13642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-ofr 6245  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-hom 13480  df-cco 13481  df-prds 13598  df-pws 13600
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