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Theorem pwsle 13407
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 2804 . . . . . . 7  |-  f  e. 
_V
2 vex 2804 . . . . . . 7  |-  g  e. 
_V
31, 2prss 3785 . . . . . 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 2296 . . . . . . . . . 10  |-  (Scalar `  R )  =  (Scalar `  R )
75, 6pwsval 13401 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
87fveq2d 5545 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  Y
)  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
94, 8syl5eq 2340 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
109sseq2d 3219 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( { f ,  g }  C_  B  <->  { f ,  g } 
C_  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
113, 10syl5bb 248 . . . . 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 685 . . . 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 730 . . . . . . . . . . . 12  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  ->  R  e.  V )
14 fvconst2g 5743 . . . . . . . . . . . 12  |-  ( ( R  e.  V  /\  x  e.  I )  ->  ( ( I  X.  { R } ) `  x )  =  R )
1513, 14sylan 457 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  (
( I  X.  { R } ) `  x
)  =  R )
1615fveq2d 5545 . . . . . . . . . 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 2346 . . . . . . . . 9  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  ( le `  ( ( I  X.  { R }
) `  x )
)  =  O )
1918breqd 4050 . . . . . . . 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 2572 . . . . . . 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 2296 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
22 simplr 731 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  ->  I  e.  W )
23 simprl 732 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f  e.  B )
245, 21, 4, 13, 22, 23pwselbas 13404 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f : I --> ( Base `  R ) )
25 ffn 5405 . . . . . . . . 9  |-  ( f : I --> ( Base `  R )  ->  f  Fn  I )
2624, 25syl 15 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f  Fn  I )
27 simprr 733 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g  e.  B )
285, 21, 4, 13, 22, 27pwselbas 13404 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g : I --> ( Base `  R ) )
29 ffn 5405 . . . . . . . . 9  |-  ( g : I --> ( Base `  R )  ->  g  Fn  I )
3028, 29syl 15 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g  Fn  I )
31 inidm 3391 . . . . . . . 8  |-  ( I  i^i  I )  =  I
32 eqidd 2297 . . . . . . . 8  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
33 eqidd 2297 . . . . . . . 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 6102 . . . . . . 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 247 . . . . . 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 622 . . . . 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 4767 . . . . . 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 936 . . . . . 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 240 . . . . 5  |-  ( f (  o R O  i^i  ( B  X.  B ) ) g  <-> 
( ( f  e.  B  /\  g  e.  B )  /\  f  o R O g ) )
4036, 39syl6bbr 254 . . . 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 246 . . 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 4098 . 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 5545 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( le `  Y
)  =  ( le
`  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
4543, 44syl5eq 2340 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  -> 
.<_  =  ( le `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
46 eqid 2296 . . . 4  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
47 fvex 5555 . . . . 5  |-  (Scalar `  R )  e.  _V
4847a1i 10 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  R )  e.  _V )
49 simpr 447 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  I  e.  W )
50 snex 4232 . . . . 5  |-  { R }  e.  _V
51 xpexg 4816 . . . . 5  |-  ( ( I  e.  W  /\  { R }  e.  _V )  ->  ( I  X.  { R } )  e. 
_V )
5249, 50, 51sylancl 643 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { R } )  e.  _V )
53 eqid 2296 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
54 snnzg 3756 . . . . . 6  |-  ( R  e.  V  ->  { R }  =/=  (/) )
5554adantr 451 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { R }  =/=  (/) )
56 dmxp 4913 . . . . 5  |-  ( { R }  =/=  (/)  ->  dom  ( I  X.  { R } )  =  I )
5755, 56syl 15 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  dom  ( I  X.  { R } )  =  I )
58 eqid 2296 . . . 4  |-  ( le
`  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  ( le `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
5946, 48, 52, 53, 57, 58prdsle 13377 . . 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 2328 . 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 3403 . . . . 5  |-  (  o R O  i^i  ( B  X.  B ) ) 
C_  ( B  X.  B )
62 relxp 4810 . . . . 5  |-  Rel  ( B  X.  B )
63 relss 4791 . . . . 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 17 . . . 4  |-  Rel  (  o R O  i^i  ( B  X.  B ) )
6564a1i 10 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Rel  (  o R O  i^i  ( B  X.  B ) ) )
66 dfrel4v 5141 . . 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 188 . 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 2338 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   {cpr 3654   class class class wbr 4039   {copab 4092    X. cxp 4703   dom cdm 4705   Rel wrel 4710    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Rcofr 6093   Basecbs 13164  Scalarcsca 13227   lecple 13231   X_scprds 13362    ^s cpws 13363
This theorem is referenced by:  pwsleval  13408
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-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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-nel 2462  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-int 3879  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-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  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-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366
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