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Theorem pwsle 14747
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 )  -> 
.<_  =  (  oR 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 3116 . . . . . . 7  |-  f  e. 
_V
2 vex 3116 . . . . . . 7  |-  g  e. 
_V
31, 2prss 4181 . . . . . 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 2467 . . . . . . . . . 10  |-  (Scalar `  R )  =  (Scalar `  R )
75, 6pwsval 14741 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
87fveq2d 5870 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  Y
)  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
94, 8syl5eq 2520 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
109sseq2d 3532 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( { f ,  g }  C_  B  <->  { f ,  g } 
C_  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
113, 10syl5bb 257 . . . . 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 704 . . . 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 753 . . . . . . . . . . . 12  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  ->  R  e.  V )
14 fvconst2g 6114 . . . . . . . . . . . 12  |-  ( ( R  e.  V  /\  x  e.  I )  ->  ( ( I  X.  { R } ) `  x )  =  R )
1513, 14sylan 471 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  (
( I  X.  { R } ) `  x
)  =  R )
1615fveq2d 5870 . . . . . . . . . 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 2526 . . . . . . . . 9  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  ( le `  ( ( I  X.  { R }
) `  x )
)  =  O )
1918breqd 4458 . . . . . . . 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 2900 . . . . . . 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 2467 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
22 simplr 754 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  ->  I  e.  W )
23 simprl 755 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f  e.  B )
245, 21, 4, 13, 22, 23pwselbas 14744 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f : I --> ( Base `  R ) )
25 ffn 5731 . . . . . . . . 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 756 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g  e.  B )
285, 21, 4, 13, 22, 27pwselbas 14744 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g : I --> ( Base `  R ) )
29 ffn 5731 . . . . . . . . 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 3707 . . . . . . . 8  |-  ( I  i^i  I )  =  I
32 eqidd 2468 . . . . . . . 8  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
33 eqidd 2468 . . . . . . . 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 6532 . . . . . . 7  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
( f  oR O g  <->  A. x  e.  I  ( f `  x ) O ( g `  x ) ) )
3520, 34bitr4d 256 . . . . . 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  oR O g ) )
3635pm5.32da 641 . . . . 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  oR O g ) ) )
37 brinxp2 5061 . . . . . 6  |-  ( f (  oR O  i^i  ( B  X.  B ) ) g  <-> 
( f  e.  B  /\  g  e.  B  /\  f  oR
O g ) )
38 df-3an 975 . . . . . 6  |-  ( ( f  e.  B  /\  g  e.  B  /\  f  oR O g )  <->  ( ( f  e.  B  /\  g  e.  B )  /\  f  oR O g ) )
3937, 38bitri 249 . . . . 5  |-  ( f (  oR O  i^i  ( B  X.  B ) ) g  <-> 
( ( f  e.  B  /\  g  e.  B )  /\  f  oR O g ) )
4036, 39syl6bbr 263 . . . 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 (  oR O  i^i  ( B  X.  B ) ) g ) )
4112, 40bitr3d 255 . . 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 (  oR O  i^i  ( B  X.  B
) ) g ) )
4241opabbidv 4510 . 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 (  oR O  i^i  ( B  X.  B ) ) g } )
43 pwsle.l . . . 4  |-  .<_  =  ( le `  Y )
447fveq2d 5870 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( le `  Y
)  =  ( le
`  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
4543, 44syl5eq 2520 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  -> 
.<_  =  ( le `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
46 eqid 2467 . . . 4  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
47 fvex 5876 . . . . 5  |-  (Scalar `  R )  e.  _V
4847a1i 11 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  R )  e.  _V )
49 simpr 461 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  I  e.  W )
50 snex 4688 . . . . 5  |-  { R }  e.  _V
51 xpexg 6586 . . . . 5  |-  ( ( I  e.  W  /\  { R }  e.  _V )  ->  ( I  X.  { R } )  e. 
_V )
5249, 50, 51sylancl 662 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { R } )  e.  _V )
53 eqid 2467 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
54 snnzg 4144 . . . . . 6  |-  ( R  e.  V  ->  { R }  =/=  (/) )
5554adantr 465 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { R }  =/=  (/) )
56 dmxp 5221 . . . . 5  |-  ( { R }  =/=  (/)  ->  dom  ( I  X.  { R } )  =  I )
5755, 56syl 16 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  dom  ( I  X.  { R } )  =  I )
58 eqid 2467 . . . 4  |-  ( le
`  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  ( le `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
5946, 48, 52, 53, 57, 58prdsle 14717 . . 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 2508 . 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 3719 . . . . 5  |-  (  oR O  i^i  ( B  X.  B ) ) 
C_  ( B  X.  B )
62 relxp 5110 . . . . 5  |-  Rel  ( B  X.  B )
63 relss 5090 . . . . 5  |-  ( (  oR O  i^i  ( B  X.  B
) )  C_  ( B  X.  B )  -> 
( Rel  ( B  X.  B )  ->  Rel  (  oR O  i^i  ( B  X.  B
) ) ) )
6461, 62, 63mp2 9 . . . 4  |-  Rel  (  oR O  i^i  ( B  X.  B
) )
6564a1i 11 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Rel  (  oR O  i^i  ( B  X.  B ) ) )
66 dfrel4v 5458 . . 3  |-  ( Rel  (  oR O  i^i  ( B  X.  B ) )  <->  (  oR O  i^i  ( B  X.  B ) )  =  { <. f ,  g >.  |  f (  oR O  i^i  ( B  X.  B ) ) g } )
6765, 66sylib 196 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (  oR O  i^i  ( B  X.  B ) )  =  { <. f ,  g
>.  |  f (  oR O  i^i  ( B  X.  B
) ) g } )
6842, 60, 673eqtr4d 2518 1  |-  ( ( R  e.  V  /\  I  e.  W )  -> 
.<_  =  (  oR O  i^i  ( B  X.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   {cpr 4029   class class class wbr 4447   {copab 4504    X. cxp 4997   dom cdm 4999   Rel wrel 5004    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    oRcofr 6523   Basecbs 14490  Scalarcsca 14558   lecple 14562   X_scprds 14701    ^s cpws 14702
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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-nel 2665  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-int 4283  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-ofr 6525  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-hom 14579  df-cco 14580  df-prds 14703  df-pws 14705
This theorem is referenced by:  pwsleval  14748
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