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Theorem pwsle 14871
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 3098 . . . . . . 7  |-  f  e. 
_V
2 vex 3098 . . . . . . 7  |-  g  e. 
_V
31, 2prss 4169 . . . . . 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 2443 . . . . . . . . . 10  |-  (Scalar `  R )  =  (Scalar `  R )
75, 6pwsval 14865 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
87fveq2d 5860 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  Y
)  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
94, 8syl5eq 2496 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
109sseq2d 3517 . . . . . 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 6109 . . . . . . . . . . . 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 5860 . . . . . . . . . 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 2502 . . . . . . . . 9  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  ( le `  ( ( I  X.  { R }
) `  x )
)  =  O )
1918breqd 4448 . . . . . . . 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 2879 . . . . . . 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 2443 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
22 simplr 755 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  ->  I  e.  W )
23 simprl 756 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f  e.  B )
245, 21, 4, 13, 22, 23pwselbas 14868 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f : I --> ( Base `  R ) )
25 ffn 5721 . . . . . . . . 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 757 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g  e.  B )
285, 21, 4, 13, 22, 27pwselbas 14868 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g : I --> ( Base `  R ) )
29 ffn 5721 . . . . . . . . 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 3692 . . . . . . . 8  |-  ( I  i^i  I )  =  I
32 eqidd 2444 . . . . . . . 8  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
33 eqidd 2444 . . . . . . . 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 6533 . . . . . . 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 5051 . . . . . 6  |-  ( f (  oR O  i^i  ( B  X.  B ) ) g  <-> 
( f  e.  B  /\  g  e.  B  /\  f  oR
O g ) )
38 df-3an 976 . . . . . 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 4500 . 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 5860 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( le `  Y
)  =  ( le
`  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
4543, 44syl5eq 2496 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  -> 
.<_  =  ( le `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
46 eqid 2443 . . . 4  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
47 fvex 5866 . . . . 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 4678 . . . . 5  |-  { R }  e.  _V
51 xpexg 6587 . . . . 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 2443 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
54 snnzg 4132 . . . . . 6  |-  ( R  e.  V  ->  { R }  =/=  (/) )
5554adantr 465 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { R }  =/=  (/) )
56 dmxp 5211 . . . . 5  |-  ( { R }  =/=  (/)  ->  dom  ( I  X.  { R } )  =  I )
5755, 56syl 16 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  dom  ( I  X.  { R } )  =  I )
58 eqid 2443 . . . 4  |-  ( le
`  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  ( le `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
5946, 48, 52, 53, 57, 58prdsle 14841 . . 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 2484 . 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 3704 . . . . 5  |-  (  oR O  i^i  ( B  X.  B ) ) 
C_  ( B  X.  B )
62 relxp 5100 . . . . 5  |-  Rel  ( B  X.  B )
63 relss 5080 . . . . 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 5448 . . 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 2494 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095    i^i cin 3460    C_ wss 3461   (/)c0 3770   {csn 4014   {cpr 4016   class class class wbr 4437   {copab 4494    X. cxp 4987   dom cdm 4989   Rel wrel 4994    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    oRcofr 6524   Basecbs 14614  Scalarcsca 14682   lecple 14686   X_scprds 14825    ^s cpws 14826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-fz 11684  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-plusg 14692  df-mulr 14693  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-hom 14703  df-cco 14704  df-prds 14827  df-pws 14829
This theorem is referenced by:  pwsleval  14872
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