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Theorem pwsle 15108
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 3064 . . . . . . 7  |-  f  e. 
_V
2 vex 3064 . . . . . . 7  |-  g  e. 
_V
31, 2prss 4128 . . . . . 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 2404 . . . . . . . . . 10  |-  (Scalar `  R )  =  (Scalar `  R )
75, 6pwsval 15102 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
87fveq2d 5855 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  Y
)  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
94, 8syl5eq 2457 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
109sseq2d 3472 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( { f ,  g }  C_  B  <->  { f ,  g } 
C_  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
113, 10syl5bb 259 . . . . 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 705 . . . 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 754 . . . . . . . . . . . 12  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  ->  R  e.  V )
14 fvconst2g 6107 . . . . . . . . . . . 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 5855 . . . . . . . . . 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 2463 . . . . . . . . 9  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  ( le `  ( ( I  X.  { R }
) `  x )
)  =  O )
1918breqd 4408 . . . . . . . 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 2842 . . . . . . 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 2404 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
22 simplr 756 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  ->  I  e.  W )
23 simprl 758 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f  e.  B )
245, 21, 4, 13, 22, 23pwselbas 15105 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f : I --> ( Base `  R ) )
25 ffn 5716 . . . . . . . . 9  |-  ( f : I --> ( Base `  R )  ->  f  Fn  I )
2624, 25syl 17 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
f  Fn  I )
27 simprr 760 . . . . . . . . . 10  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g  e.  B )
285, 21, 4, 13, 22, 27pwselbas 15105 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g : I --> ( Base `  R ) )
29 ffn 5716 . . . . . . . . 9  |-  ( g : I --> ( Base `  R )  ->  g  Fn  I )
3028, 29syl 17 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  ( f  e.  B  /\  g  e.  B ) )  -> 
g  Fn  I )
31 inidm 3650 . . . . . . . 8  |-  ( I  i^i  I )  =  I
32 eqidd 2405 . . . . . . . 8  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  (
f  e.  B  /\  g  e.  B )
)  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
33 eqidd 2405 . . . . . . . 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 6531 . . . . . . 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 258 . . . . . 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 4887 . . . . . 6  |-  ( f (  oR O  i^i  ( B  X.  B ) ) g  <-> 
( f  e.  B  /\  g  e.  B  /\  f  oR
O g ) )
38 df-3an 978 . . . . . 6  |-  ( ( f  e.  B  /\  g  e.  B  /\  f  oR O g )  <->  ( ( f  e.  B  /\  g  e.  B )  /\  f  oR O g ) )
3937, 38bitri 251 . . . . 5  |-  ( f (  oR O  i^i  ( B  X.  B ) ) g  <-> 
( ( f  e.  B  /\  g  e.  B )  /\  f  oR O g ) )
4036, 39syl6bbr 265 . . . 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 257 . . 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 4460 . 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 5855 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( le `  Y
)  =  ( le
`  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
4543, 44syl5eq 2457 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  -> 
.<_  =  ( le `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
46 eqid 2404 . . . 4  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
47 fvex 5861 . . . . 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 4634 . . . . 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 2404 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
54 snnzg 4091 . . . . . 6  |-  ( R  e.  V  ->  { R }  =/=  (/) )
5554adantr 465 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { R }  =/=  (/) )
56 dmxp 5044 . . . . 5  |-  ( { R }  =/=  (/)  ->  dom  ( I  X.  { R } )  =  I )
5755, 56syl 17 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  dom  ( I  X.  { R } )  =  I )
58 eqid 2404 . . . 4  |-  ( le
`  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  ( le `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
5946, 48, 52, 53, 57, 58prdsle 15078 . . 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 2445 . 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 3662 . . . . 5  |-  (  oR O  i^i  ( B  X.  B ) ) 
C_  ( B  X.  B )
62 relxp 4933 . . . . 5  |-  Rel  ( B  X.  B )
63 relss 4913 . . . . 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 5277 . . 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 198 . 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 2455 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 976    = wceq 1407    e. wcel 1844    =/= wne 2600   A.wral 2756   _Vcvv 3061    i^i cin 3415    C_ wss 3416   (/)c0 3740   {csn 3974   {cpr 3976   class class class wbr 4397   {copab 4454    X. cxp 4823   dom cdm 4825   Rel wrel 4830    Fn wfn 5566   -->wf 5567   ` cfv 5571  (class class class)co 6280    oRcofr 6522   Basecbs 14843  Scalarcsca 14914   lecple 14918   X_scprds 15062    ^s cpws 15063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-ofr 6524  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-fz 11729  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-plusg 14924  df-mulr 14925  df-sca 14927  df-vsca 14928  df-ip 14929  df-tset 14930  df-ple 14931  df-ds 14933  df-hom 14935  df-cco 14936  df-prds 15064  df-pws 15066
This theorem is referenced by:  pwsleval  15109
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