MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lsppropd Unicode version

Theorem lsppropd 16049
Description: If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
lsspropd.b1  |-  ( ph  ->  B  =  ( Base `  K ) )
lsspropd.b2  |-  ( ph  ->  B  =  ( Base `  L ) )
lsspropd.w  |-  ( ph  ->  B  C_  W )
lsspropd.p  |-  ( (
ph  /\  ( x  e.  W  /\  y  e.  W ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
lsspropd.s1  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  B ) )  -> 
( x ( .s
`  K ) y )  e.  W )
lsspropd.s2  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  B ) )  -> 
( x ( .s
`  K ) y )  =  ( x ( .s `  L
) y ) )
lsspropd.p1  |-  ( ph  ->  P  =  ( Base `  (Scalar `  K )
) )
lsspropd.p2  |-  ( ph  ->  P  =  ( Base `  (Scalar `  L )
) )
lsspropd.v1  |-  ( ph  ->  K  e.  _V )
lsspropd.v2  |-  ( ph  ->  L  e.  _V )
Assertion
Ref Expression
lsppropd  |-  ( ph  ->  ( LSpan `  K )  =  ( LSpan `  L
) )
Distinct variable groups:    x, y, B    x, K, y    ph, x, y    x, W, y    x, L, y    x, P, y

Proof of Theorem lsppropd
Dummy variables  s 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsspropd.b1 . . . . 5  |-  ( ph  ->  B  =  ( Base `  K ) )
2 lsspropd.b2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
31, 2eqtr3d 2438 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
43pweqd 3764 . . 3  |-  ( ph  ->  ~P ( Base `  K
)  =  ~P ( Base `  L ) )
5 lsspropd.w . . . . . 6  |-  ( ph  ->  B  C_  W )
6 lsspropd.p . . . . . 6  |-  ( (
ph  /\  ( x  e.  W  /\  y  e.  W ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
7 lsspropd.s1 . . . . . 6  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  B ) )  -> 
( x ( .s
`  K ) y )  e.  W )
8 lsspropd.s2 . . . . . 6  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  B ) )  -> 
( x ( .s
`  K ) y )  =  ( x ( .s `  L
) y ) )
9 lsspropd.p1 . . . . . 6  |-  ( ph  ->  P  =  ( Base `  (Scalar `  K )
) )
10 lsspropd.p2 . . . . . 6  |-  ( ph  ->  P  =  ( Base `  (Scalar `  L )
) )
111, 2, 5, 6, 7, 8, 9, 10lsspropd 16048 . . . . 5  |-  ( ph  ->  ( LSubSp `  K )  =  ( LSubSp `  L
) )
12 rabeq 2910 . . . . 5  |-  ( (
LSubSp `  K )  =  ( LSubSp `  L )  ->  { t  e.  (
LSubSp `  K )  |  s  C_  t }  =  { t  e.  (
LSubSp `  L )  |  s  C_  t }
)
1311, 12syl 16 . . . 4  |-  ( ph  ->  { t  e.  (
LSubSp `  K )  |  s  C_  t }  =  { t  e.  (
LSubSp `  L )  |  s  C_  t }
)
1413inteqd 4015 . . 3  |-  ( ph  ->  |^| { t  e.  ( LSubSp `  K )  |  s  C_  t }  =  |^| { t  e.  ( LSubSp `  L
)  |  s  C_  t } )
154, 14mpteq12dv 4247 . 2  |-  ( ph  ->  ( s  e.  ~P ( Base `  K )  |-> 
|^| { t  e.  (
LSubSp `  K )  |  s  C_  t }
)  =  ( s  e.  ~P ( Base `  L )  |->  |^| { t  e.  ( LSubSp `  L
)  |  s  C_  t } ) )
16 lsspropd.v1 . . 3  |-  ( ph  ->  K  e.  _V )
17 eqid 2404 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2404 . . . 4  |-  ( LSubSp `  K )  =  (
LSubSp `  K )
19 eqid 2404 . . . 4  |-  ( LSpan `  K )  =  (
LSpan `  K )
2017, 18, 19lspfval 16004 . . 3  |-  ( K  e.  _V  ->  ( LSpan `  K )  =  ( s  e.  ~P ( Base `  K )  |-> 
|^| { t  e.  (
LSubSp `  K )  |  s  C_  t }
) )
2116, 20syl 16 . 2  |-  ( ph  ->  ( LSpan `  K )  =  ( s  e. 
~P ( Base `  K
)  |->  |^| { t  e.  ( LSubSp `  K )  |  s  C_  t } ) )
22 lsspropd.v2 . . 3  |-  ( ph  ->  L  e.  _V )
23 eqid 2404 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
24 eqid 2404 . . . 4  |-  ( LSubSp `  L )  =  (
LSubSp `  L )
25 eqid 2404 . . . 4  |-  ( LSpan `  L )  =  (
LSpan `  L )
2623, 24, 25lspfval 16004 . . 3  |-  ( L  e.  _V  ->  ( LSpan `  L )  =  ( s  e.  ~P ( Base `  L )  |-> 
|^| { t  e.  (
LSubSp `  L )  |  s  C_  t }
) )
2722, 26syl 16 . 2  |-  ( ph  ->  ( LSpan `  L )  =  ( s  e. 
~P ( Base `  L
)  |->  |^| { t  e.  ( LSubSp `  L )  |  s  C_  t } ) )
2815, 21, 273eqtr4d 2446 1  |-  ( ph  ->  ( LSpan `  K )  =  ( LSpan `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   |^|cint 4010    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   .scvsca 13488   LSubSpclss 15963   LSpanclspn 16002
This theorem is referenced by:  lbspropd  16126  lidlrsppropd  16256
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-lss 15964  df-lsp 16003
  Copyright terms: Public domain W3C validator