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Theorem lsppropd 17099
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 2477 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
43pweqd 3865 . . 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 17098 . . . . 5  |-  ( ph  ->  ( LSubSp `  K )  =  ( LSubSp `  L
) )
12 rabeq 2966 . . . . 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 4133 . . 3  |-  ( ph  ->  |^| { t  e.  ( LSubSp `  K )  |  s  C_  t }  =  |^| { t  e.  ( LSubSp `  L
)  |  s  C_  t } )
154, 14mpteq12dv 4370 . 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 2443 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2443 . . . 4  |-  ( LSubSp `  K )  =  (
LSubSp `  K )
19 eqid 2443 . . . 4  |-  ( LSpan `  K )  =  (
LSpan `  K )
2017, 18, 19lspfval 17054 . . 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 2443 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
24 eqid 2443 . . . 4  |-  ( LSubSp `  L )  =  (
LSubSp `  L )
25 eqid 2443 . . . 4  |-  ( LSpan `  L )  =  (
LSpan `  L )
2623, 24, 25lspfval 17054 . . 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 2485 1  |-  ( ph  ->  ( LSpan `  K )  =  ( LSpan `  L
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972    C_ wss 3328   ~Pcpw 3860   |^|cint 4128    e. cmpt 4350   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238  Scalarcsca 14241   .scvsca 14242   LSubSpclss 17013   LSpanclspn 17052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-lss 17014  df-lsp 17053
This theorem is referenced by:  lbspropd  17180  lidlrsppropd  17312
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