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Theorem lsppropd 15985
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 2400 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
43pweqd 3719 . . 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 15984 . . . . 5  |-  ( ph  ->  ( LSubSp `  K )  =  ( LSubSp `  L
) )
12 rabeq 2867 . . . . 5  |-  ( (
LSubSp `  K )  =  ( LSubSp `  L )  ->  { t  e.  (
LSubSp `  K )  |  s  C_  t }  =  { t  e.  (
LSubSp `  L )  |  s  C_  t }
)
1311, 12syl 15 . . . 4  |-  ( ph  ->  { t  e.  (
LSubSp `  K )  |  s  C_  t }  =  { t  e.  (
LSubSp `  L )  |  s  C_  t }
)
1413inteqd 3969 . . 3  |-  ( ph  ->  |^| { t  e.  ( LSubSp `  K )  |  s  C_  t }  =  |^| { t  e.  ( LSubSp `  L
)  |  s  C_  t } )
154, 14mpteq12dv 4200 . 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 2366 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2366 . . . 4  |-  ( LSubSp `  K )  =  (
LSubSp `  K )
19 eqid 2366 . . . 4  |-  ( LSpan `  K )  =  (
LSpan `  K )
2017, 18, 19lspfval 15940 . . 3  |-  ( K  e.  _V  ->  ( LSpan `  K )  =  ( s  e.  ~P ( Base `  K )  |-> 
|^| { t  e.  (
LSubSp `  K )  |  s  C_  t }
) )
2116, 20syl 15 . 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 2366 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
24 eqid 2366 . . . 4  |-  ( LSubSp `  L )  =  (
LSubSp `  L )
25 eqid 2366 . . . 4  |-  ( LSpan `  L )  =  (
LSpan `  L )
2623, 24, 25lspfval 15940 . . 3  |-  ( L  e.  _V  ->  ( LSpan `  L )  =  ( s  e.  ~P ( Base `  L )  |-> 
|^| { t  e.  (
LSubSp `  L )  |  s  C_  t }
) )
2722, 26syl 15 . 2  |-  ( ph  ->  ( LSpan `  L )  =  ( s  e. 
~P ( Base `  L
)  |->  |^| { t  e.  ( LSubSp `  L )  |  s  C_  t } ) )
2815, 21, 273eqtr4d 2408 1  |-  ( ph  ->  ( LSpan `  K )  =  ( LSpan `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   {crab 2632   _Vcvv 2873    C_ wss 3238   ~Pcpw 3714   |^|cint 3964    e. cmpt 4179   ` cfv 5358  (class class class)co 5981   Basecbs 13356   +g cplusg 13416  Scalarcsca 13419   .scvsca 13420   LSubSpclss 15899   LSpanclspn 15938
This theorem is referenced by:  lbspropd  16062  lidlrsppropd  16192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-lss 15900  df-lsp 15939
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