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Theorem lsppropd 16099
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 2472 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
43pweqd 3806 . . 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 16098 . . . . 5  |-  ( ph  ->  ( LSubSp `  K )  =  ( LSubSp `  L
) )
12 rabeq 2952 . . . . 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 4057 . . 3  |-  ( ph  ->  |^| { t  e.  ( LSubSp `  K )  |  s  C_  t }  =  |^| { t  e.  ( LSubSp `  L
)  |  s  C_  t } )
154, 14mpteq12dv 4290 . 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 2438 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2438 . . . 4  |-  ( LSubSp `  K )  =  (
LSubSp `  K )
19 eqid 2438 . . . 4  |-  ( LSpan `  K )  =  (
LSpan `  K )
2017, 18, 19lspfval 16054 . . 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 2438 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
24 eqid 2438 . . . 4  |-  ( LSubSp `  L )  =  (
LSubSp `  L )
25 eqid 2438 . . . 4  |-  ( LSpan `  L )  =  (
LSpan `  L )
2623, 24, 25lspfval 16054 . . 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 2480 1  |-  ( ph  ->  ( LSpan `  K )  =  ( LSpan `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   |^|cint 4052    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   Basecbs 13474   +g cplusg 13534  Scalarcsca 13537   .scvsca 13538   LSubSpclss 16013   LSpanclspn 16052
This theorem is referenced by:  lbspropd  16176  lidlrsppropd  16306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-lss 16014  df-lsp 16053
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