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Theorem lsppropd 15775
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 2317 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
43pweqd 3630 . . 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 15774 . . . . 5  |-  ( ph  ->  ( LSubSp `  K )  =  ( LSubSp `  L
) )
12 rabeq 2782 . . . . 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 3867 . . 3  |-  ( ph  ->  |^| { t  e.  ( LSubSp `  K )  |  s  C_  t }  =  |^| { t  e.  ( LSubSp `  L
)  |  s  C_  t } )
154, 14mpteq12dv 4098 . 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 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2283 . . . 4  |-  ( LSubSp `  K )  =  (
LSubSp `  K )
19 eqid 2283 . . . 4  |-  ( LSpan `  K )  =  (
LSpan `  K )
2017, 18, 19lspfval 15730 . . 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 2283 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
24 eqid 2283 . . . 4  |-  ( LSubSp `  L )  =  (
LSubSp `  L )
25 eqid 2283 . . . 4  |-  ( LSpan `  L )  =  (
LSpan `  L )
2623, 24, 25lspfval 15730 . . 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 2325 1  |-  ( ph  ->  ( LSpan `  K )  =  ( LSpan `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   |^|cint 3862    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   LSubSpclss 15689   LSpanclspn 15728
This theorem is referenced by:  lbspropd  15852  lidlrsppropd  15982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-lss 15690  df-lsp 15729
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