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

Theorem lsmpropd 16484
Description: If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.)
Hypotheses
Ref Expression
lsmpropd.b1  |-  ( ph  ->  B  =  ( Base `  K ) )
lsmpropd.b2  |-  ( ph  ->  B  =  ( Base `  L ) )
lsmpropd.p  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
lsmpropd.v1  |-  ( ph  ->  K  e.  _V )
lsmpropd.v2  |-  ( ph  ->  L  e.  _V )
Assertion
Ref Expression
lsmpropd  |-  ( ph  ->  ( LSSum `  K )  =  ( LSSum `  L
) )
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y

Proof of Theorem lsmpropd
Dummy variables  u  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 1021 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  ph )
2 simp12 1022 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  t  e.  ~P B
)
32elpwid 4013 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  t  C_  B )
4 simp2 992 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  x  e.  t )
53, 4sseldd 3498 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  x  e.  B )
6 simp13 1023 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  u  e.  ~P B
)
76elpwid 4013 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  u  C_  B )
8 simp3 993 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  y  e.  u )
97, 8sseldd 3498 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  y  e.  B )
10 lsmpropd.p . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
111, 5, 9, 10syl12anc 1221 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
1211mpt2eq3dva 6336 . . . . 5  |-  ( (
ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  -> 
( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  K ) y ) )  =  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  L
) y ) ) )
1312rneqd 5221 . . . 4  |-  ( (
ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  ->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  K ) y ) )  =  ran  (
x  e.  t ,  y  e.  u  |->  ( x ( +g  `  L
) y ) ) )
1413mpt2eq3dva 6336 . . 3  |-  ( ph  ->  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x ( +g  `  K
) y ) ) )  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  L ) y ) ) ) )
15 lsmpropd.b1 . . . . 5  |-  ( ph  ->  B  =  ( Base `  K ) )
1615pweqd 4008 . . . 4  |-  ( ph  ->  ~P B  =  ~P ( Base `  K )
)
17 mpt2eq12 6332 . . . 4  |-  ( ( ~P B  =  ~P ( Base `  K )  /\  ~P B  =  ~P ( Base `  K )
)  ->  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x ( +g  `  K
) y ) ) )  =  ( t  e.  ~P ( Base `  K ) ,  u  e.  ~P ( Base `  K
)  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  K ) y ) ) ) )
1816, 16, 17syl2anc 661 . . 3  |-  ( ph  ->  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x ( +g  `  K
) y ) ) )  =  ( t  e.  ~P ( Base `  K ) ,  u  e.  ~P ( Base `  K
)  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  K ) y ) ) ) )
19 lsmpropd.b2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
2019pweqd 4008 . . . 4  |-  ( ph  ->  ~P B  =  ~P ( Base `  L )
)
21 mpt2eq12 6332 . . . 4  |-  ( ( ~P B  =  ~P ( Base `  L )  /\  ~P B  =  ~P ( Base `  L )
)  ->  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x ( +g  `  L
) y ) ) )  =  ( t  e.  ~P ( Base `  L ) ,  u  e.  ~P ( Base `  L
)  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  L ) y ) ) ) )
2220, 20, 21syl2anc 661 . . 3  |-  ( ph  ->  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x ( +g  `  L
) y ) ) )  =  ( t  e.  ~P ( Base `  L ) ,  u  e.  ~P ( Base `  L
)  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  L ) y ) ) ) )
2314, 18, 223eqtr3d 2509 . 2  |-  ( ph  ->  ( t  e.  ~P ( Base `  K ) ,  u  e.  ~P ( Base `  K )  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  K ) y ) ) )  =  ( t  e. 
~P ( Base `  L
) ,  u  e. 
~P ( Base `  L
)  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  L ) y ) ) ) )
24 lsmpropd.v1 . . 3  |-  ( ph  ->  K  e.  _V )
25 eqid 2460 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
26 eqid 2460 . . . 4  |-  ( +g  `  K )  =  ( +g  `  K )
27 eqid 2460 . . . 4  |-  ( LSSum `  K )  =  (
LSSum `  K )
2825, 26, 27lsmfval 16447 . . 3  |-  ( K  e.  _V  ->  ( LSSum `  K )  =  ( t  e.  ~P ( Base `  K ) ,  u  e.  ~P ( Base `  K )  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  K ) y ) ) ) )
2924, 28syl 16 . 2  |-  ( ph  ->  ( LSSum `  K )  =  ( t  e. 
~P ( Base `  K
) ,  u  e. 
~P ( Base `  K
)  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  K ) y ) ) ) )
30 lsmpropd.v2 . . 3  |-  ( ph  ->  L  e.  _V )
31 eqid 2460 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
32 eqid 2460 . . . 4  |-  ( +g  `  L )  =  ( +g  `  L )
33 eqid 2460 . . . 4  |-  ( LSSum `  L )  =  (
LSSum `  L )
3431, 32, 33lsmfval 16447 . . 3  |-  ( L  e.  _V  ->  ( LSSum `  L )  =  ( t  e.  ~P ( Base `  L ) ,  u  e.  ~P ( Base `  L )  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  L ) y ) ) ) )
3530, 34syl 16 . 2  |-  ( ph  ->  ( LSSum `  L )  =  ( t  e. 
~P ( Base `  L
) ,  u  e. 
~P ( Base `  L
)  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x ( +g  `  L ) y ) ) ) )
3623, 29, 353eqtr4d 2511 1  |-  ( ph  ->  ( LSSum `  K )  =  ( LSSum `  L
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3106   ~Pcpw 4003   ran crn 4993   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   Basecbs 14479   +g cplusg 14544   LSSumclsm 16443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-lsm 16445
This theorem is referenced by:  hlhillsm  36631
  Copyright terms: Public domain W3C validator