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

Theorem lsmpropd 14986
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 985 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  ph )
2 simp12 986 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  t  e.  ~P B
)
3 elpwi 3633 . . . . . . . . 9  |-  ( t  e.  ~P B  -> 
t  C_  B )
42, 3syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  t  C_  B )
5 simp2 956 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  x  e.  t )
64, 5sseldd 3181 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  x  e.  B )
7 simp13 987 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  u  e.  ~P B
)
8 elpwi 3633 . . . . . . . . 9  |-  ( u  e.  ~P B  ->  u  C_  B )
97, 8syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  u  C_  B )
10 simp3 957 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  y  e.  u )
119, 10sseldd 3181 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  y  e.  B )
12 lsmpropd.p . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
131, 6, 11, 12syl12anc 1180 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
1413mpt2eq3dva 5912 . . . . 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 ) ) )
1514rneqd 4906 . . . 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 ) ) )
1615mpt2eq3dva 5912 . . 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 ) ) ) )
17 lsmpropd.b1 . . . . 5  |-  ( ph  ->  B  =  ( Base `  K ) )
1817pweqd 3630 . . . 4  |-  ( ph  ->  ~P B  =  ~P ( Base `  K )
)
19 mpt2eq12 5908 . . . 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 ) ) ) )
2018, 18, 19syl2anc 642 . . 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 ) ) ) )
21 lsmpropd.b2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
2221pweqd 3630 . . . 4  |-  ( ph  ->  ~P B  =  ~P ( Base `  L )
)
23 mpt2eq12 5908 . . . 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 ) ) ) )
2422, 22, 23syl2anc 642 . . 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 ) ) ) )
2516, 20, 243eqtr3d 2323 . 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 ) ) ) )
26 lsmpropd.v1 . . 3  |-  ( ph  ->  K  e.  _V )
27 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
28 eqid 2283 . . . 4  |-  ( +g  `  K )  =  ( +g  `  K )
29 eqid 2283 . . . 4  |-  ( LSSum `  K )  =  (
LSSum `  K )
3027, 28, 29lsmfval 14949 . . 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 ) ) ) )
3126, 30syl 15 . 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 ) ) ) )
32 lsmpropd.v2 . . 3  |-  ( ph  ->  L  e.  _V )
33 eqid 2283 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
34 eqid 2283 . . . 4  |-  ( +g  `  L )  =  ( +g  `  L )
35 eqid 2283 . . . 4  |-  ( LSSum `  L )  =  (
LSSum `  L )
3633, 34, 35lsmfval 14949 . . 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 ) ) ) )
3732, 36syl 15 . 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 ) ) ) )
3825, 31, 373eqtr4d 2325 1  |-  ( ph  ->  ( LSSum `  K )  =  ( LSSum `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   +g cplusg 13208   LSSumclsm 14945
This theorem is referenced by:  hlhillsm  32149
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  ax-un 4512
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-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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-lsm 14947
  Copyright terms: Public domain W3C validator