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Theorem lsmpropd 16897
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 1024 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  ph )
2 simp12 1025 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  t  e.  ~P B
)
32elpwid 4009 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  t  C_  B )
4 simp2 995 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  x  e.  t )
53, 4sseldd 3490 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  x  e.  B )
6 simp13 1026 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  u  e.  ~P B
)
76elpwid 4009 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  u  C_  B )
8 simp3 996 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  y  e.  u )
97, 8sseldd 3490 . . . . . . 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 1224 . . . . . 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 6334 . . . . 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 5219 . . . 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 6334 . . 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 4004 . . . 4  |-  ( ph  ->  ~P B  =  ~P ( Base `  K )
)
17 mpt2eq12 6330 . . . 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 659 . . 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 4004 . . . 4  |-  ( ph  ->  ~P B  =  ~P ( Base `  L )
)
21 mpt2eq12 6330 . . . 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 659 . . 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 2503 . 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 2454 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
26 eqid 2454 . . . 4  |-  ( +g  `  K )  =  ( +g  `  K )
27 eqid 2454 . . . 4  |-  ( LSSum `  K )  =  (
LSSum `  K )
2825, 26, 27lsmfval 16860 . . 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 2454 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
32 eqid 2454 . . . 4  |-  ( +g  `  L )  =  ( +g  `  L )
33 eqid 2454 . . . 4  |-  ( LSSum `  L )  =  (
LSSum `  L )
3431, 32, 33lsmfval 16860 . . 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 2505 1  |-  ( ph  ->  ( LSSum `  K )  =  ( LSSum `  L
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106   ~Pcpw 3999   ran crn 4989   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Basecbs 14719   +g cplusg 14787   LSSumclsm 16856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-lsm 16858
This theorem is referenced by:  hlhillsm  38102
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