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Theorem lsmpropd 16193
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 1018 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  ph )
2 simp12 1019 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  t  e.  ~P B
)
32elpwid 3889 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  t  C_  B )
4 simp2 989 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  x  e.  t )
53, 4sseldd 3376 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  x  e.  B )
6 simp13 1020 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  u  e.  ~P B
)
76elpwid 3889 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  u  C_  B )
8 simp3 990 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  y  e.  u )
97, 8sseldd 3376 . . . . . . 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 1216 . . . . . 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 6169 . . . . 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 5086 . . . 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 6169 . . 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 3884 . . . 4  |-  ( ph  ->  ~P B  =  ~P ( Base `  K )
)
17 mpt2eq12 6165 . . . 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 3884 . . . 4  |-  ( ph  ->  ~P B  =  ~P ( Base `  L )
)
21 mpt2eq12 6165 . . . 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 2483 . 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 2443 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
26 eqid 2443 . . . 4  |-  ( +g  `  K )  =  ( +g  `  K )
27 eqid 2443 . . . 4  |-  ( LSSum `  K )  =  (
LSSum `  K )
2825, 26, 27lsmfval 16156 . . 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 2443 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
32 eqid 2443 . . . 4  |-  ( +g  `  L )  =  ( +g  `  L )
33 eqid 2443 . . . 4  |-  ( LSSum `  L )  =  (
LSSum `  L )
3431, 32, 33lsmfval 16156 . . 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 2485 1  |-  ( ph  ->  ( LSSum `  K )  =  ( LSSum `  L
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2991   ~Pcpw 3879   ran crn 4860   ` cfv 5437  (class class class)co 6110    e. cmpt2 6112   Basecbs 14193   +g cplusg 14257   LSSumclsm 16152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-lsm 16154
This theorem is referenced by:  hlhillsm  35627
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