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Theorem lsmpropd 15272
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 987 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  ph )
2 simp12 988 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  t  e.  ~P B
)
32elpwid 3776 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  t  C_  B )
4 simp2 958 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  x  e.  t )
53, 4sseldd 3317 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  x  e.  B )
6 simp13 989 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  u  e.  ~P B
)
76elpwid 3776 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  u  C_  B )
8 simp3 959 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ~P B  /\  u  e.  ~P B )  /\  x  e.  t  /\  y  e.  u )  ->  y  e.  u )
97, 8sseldd 3317 . . . . . . 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 1182 . . . . . 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 6105 . . . . 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 5064 . . . 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 6105 . . 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 3772 . . . 4  |-  ( ph  ->  ~P B  =  ~P ( Base `  K )
)
17 mpt2eq12 6101 . . . 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 643 . . 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 3772 . . . 4  |-  ( ph  ->  ~P B  =  ~P ( Base `  L )
)
21 mpt2eq12 6101 . . . 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 643 . . 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 2452 . 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 2412 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
26 eqid 2412 . . . 4  |-  ( +g  `  K )  =  ( +g  `  K )
27 eqid 2412 . . . 4  |-  ( LSSum `  K )  =  (
LSSum `  K )
2825, 26, 27lsmfval 15235 . . 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 2412 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
32 eqid 2412 . . . 4  |-  ( +g  `  L )  =  ( +g  `  L )
33 eqid 2412 . . . 4  |-  ( LSSum `  L )  =  (
LSSum `  L )
3431, 32, 33lsmfval 15235 . . 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 2454 1  |-  ( ph  ->  ( LSSum `  K )  =  ( LSSum `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924   ~Pcpw 3767   ran crn 4846   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   Basecbs 13432   +g cplusg 13492   LSSumclsm 15231
This theorem is referenced by:  hlhillsm  32454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-lsm 15233
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