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Theorem homco1 22397
Description: Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
homco1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U )  =  ( A  .op  ( T  o.  U
) ) )

Proof of Theorem homco1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fvco3 5612 . . . . . 6  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( A 
.op  T )  o.  U ) `  x
)  =  ( ( A  .op  T ) `
 ( U `  x ) ) )
213ad2antl3 1119 . . . . 5  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T )  o.  U ) `  x )  =  ( ( A  .op  T
) `  ( U `  x ) ) )
3 fvco3 5612 . . . . . . . 8  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  o.  U ) `  x
)  =  ( T `
 ( U `  x ) ) )
433ad2antl3 1119 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( T  o.  U ) `  x )  =  ( T `  ( U `
 x ) ) )
54oveq2d 5890 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  o.  U ) `  x
) )  =  ( A  .h  ( T `
 ( U `  x ) ) ) )
6 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
7 homval 22337 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  ( U `  x )  e.  ~H )  -> 
( ( A  .op  T ) `  ( U `
 x ) )  =  ( A  .h  ( T `  ( U `
 x ) ) ) )
86, 7syl3an3 1217 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  ( U : ~H --> ~H  /\  x  e.  ~H )
)  ->  ( ( A  .op  T ) `  ( U `  x ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) )
983expa 1151 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( U : ~H --> ~H  /\  x  e.  ~H ) )  ->  (
( A  .op  T
) `  ( U `  x ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) )
109exp43 595 . . . . . . 7  |-  ( A  e.  CC  ->  ( T : ~H --> ~H  ->  ( U : ~H --> ~H  ->  ( x  e.  ~H  ->  ( ( A  .op  T
) `  ( U `  x ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) ) ) ) )
11103imp1 1164 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  ( U `  x ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) )
125, 11eqtr4d 2331 . . . . 5  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  o.  U ) `  x
) )  =  ( ( A  .op  T
) `  ( U `  x ) ) )
132, 12eqtr4d 2331 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T )  o.  U ) `  x )  =  ( A  .h  ( ( T  o.  U ) `
 x ) ) )
14 fco 5414 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  o.  U
) : ~H --> ~H )
15 homval 22337 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( T  o.  U
) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  ( T  o.  U
) ) `  x
)  =  ( A  .h  ( ( T  o.  U ) `  x ) ) )
1614, 15syl3an2 1216 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  ( T  o.  U ) ) `  x )  =  ( A  .h  ( ( T  o.  U ) `
 x ) ) )
17163expia 1153 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( x  e.  ~H  ->  ( ( A  .op  ( T  o.  U ) ) `  x )  =  ( A  .h  ( ( T  o.  U ) `
 x ) ) ) )
18173impb 1147 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( x  e.  ~H  ->  ( ( A  .op  ( T  o.  U
) ) `  x
)  =  ( A  .h  ( ( T  o.  U ) `  x ) ) ) )
1918imp 418 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  ( T  o.  U ) ) `  x )  =  ( A  .h  ( ( T  o.  U ) `
 x ) ) )
2013, 19eqtr4d 2331 . . 3  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T )  o.  U ) `  x )  =  ( ( A  .op  ( T  o.  U )
) `  x )
)
2120ralrimiva 2639 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  A. x  e.  ~H  ( ( ( A 
.op  T )  o.  U ) `  x
)  =  ( ( A  .op  ( T  o.  U ) ) `
 x ) )
22 homulcl 22355 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
23 fco 5414 . . . . 5  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U ) : ~H --> ~H )
2422, 23sylan 457 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U ) : ~H --> ~H )
25243impa 1146 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U ) : ~H --> ~H )
26 homulcl 22355 . . . . 5  |-  ( ( A  e.  CC  /\  ( T  o.  U
) : ~H --> ~H )  ->  ( A  .op  ( T  o.  U )
) : ~H --> ~H )
2714, 26sylan2 460 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  .op  ( T  o.  U
) ) : ~H --> ~H )
28273impb 1147 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  o.  U )
) : ~H --> ~H )
29 hoeq 22356 . . 3  |-  ( ( ( ( A  .op  T )  o.  U ) : ~H --> ~H  /\  ( A  .op  ( T  o.  U ) ) : ~H --> ~H )  ->  ( A. x  e. 
~H  ( ( ( A  .op  T )  o.  U ) `  x )  =  ( ( A  .op  ( T  o.  U )
) `  x )  <->  ( ( A  .op  T
)  o.  U )  =  ( A  .op  ( T  o.  U
) ) ) )
3025, 28, 29syl2anc 642 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A. x  e. 
~H  ( ( ( A  .op  T )  o.  U ) `  x )  =  ( ( A  .op  ( T  o.  U )
) `  x )  <->  ( ( A  .op  T
)  o.  U )  =  ( A  .op  ( T  o.  U
) ) ) )
3121, 30mpbid 201 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U )  =  ( A  .op  ( T  o.  U
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   ~Hchil 21515    .h csm 21517    .op chot 21535
This theorem is referenced by:  opsqrlem1  22736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-hilex 21595  ax-hfvmul 21601
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-homul 22327
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