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Theorem homco1 26396
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 5942 . . . . . 6  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( A 
.op  T )  o.  U ) `  x
)  =  ( ( A  .op  T ) `
 ( U `  x ) ) )
213ad2antl3 1160 . . . . 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 5942 . . . . . . . 8  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  o.  U ) `  x
)  =  ( T `
 ( U `  x ) ) )
433ad2antl3 1160 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( T  o.  U ) `  x )  =  ( T `  ( U `
 x ) ) )
54oveq2d 6298 . . . . . 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 6017 . . . . . . . . . 10  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
7 homval 26336 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  ( U `  x )  e.  ~H )  -> 
( ( A  .op  T ) `  ( U `
 x ) )  =  ( A  .h  ( T `  ( U `
 x ) ) ) )
86, 7syl3an3 1263 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  ( U : ~H --> ~H  /\  x  e.  ~H )
)  ->  ( ( A  .op  T ) `  ( U `  x ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) )
983expa 1196 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( U : ~H --> ~H  /\  x  e.  ~H ) )  ->  (
( A  .op  T
) `  ( U `  x ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) )
109exp43 612 . . . . . . 7  |-  ( A  e.  CC  ->  ( T : ~H --> ~H  ->  ( U : ~H --> ~H  ->  ( x  e.  ~H  ->  ( ( A  .op  T
) `  ( U `  x ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) ) ) ) )
11103imp1 1209 . . . . . 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 2511 . . . . 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 2511 . . . 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 5739 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  o.  U
) : ~H --> ~H )
15 homval 26336 . . . . . . . 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 1262 . . . . . . 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 1198 . . . . . 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 1192 . . . . 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 429 . . . 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 2511 . . 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 2878 . 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 26354 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
23 fco 5739 . . . . 5  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U ) : ~H --> ~H )
2422, 23sylan 471 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U ) : ~H --> ~H )
25243impa 1191 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U ) : ~H --> ~H )
26 homulcl 26354 . . . . 5  |-  ( ( A  e.  CC  /\  ( T  o.  U
) : ~H --> ~H )  ->  ( A  .op  ( T  o.  U )
) : ~H --> ~H )
2714, 26sylan2 474 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  .op  ( T  o.  U
) ) : ~H --> ~H )
28273impb 1192 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  o.  U )
) : ~H --> ~H )
29 hoeq 26355 . . 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 661 . 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 210 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U )  =  ( A  .op  ( T  o.  U
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    o. ccom 5003   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   ~Hchil 25512    .h csm 25514    .op chot 25532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-hilex 25592  ax-hfvmul 25598
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-homul 26326
This theorem is referenced by:  opsqrlem1  26735
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