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Theorem homco1 22381
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 5596 . . . . . 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 5596 . . . . . . . 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 5874 . . . . . 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 5663 . . . . . . . . . 10  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
7 homval 22321 . . . . . . . . . 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 2318 . . . . 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 2318 . . . 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 5398 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  o.  U
) : ~H --> ~H )
15 homval 22321 . . . . . . . 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 2318 . . 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 2626 . 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 22339 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
23 fco 5398 . . . . 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 22339 . . . . 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 22340 . . 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 1623    e. wcel 1684   A.wral 2543    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   ~Hchil 21499    .h csm 21501    .op chot 21519
This theorem is referenced by:  opsqrlem1  22720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579  ax-hfvmul 21585
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-homul 22311
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