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Theorem homco1 26836
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 5851 . . . . . 6  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( A 
.op  T )  o.  U ) `  x
)  =  ( ( A  .op  T ) `
 ( U `  x ) ) )
213ad2antl3 1158 . . . . 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 5851 . . . . . . . 8  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  o.  U ) `  x
)  =  ( T `
 ( U `  x ) ) )
433ad2antl3 1158 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( T  o.  U ) `  x )  =  ( T `  ( U `
 x ) ) )
54oveq2d 6212 . . . . . 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 5931 . . . . . . . . . 10  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
7 homval 26776 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  ( U `  x )  e.  ~H )  -> 
( ( A  .op  T ) `  ( U `
 x ) )  =  ( A  .h  ( T `  ( U `
 x ) ) ) )
86, 7syl3an3 1261 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  ( U : ~H --> ~H  /\  x  e.  ~H )
)  ->  ( ( A  .op  T ) `  ( U `  x ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) )
983expa 1194 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( U : ~H --> ~H  /\  x  e.  ~H ) )  ->  (
( A  .op  T
) `  ( U `  x ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) )
109exp43 610 . . . . . . 7  |-  ( A  e.  CC  ->  ( T : ~H --> ~H  ->  ( U : ~H --> ~H  ->  ( x  e.  ~H  ->  ( ( A  .op  T
) `  ( U `  x ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) ) ) ) )
11103imp1 1207 . . . . . 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 2426 . . . . 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 2426 . . . 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 5649 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  o.  U
) : ~H --> ~H )
15 homval 26776 . . . . . . . 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 1260 . . . . . . 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 1196 . . . . . 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 1190 . . . . 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 427 . . . 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 2426 . . 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 2796 . 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 26794 . . . 4  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
23 fco 5649 . . . 4  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U ) : ~H --> ~H )
2422, 23stoic3 1617 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U ) : ~H --> ~H )
25 homulcl 26794 . . . . 5  |-  ( ( A  e.  CC  /\  ( T  o.  U
) : ~H --> ~H )  ->  ( A  .op  ( T  o.  U )
) : ~H --> ~H )
2614, 25sylan2 472 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  .op  ( T  o.  U
) ) : ~H --> ~H )
27263impb 1190 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  o.  U )
) : ~H --> ~H )
28 hoeq 26795 . . 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
) ) ) )
2924, 27, 28syl2anc 659 . 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
) ) ) )
3021, 29mpbid 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732    o. ccom 4917   -->wf 5492   ` cfv 5496  (class class class)co 6196   CCcc 9401   ~Hchil 25953    .h csm 25955    .op chot 25973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-hilex 26033  ax-hfvmul 26039
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-map 7340  df-homul 26766
This theorem is referenced by:  opsqrlem1  27175
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