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Theorem homco2 22573
Description: Move a scalar product out of a composition of operators. The operator  T must be linear, unlike homco1 22397 that works for any operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
homco2  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( T  o.  ( A  .op  U ) )  =  ( A  .op  ( T  o.  U
) ) )

Proof of Theorem homco2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl1 958 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  A  e.  CC )
2 simpl3 960 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  U : ~H --> ~H )
3 simpr 447 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  x  e.  ~H )
4 homval 22337 . . . . . 6  |-  ( ( A  e.  CC  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  U ) `  x )  =  ( A  .h  ( U `  x ) ) )
51, 2, 3, 4syl3anc 1182 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A  .op  U ) `  x )  =  ( A  .h  ( U `  x ) ) )
65fveq2d 5545 . . . 4  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T `  (
( A  .op  U
) `  x )
)  =  ( T `
 ( A  .h  ( U `  x ) ) ) )
7 homulcl 22355 . . . . . 6  |-  ( ( A  e.  CC  /\  U : ~H --> ~H )  ->  ( A  .op  U
) : ~H --> ~H )
873adant2 974 . . . . 5  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( A  .op  U
) : ~H --> ~H )
9 fvco3 5612 . . . . 5  |-  ( ( ( A  .op  U
) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  o.  ( A  .op  U ) ) `  x )  =  ( T `  ( ( A  .op  U ) `  x ) ) )
108, 9sylan 457 . . . 4  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( T  o.  ( A  .op  U ) ) `  x )  =  ( T `  ( ( A  .op  U ) `  x ) ) )
11 fvco3 5612 . . . . . . 7  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  o.  U ) `  x
)  =  ( T `
 ( U `  x ) ) )
122, 3, 11syl2anc 642 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( T  o.  U ) `  x
)  =  ( T `
 ( U `  x ) ) )
1312oveq2d 5890 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  (
( T  o.  U
) `  x )
)  =  ( A  .h  ( T `  ( U `  x ) ) ) )
14 lnopf 22455 . . . . . . . . 9  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
15143ad2ant2 977 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  ->  T : ~H --> ~H )
16 simp3 957 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  ->  U : ~H --> ~H )
17 fco 5414 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  o.  U
) : ~H --> ~H )
1815, 16, 17syl2anc 642 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( T  o.  U
) : ~H --> ~H )
1918adantr 451 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T  o.  U
) : ~H --> ~H )
20 homval 22337 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T  o.  U
) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  ( T  o.  U
) ) `  x
)  =  ( A  .h  ( ( T  o.  U ) `  x ) ) )
211, 19, 3, 20syl3anc 1182 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A  .op  ( T  o.  U
) ) `  x
)  =  ( A  .h  ( ( T  o.  U ) `  x ) ) )
22 simpl2 959 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  T  e.  LinOp )
23 ffvelrn 5679 . . . . . . 7  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
2416, 23sylan 457 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
25 lnopmul 22563 . . . . . 6  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  ( U `
 x )  e. 
~H )  ->  ( T `  ( A  .h  ( U `  x
) ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) )
2622, 1, 24, 25syl3anc 1182 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T `  ( A  .h  ( U `  x ) ) )  =  ( A  .h  ( T `  ( U `
 x ) ) ) )
2713, 21, 263eqtr4d 2338 . . . 4  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A  .op  ( T  o.  U
) ) `  x
)  =  ( T `
 ( A  .h  ( U `  x ) ) ) )
286, 10, 273eqtr4d 2338 . . 3  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( T  o.  ( A  .op  U ) ) `  x )  =  ( ( A 
.op  ( T  o.  U ) ) `  x ) )
2928ralrimiva 2639 . 2  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  ->  A. x  e.  ~H  ( ( T  o.  ( A  .op  U ) ) `  x )  =  ( ( A 
.op  ( T  o.  U ) ) `  x ) )
30 fco 5414 . . . 4  |-  ( ( T : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )  ->  ( T  o.  ( A  .op  U ) ) : ~H --> ~H )
3115, 8, 30syl2anc 642 . . 3  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( T  o.  ( A  .op  U ) ) : ~H --> ~H )
32 simp1 955 . . . 4  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  ->  A  e.  CC )
33 homulcl 22355 . . . 4  |-  ( ( A  e.  CC  /\  ( T  o.  U
) : ~H --> ~H )  ->  ( A  .op  ( T  o.  U )
) : ~H --> ~H )
3432, 18, 33syl2anc 642 . . 3  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( A  .op  ( T  o.  U )
) : ~H --> ~H )
35 hoeq 22356 . . 3  |-  ( ( ( T  o.  ( A  .op  U ) ) : ~H --> ~H  /\  ( A  .op  ( T  o.  U ) ) : ~H --> ~H )  ->  ( A. x  e. 
~H  ( ( T  o.  ( A  .op  U ) ) `  x
)  =  ( ( A  .op  ( T  o.  U ) ) `
 x )  <->  ( T  o.  ( A  .op  U
) )  =  ( A  .op  ( T  o.  U ) ) ) )
3631, 34, 35syl2anc 642 . 2  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( A. x  e. 
~H  ( ( T  o.  ( A  .op  U ) ) `  x
)  =  ( ( A  .op  ( T  o.  U ) ) `
 x )  <->  ( T  o.  ( A  .op  U
) )  =  ( A  .op  ( T  o.  U ) ) ) )
3729, 36mpbid 201 1  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( T  o.  ( A  .op  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   LinOpclo 21543
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-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-hilex 21595  ax-hfvadd 21596  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvdistr2 21605  ax-hvmul0 21606
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  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-po 4330  df-so 4331  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-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-hvsub 21567  df-homul 22327  df-lnop 22437
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