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Theorem homco2 23328
Description: Move a scalar product out of a composition of operators. The operator  T must be linear, unlike homco1 23152 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 960 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  A  e.  CC )
2 simpl3 962 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  U : ~H --> ~H )
3 simpr 448 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  x  e.  ~H )
4 homval 23092 . . . . . 6  |-  ( ( A  e.  CC  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  U ) `  x )  =  ( A  .h  ( U `  x ) ) )
51, 2, 3, 4syl3anc 1184 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A  .op  U ) `  x )  =  ( A  .h  ( U `  x ) ) )
65fveq2d 5672 . . . 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 23110 . . . . . 6  |-  ( ( A  e.  CC  /\  U : ~H --> ~H )  ->  ( A  .op  U
) : ~H --> ~H )
873adant2 976 . . . . 5  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( A  .op  U
) : ~H --> ~H )
9 fvco3 5739 . . . . 5  |-  ( ( ( A  .op  U
) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  o.  ( A  .op  U ) ) `  x )  =  ( T `  ( ( A  .op  U ) `  x ) ) )
108, 9sylan 458 . . . 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 5739 . . . . . . 7  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  o.  U ) `  x
)  =  ( T `
 ( U `  x ) ) )
122, 3, 11syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( T  o.  U ) `  x
)  =  ( T `
 ( U `  x ) ) )
1312oveq2d 6036 . . . . 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 23210 . . . . . . . . 9  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
15143ad2ant2 979 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  ->  T : ~H --> ~H )
16 simp3 959 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  ->  U : ~H --> ~H )
17 fco 5540 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  o.  U
) : ~H --> ~H )
1815, 16, 17syl2anc 643 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( T  o.  U
) : ~H --> ~H )
1918adantr 452 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T  o.  U
) : ~H --> ~H )
20 homval 23092 . . . . . 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 1184 . . . . 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 961 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  T  e.  LinOp )
2316ffvelrnda 5809 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
24 lnopmul 23318 . . . . . 6  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  ( U `
 x )  e. 
~H )  ->  ( T `  ( A  .h  ( U `  x
) ) )  =  ( A  .h  ( T `  ( U `  x ) ) ) )
2522, 1, 23, 24syl3anc 1184 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T `  ( A  .h  ( U `  x ) ) )  =  ( A  .h  ( T `  ( U `
 x ) ) ) )
2613, 21, 253eqtr4d 2429 . . . 4  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A  .op  ( T  o.  U
) ) `  x
)  =  ( T `
 ( A  .h  ( U `  x ) ) ) )
276, 10, 263eqtr4d 2429 . . 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 ) )
2827ralrimiva 2732 . 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 ) )
29 fco 5540 . . . 4  |-  ( ( T : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )  ->  ( T  o.  ( A  .op  U ) ) : ~H --> ~H )
3015, 8, 29syl2anc 643 . . 3  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( T  o.  ( A  .op  U ) ) : ~H --> ~H )
31 simp1 957 . . . 4  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  ->  A  e.  CC )
32 homulcl 23110 . . . 4  |-  ( ( A  e.  CC  /\  ( T  o.  U
) : ~H --> ~H )  ->  ( A  .op  ( T  o.  U )
) : ~H --> ~H )
3331, 18, 32syl2anc 643 . . 3  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  -> 
( A  .op  ( T  o.  U )
) : ~H --> ~H )
34 hoeq 23111 . . 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 ) ) ) )
3530, 33, 34syl2anc 643 . 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 ) ) ) )
3628, 35mpbid 202 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649    o. ccom 4822   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   ~Hchil 22270    .h csm 22272    .op chot 22290   LinOpclo 22298
This theorem is referenced by:  opsqrlem1  23491
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-hilex 22350  ax-hfvadd 22351  ax-hvass 22353  ax-hv0cl 22354  ax-hvaddid 22355  ax-hfvmul 22356  ax-hvmulid 22357  ax-hvdistr2 22360  ax-hvmul0 22361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-sub 9225  df-neg 9226  df-hvsub 22322  df-homul 23082  df-lnop 23192
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