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Theorem homco2 23480
Description: Move a scalar product out of a composition of operators. The operator  T must be linear, unlike homco1 23304 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 23244 . . . . . 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 5732 . . . 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 23262 . . . . . 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 5800 . . . . 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 5800 . . . . . . 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 6097 . . . . 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 23362 . . . . . . . . 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 5600 . . . . . . . 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 23244 . . . . . 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 5870 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
24 lnopmul 23470 . . . . . 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 2478 . . . 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 2478 . . 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 2789 . 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 5600 . . . 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 23262 . . . 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 23263 . . 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 1652    e. wcel 1725   A.wral 2705    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   ~Hchil 22422    .h csm 22424    .op chot 22442   LinOpclo 22450
This theorem is referenced by:  opsqrlem1  23643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-hilex 22502  ax-hfvadd 22503  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvdistr2 22512  ax-hvmul0 22513
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-hvsub 22474  df-homul 23234  df-lnop 23344
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