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Theorem homco2 23441
Description: Move a scalar product out of a composition of operators. The operator  T must be linear, unlike homco1 23265 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 23205 . . . . . 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 5699 . . . 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 23223 . . . . . 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 5767 . . . . 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 5767 . . . . . . 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 6064 . . . . 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 23323 . . . . . . . . 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 5567 . . . . . . . 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 23205 . . . . . 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 5837 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
24 lnopmul 23431 . . . . . 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 2454 . . . 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 2454 . . 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 2757 . 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 5567 . . . 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 23223 . . . 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 23224 . . 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 1721   A.wral 2674    o. ccom 4849   -->wf 5417   ` cfv 5421  (class class class)co 6048   CCcc 8952   ~Hchil 22383    .h csm 22385    .op chot 22403   LinOpclo 22411
This theorem is referenced by:  opsqrlem1  23604
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-hilex 22463  ax-hfvadd 22464  ax-hvass 22466  ax-hv0cl 22467  ax-hvaddid 22468  ax-hfvmul 22469  ax-hvmulid 22470  ax-hvdistr2 22473  ax-hvmul0 22474
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-ltxr 9089  df-sub 9257  df-neg 9258  df-hvsub 22435  df-homul 23195  df-lnop 23305
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