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Theorem homco2 22557
Description: Move a scalar product out of a composition of operators. The operator  T must be linear, unlike homco1 22381 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 22321 . . . . . 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 5529 . . . 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 22339 . . . . . 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 5596 . . . . 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 5596 . . . . . . 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 5874 . . . . 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 22439 . . . . . . . . 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 5398 . . . . . . . 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 22321 . . . . . 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 5663 . . . . . . 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 22547 . . . . . 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 2325 . . . 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 2325 . . 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 2626 . 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 5398 . . . 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 22339 . . . 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 22340 . . 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 1623    e. wcel 1684   A.wral 2543    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   ~Hchil 21499    .h csm 21501    .op chot 21519   LinOpclo 21527
This theorem is referenced by:  opsqrlem1  22720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-hilex 21579  ax-hfvadd 21580  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvdistr2 21589  ax-hvmul0 21590
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-hvsub 21551  df-homul 22311  df-lnop 22421
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