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Theorem hocsubdiri 23273
Description: Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hods.1  |-  R : ~H
--> ~H
hods.2  |-  S : ~H
--> ~H
hods.3  |-  T : ~H
--> ~H
Assertion
Ref Expression
hocsubdiri  |-  ( ( R  -op  S )  o.  T )  =  ( ( R  o.  T )  -op  ( S  o.  T )
)

Proof of Theorem hocsubdiri
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hods.1 . . . . . 6  |-  R : ~H
--> ~H
2 hods.2 . . . . . 6  |-  S : ~H
--> ~H
31, 2hosubcli 23262 . . . . 5  |-  ( R  -op  S ) : ~H --> ~H
4 hods.3 . . . . 5  |-  T : ~H
--> ~H
53, 4hocoi 23257 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  -op  S )  o.  T ) `
 x )  =  ( ( R  -op  S ) `  ( T `
 x ) ) )
61, 4hocofi 23259 . . . . . 6  |-  ( R  o.  T ) : ~H --> ~H
72, 4hocofi 23259 . . . . . 6  |-  ( S  o.  T ) : ~H --> ~H
8 hodval 23235 . . . . . 6  |-  ( ( ( R  o.  T
) : ~H --> ~H  /\  ( S  o.  T
) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( R  o.  T )  -op  ( S  o.  T
) ) `  x
)  =  ( ( ( R  o.  T
) `  x )  -h  ( ( S  o.  T ) `  x
) ) )
96, 7, 8mp3an12 1269 . . . . 5  |-  ( x  e.  ~H  ->  (
( ( R  o.  T )  -op  ( S  o.  T )
) `  x )  =  ( ( ( R  o.  T ) `
 x )  -h  ( ( S  o.  T ) `  x
) ) )
104ffvelrni 5861 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
11 hodval 23235 . . . . . . . 8  |-  ( ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  ( T `  x )  e.  ~H )  -> 
( ( R  -op  S ) `  ( T `
 x ) )  =  ( ( R `
 ( T `  x ) )  -h  ( S `  ( T `  x )
) ) )
121, 2, 11mp3an12 1269 . . . . . . 7  |-  ( ( T `  x )  e.  ~H  ->  (
( R  -op  S
) `  ( T `  x ) )  =  ( ( R `  ( T `  x ) )  -h  ( S `
 ( T `  x ) ) ) )
1310, 12syl 16 . . . . . 6  |-  ( x  e.  ~H  ->  (
( R  -op  S
) `  ( T `  x ) )  =  ( ( R `  ( T `  x ) )  -h  ( S `
 ( T `  x ) ) ) )
141, 4hocoi 23257 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( R  o.  T
) `  x )  =  ( R `  ( T `  x ) ) )
152, 4hocoi 23257 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( S  o.  T
) `  x )  =  ( S `  ( T `  x ) ) )
1614, 15oveq12d 6091 . . . . . 6  |-  ( x  e.  ~H  ->  (
( ( R  o.  T ) `  x
)  -h  ( ( S  o.  T ) `
 x ) )  =  ( ( R `
 ( T `  x ) )  -h  ( S `  ( T `  x )
) ) )
1713, 16eqtr4d 2470 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  -op  S
) `  ( T `  x ) )  =  ( ( ( R  o.  T ) `  x )  -h  (
( S  o.  T
) `  x )
) )
189, 17eqtr4d 2470 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  o.  T )  -op  ( S  o.  T )
) `  x )  =  ( ( R  -op  S ) `  ( T `  x ) ) )
195, 18eqtr4d 2470 . . 3  |-  ( x  e.  ~H  ->  (
( ( R  -op  S )  o.  T ) `
 x )  =  ( ( ( R  o.  T )  -op  ( S  o.  T
) ) `  x
) )
2019rgen 2763 . 2  |-  A. x  e.  ~H  ( ( ( R  -op  S )  o.  T ) `  x )  =  ( ( ( R  o.  T )  -op  ( S  o.  T )
) `  x )
213, 4hocofi 23259 . . 3  |-  ( ( R  -op  S )  o.  T ) : ~H --> ~H
226, 7hosubcli 23262 . . 3  |-  ( ( R  o.  T )  -op  ( S  o.  T ) ) : ~H --> ~H
2321, 22hoeqi 23254 . 2  |-  ( A. x  e.  ~H  (
( ( R  -op  S )  o.  T ) `
 x )  =  ( ( ( R  o.  T )  -op  ( S  o.  T
) ) `  x
)  <->  ( ( R  -op  S )  o.  T )  =  ( ( R  o.  T
)  -op  ( S  o.  T ) ) )
2420, 23mpbi 200 1  |-  ( ( R  -op  S )  o.  T )  =  ( ( R  o.  T )  -op  ( S  o.  T )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   A.wral 2697    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073   ~Hchil 22412    -h cmv 22418    -op chod 22433
This theorem is referenced by:  hocsubdir  23278  unierri  23597  pjclem3  23690
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-hilex 22492  ax-hfvadd 22493  ax-hfvmul 22498
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-ltxr 9115  df-sub 9283  df-neg 9284  df-hvsub 22464  df-hodif 23225
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