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Theorem hocsubdiri 23133
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 23122 . . . . 5  |-  ( R  -op  S ) : ~H --> ~H
4 hods.3 . . . . 5  |-  T : ~H
--> ~H
53, 4hocoi 23117 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  -op  S )  o.  T ) `
 x )  =  ( ( R  -op  S ) `  ( T `
 x ) ) )
61, 4hocofi 23119 . . . . . 6  |-  ( R  o.  T ) : ~H --> ~H
72, 4hocofi 23119 . . . . . 6  |-  ( S  o.  T ) : ~H --> ~H
8 hodval 23095 . . . . . 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 5810 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
11 hodval 23095 . . . . . . . 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 23117 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( R  o.  T
) `  x )  =  ( R `  ( T `  x ) ) )
152, 4hocoi 23117 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( S  o.  T
) `  x )  =  ( S `  ( T `  x ) ) )
1614, 15oveq12d 6040 . . . . . 6  |-  ( x  e.  ~H  ->  (
( ( R  o.  T ) `  x
)  -h  ( ( S  o.  T ) `
 x ) )  =  ( ( R `
 ( T `  x ) )  -h  ( S `  ( T `  x )
) ) )
1713, 16eqtr4d 2424 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  -op  S
) `  ( T `  x ) )  =  ( ( ( R  o.  T ) `  x )  -h  (
( S  o.  T
) `  x )
) )
189, 17eqtr4d 2424 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  o.  T )  -op  ( S  o.  T )
) `  x )  =  ( ( R  -op  S ) `  ( T `  x ) ) )
195, 18eqtr4d 2424 . . 3  |-  ( x  e.  ~H  ->  (
( ( R  -op  S )  o.  T ) `
 x )  =  ( ( ( R  o.  T )  -op  ( S  o.  T
) ) `  x
) )
2019rgen 2716 . 2  |-  A. x  e.  ~H  ( ( ( R  -op  S )  o.  T ) `  x )  =  ( ( ( R  o.  T )  -op  ( S  o.  T )
) `  x )
213, 4hocofi 23119 . . 3  |-  ( ( R  -op  S )  o.  T ) : ~H --> ~H
226, 7hosubcli 23122 . . 3  |-  ( ( R  o.  T )  -op  ( S  o.  T ) ) : ~H --> ~H
2321, 22hoeqi 23114 . 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 1649    e. wcel 1717   A.wral 2651    o. ccom 4824   -->wf 5392   ` cfv 5396  (class class class)co 6022   ~Hchil 22272    -h cmv 22278    -op chod 22293
This theorem is referenced by:  hocsubdir  23138  unierri  23457  pjclem3  23550
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-hilex 22352  ax-hfvadd 22353  ax-hfvmul 22358
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-ltxr 9060  df-sub 9227  df-neg 9228  df-hvsub 22324  df-hodif 23085
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