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Theorem hoddii 23492
Description: Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 23283 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoddi.1  |-  R  e. 
LinOp
hoddi.2  |-  S : ~H
--> ~H
hoddi.3  |-  T : ~H
--> ~H
Assertion
Ref Expression
hoddii  |-  ( R  o.  ( S  -op  T ) )  =  ( ( R  o.  S
)  -op  ( R  o.  T ) )

Proof of Theorem hoddii
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hoddi.2 . . . . . . 7  |-  S : ~H
--> ~H
21ffvelrni 5869 . . . . . 6  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
3 hoddi.3 . . . . . . 7  |-  T : ~H
--> ~H
43ffvelrni 5869 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
5 hoddi.1 . . . . . . 7  |-  R  e. 
LinOp
65lnopsubi 23477 . . . . . 6  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( R `  (
( S `  x
)  -h  ( T `
 x ) ) )  =  ( ( R `  ( S `
 x ) )  -h  ( R `  ( T `  x ) ) ) )
72, 4, 6syl2anc 643 . . . . 5  |-  ( x  e.  ~H  ->  ( R `  ( ( S `  x )  -h  ( T `  x
) ) )  =  ( ( R `  ( S `  x ) )  -h  ( R `
 ( T `  x ) ) ) )
8 hodval 23245 . . . . . . 7  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  -op  T ) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
91, 3, 8mp3an12 1269 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S  -op  T
) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
109fveq2d 5732 . . . . 5  |-  ( x  e.  ~H  ->  ( R `  ( ( S  -op  T ) `  x ) )  =  ( R `  (
( S `  x
)  -h  ( T `
 x ) ) ) )
115lnopfi 23472 . . . . . . 7  |-  R : ~H
--> ~H
1211, 1hocoi 23267 . . . . . 6  |-  ( x  e.  ~H  ->  (
( R  o.  S
) `  x )  =  ( R `  ( S `  x ) ) )
1311, 3hocoi 23267 . . . . . 6  |-  ( x  e.  ~H  ->  (
( R  o.  T
) `  x )  =  ( R `  ( T `  x ) ) )
1412, 13oveq12d 6099 . . . . 5  |-  ( x  e.  ~H  ->  (
( ( R  o.  S ) `  x
)  -h  ( ( R  o.  T ) `
 x ) )  =  ( ( R `
 ( S `  x ) )  -h  ( R `  ( T `  x )
) ) )
157, 10, 143eqtr4d 2478 . . . 4  |-  ( x  e.  ~H  ->  ( R `  ( ( S  -op  T ) `  x ) )  =  ( ( ( R  o.  S ) `  x )  -h  (
( R  o.  T
) `  x )
) )
161, 3hosubcli 23272 . . . . 5  |-  ( S  -op  T ) : ~H --> ~H
1711, 16hocoi 23267 . . . 4  |-  ( x  e.  ~H  ->  (
( R  o.  ( S  -op  T ) ) `
 x )  =  ( R `  (
( S  -op  T
) `  x )
) )
1811, 1hocofi 23269 . . . . 5  |-  ( R  o.  S ) : ~H --> ~H
1911, 3hocofi 23269 . . . . 5  |-  ( R  o.  T ) : ~H --> ~H
20 hodval 23245 . . . . 5  |-  ( ( ( R  o.  S
) : ~H --> ~H  /\  ( R  o.  T
) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( R  o.  S )  -op  ( R  o.  T
) ) `  x
)  =  ( ( ( R  o.  S
) `  x )  -h  ( ( R  o.  T ) `  x
) ) )
2118, 19, 20mp3an12 1269 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  o.  S )  -op  ( R  o.  T )
) `  x )  =  ( ( ( R  o.  S ) `
 x )  -h  ( ( R  o.  T ) `  x
) ) )
2215, 17, 213eqtr4d 2478 . . 3  |-  ( x  e.  ~H  ->  (
( R  o.  ( S  -op  T ) ) `
 x )  =  ( ( ( R  o.  S )  -op  ( R  o.  T
) ) `  x
) )
2322rgen 2771 . 2  |-  A. x  e.  ~H  ( ( R  o.  ( S  -op  T ) ) `  x
)  =  ( ( ( R  o.  S
)  -op  ( R  o.  T ) ) `  x )
2411, 16hocofi 23269 . . 3  |-  ( R  o.  ( S  -op  T ) ) : ~H --> ~H
2518, 19hosubcli 23272 . . 3  |-  ( ( R  o.  S )  -op  ( R  o.  T ) ) : ~H --> ~H
2624, 25hoeqi 23264 . 2  |-  ( A. x  e.  ~H  (
( R  o.  ( S  -op  T ) ) `
 x )  =  ( ( ( R  o.  S )  -op  ( R  o.  T
) ) `  x
)  <->  ( R  o.  ( S  -op  T ) )  =  ( ( R  o.  S )  -op  ( R  o.  T ) ) )
2723, 26mpbi 200 1  |-  ( R  o.  ( S  -op  T ) )  =  ( ( R  o.  S
)  -op  ( R  o.  T ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   A.wral 2705    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   ~Hchil 22422    -h cmv 22428    -op chod 22443   LinOpclo 22450
This theorem is referenced by:  hoddi  23493  unierri  23607
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-hodif 23235  df-lnop 23344
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