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Theorem lnopsubmuli 23328
Description: Subtraction/product property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnopl.1  |-  T  e. 
LinOp
Assertion
Ref Expression
lnopsubmuli  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C
) ) )  =  ( ( T `  B )  -h  ( A  .h  ( T `  C ) ) ) )

Proof of Theorem lnopsubmuli
StepHypRef Expression
1 hvmulcl 22366 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
2 lnopl.1 . . . . . 6  |-  T  e. 
LinOp
32lnopsubi 23327 . . . . 5  |-  ( ( B  e.  ~H  /\  ( A  .h  C
)  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C ) ) )  =  ( ( T `
 B )  -h  ( T `  ( A  .h  C )
) ) )
41, 3sylan2 461 . . . 4  |-  ( ( B  e.  ~H  /\  ( A  e.  CC  /\  C  e.  ~H )
)  ->  ( T `  ( B  -h  ( A  .h  C )
) )  =  ( ( T `  B
)  -h  ( T `
 ( A  .h  C ) ) ) )
543impb 1149 . . 3  |-  ( ( B  e.  ~H  /\  A  e.  CC  /\  C  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C
) ) )  =  ( ( T `  B )  -h  ( T `  ( A  .h  C ) ) ) )
653com12 1157 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C
) ) )  =  ( ( T `  B )  -h  ( T `  ( A  .h  C ) ) ) )
72lnopmuli 23325 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( T `  ( A  .h  C )
)  =  ( A  .h  ( T `  C ) ) )
87oveq2d 6038 . . 3  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( ( T `  B )  -h  ( T `  ( A  .h  C ) ) )  =  ( ( T `
 B )  -h  ( A  .h  ( T `  C )
) ) )
983adant2 976 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( T `  B
)  -h  ( T `
 ( A  .h  C ) ) )  =  ( ( T `
 B )  -h  ( A  .h  ( T `  C )
) ) )
106, 9eqtrd 2421 1  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C
) ) )  =  ( ( T `  B )  -h  ( A  .h  ( T `  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5396  (class class class)co 6022   CCcc 8923   ~Hchil 22272    .h csm 22274    -h cmv 22278   LinOpclo 22300
This theorem is referenced by:  lnopeq0lem1  23358  lnophmlem2  23370
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-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-hvass 22355  ax-hv0cl 22356  ax-hvaddid 22357  ax-hfvmul 22358  ax-hvmulid 22359  ax-hvdistr2 22362  ax-hvmul0 22363
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-lnop 23194
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