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Theorem lnopsubmuli 26766
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 25802 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
2 lnopl.1 . . . . . 6  |-  T  e. 
LinOp
32lnopsubi 26765 . . . . 5  |-  ( ( B  e.  ~H  /\  ( A  .h  C
)  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C ) ) )  =  ( ( T `
 B )  -h  ( T `  ( A  .h  C )
) ) )
41, 3sylan2 474 . . . 4  |-  ( ( B  e.  ~H  /\  ( A  e.  CC  /\  C  e.  ~H )
)  ->  ( T `  ( B  -h  ( A  .h  C )
) )  =  ( ( T `  B
)  -h  ( T `
 ( A  .h  C ) ) ) )
543impb 1193 . . 3  |-  ( ( B  e.  ~H  /\  A  e.  CC  /\  C  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C
) ) )  =  ( ( T `  B )  -h  ( T `  ( A  .h  C ) ) ) )
653com12 1201 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C
) ) )  =  ( ( T `  B )  -h  ( T `  ( A  .h  C ) ) ) )
72lnopmuli 26763 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( T `  ( A  .h  C )
)  =  ( A  .h  ( T `  C ) ) )
87oveq2d 6297 . . 3  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( ( T `  B )  -h  ( T `  ( A  .h  C ) ) )  =  ( ( T `
 B )  -h  ( A  .h  ( T `  C )
) ) )
983adant2 1016 . 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 2484 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   CCcc 9493   ~Hchil 25708    .h csm 25710    -h cmv 25714   LinOpclo 25736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-hilex 25788  ax-hfvadd 25789  ax-hvass 25791  ax-hv0cl 25792  ax-hvaddid 25793  ax-hfvmul 25794  ax-hvmulid 25795  ax-hvdistr2 25798  ax-hvmul0 25799
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-ltxr 9636  df-sub 9812  df-neg 9813  df-hvsub 25760  df-lnop 26632
This theorem is referenced by:  lnopeq0lem1  26796  lnophmlem2  26808
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