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Theorem lnopsubmuli 25330
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 24366 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
2 lnopl.1 . . . . . 6  |-  T  e. 
LinOp
32lnopsubi 25329 . . . . 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 1183 . . 3  |-  ( ( B  e.  ~H  /\  A  e.  CC  /\  C  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C
) ) )  =  ( ( T `  B )  -h  ( T `  ( A  .h  C ) ) ) )
653com12 1191 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C
) ) )  =  ( ( T `  B )  -h  ( T `  ( A  .h  C ) ) ) )
72lnopmuli 25327 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( T `  ( A  .h  C )
)  =  ( A  .h  ( T `  C ) ) )
87oveq2d 6102 . . 3  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( ( T `  B )  -h  ( T `  ( A  .h  C ) ) )  =  ( ( T `
 B )  -h  ( A  .h  ( T `  C )
) ) )
983adant2 1007 . 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 2470 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 965    = wceq 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086   CCcc 9272   ~Hchil 24272    .h csm 24274    -h cmv 24278   LinOpclo 24300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-hilex 24352  ax-hfvadd 24353  ax-hvass 24355  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359  ax-hvdistr2 24362  ax-hvmul0 24363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-ltxr 9415  df-sub 9589  df-neg 9590  df-hvsub 24324  df-lnop 25196
This theorem is referenced by:  lnopeq0lem1  25360  lnophmlem2  25372
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