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Theorem lnopmul 26548
Description: Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnopmul  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  ( T `  ( A  .h  B ) )  =  ( A  .h  ( T `  B )
) )

Proof of Theorem lnopmul
StepHypRef Expression
1 ax-hv0cl 25582 . . . 4  |-  0h  e.  ~H
2 lnopl 26495 . . . 4  |-  ( ( ( T  e.  LinOp  /\  A  e.  CC )  /\  ( B  e. 
~H  /\  0h  e.  ~H ) )  ->  ( T `  ( ( A  .h  B )  +h  0h ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `
 0h ) ) )
31, 2mpanr2 684 . . 3  |-  ( ( ( T  e.  LinOp  /\  A  e.  CC )  /\  B  e.  ~H )  ->  ( T `  ( ( A  .h  B )  +h  0h ) )  =  ( ( A  .h  ( T `  B )
)  +h  ( T `
 0h ) ) )
433impa 1186 . 2  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  ( T `  ( ( A  .h  B )  +h  0h ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `
 0h ) ) )
5 hvmulcl 25592 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
6 ax-hvaddid 25583 . . . . 5  |-  ( ( A  .h  B )  e.  ~H  ->  (
( A  .h  B
)  +h  0h )  =  ( A  .h  B ) )
75, 6syl 16 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  .h  B )  +h  0h )  =  ( A  .h  B ) )
873adant1 1009 . . 3  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  (
( A  .h  B
)  +h  0h )  =  ( A  .h  B ) )
98fveq2d 5861 . 2  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  ( T `  ( ( A  .h  B )  +h  0h ) )  =  ( T `  ( A  .h  B )
) )
10 lnop0 26547 . . . . 5  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  0h )
1110oveq2d 6291 . . . 4  |-  ( T  e.  LinOp  ->  ( ( A  .h  ( T `  B ) )  +h  ( T `  0h ) )  =  ( ( A  .h  ( T `  B )
)  +h  0h )
)
12113ad2ant1 1012 . . 3  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  (
( A  .h  ( T `  B )
)  +h  ( T `
 0h ) )  =  ( ( A  .h  ( T `  B ) )  +h 
0h ) )
13 lnopf 26440 . . . . . . . 8  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
1413ffvelrnda 6012 . . . . . . 7  |-  ( ( T  e.  LinOp  /\  B  e.  ~H )  ->  ( T `  B )  e.  ~H )
15 hvmulcl 25592 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( T `  B )  e.  ~H )  -> 
( A  .h  ( T `  B )
)  e.  ~H )
1614, 15sylan2 474 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T  e.  LinOp  /\  B  e.  ~H ) )  -> 
( A  .h  ( T `  B )
)  e.  ~H )
17163impb 1187 . . . . 5  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  B  e.  ~H )  ->  ( A  .h  ( T `  B ) )  e. 
~H )
18173com12 1195 . . . 4  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  ( A  .h  ( T `  B ) )  e. 
~H )
19 ax-hvaddid 25583 . . . 4  |-  ( ( A  .h  ( T `
 B ) )  e.  ~H  ->  (
( A  .h  ( T `  B )
)  +h  0h )  =  ( A  .h  ( T `  B ) ) )
2018, 19syl 16 . . 3  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  (
( A  .h  ( T `  B )
)  +h  0h )  =  ( A  .h  ( T `  B ) ) )
2112, 20eqtrd 2501 . 2  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  (
( A  .h  ( T `  B )
)  +h  ( T `
 0h ) )  =  ( A  .h  ( T `  B ) ) )
224, 9, 213eqtr3d 2509 1  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  ( T `  ( A  .h  B ) )  =  ( A  .h  ( T `  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   CCcc 9479   ~Hchil 25498    +h cva 25499    .h csm 25500   0hc0v 25503   LinOpclo 25526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-hilex 25578  ax-hfvadd 25579  ax-hvass 25581  ax-hv0cl 25582  ax-hvaddid 25583  ax-hfvmul 25584  ax-hvmulid 25585  ax-hvdistr2 25588  ax-hvmul0 25589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9796  df-neg 9797  df-hvsub 25550  df-lnop 26422
This theorem is referenced by:  lnopmuli  26553  homco2  26558
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