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Theorem lnopmul 23471
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 22507 . . . 4  |-  0h  e.  ~H
2 lnopl 23418 . . . 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 667 . . 3  |-  ( ( ( T  e.  LinOp  /\  A  e.  CC )  /\  B  e.  ~H )  ->  ( T `  ( ( A  .h  B )  +h  0h ) )  =  ( ( A  .h  ( T `  B )
)  +h  ( T `
 0h ) ) )
433impa 1149 . 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 22517 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
6 ax-hvaddid 22508 . . . . 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 976 . . 3  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  (
( A  .h  B
)  +h  0h )  =  ( A  .h  B ) )
98fveq2d 5733 . 2  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  ( T `  ( ( A  .h  B )  +h  0h ) )  =  ( T `  ( A  .h  B )
) )
10 lnop0 23470 . . . . 5  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  0h )
1110oveq2d 6098 . . . 4  |-  ( T  e.  LinOp  ->  ( ( A  .h  ( T `  B ) )  +h  ( T `  0h ) )  =  ( ( A  .h  ( T `  B )
)  +h  0h )
)
12113ad2ant1 979 . . 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 23363 . . . . . . . 8  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
1413ffvelrnda 5871 . . . . . . 7  |-  ( ( T  e.  LinOp  /\  B  e.  ~H )  ->  ( T `  B )  e.  ~H )
15 hvmulcl 22517 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( T `  B )  e.  ~H )  -> 
( A  .h  ( T `  B )
)  e.  ~H )
1614, 15sylan2 462 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T  e.  LinOp  /\  B  e.  ~H ) )  -> 
( A  .h  ( T `  B )
)  e.  ~H )
17163impb 1150 . . . . 5  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  B  e.  ~H )  ->  ( A  .h  ( T `  B ) )  e. 
~H )
18173com12 1158 . . . 4  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  ( A  .h  ( T `  B ) )  e. 
~H )
19 ax-hvaddid 22508 . . . 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 2469 . 2  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  (
( A  .h  ( T `  B )
)  +h  ( T `
 0h ) )  =  ( A  .h  ( T `  B ) ) )
224, 9, 213eqtr3d 2477 1  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  ( T `  ( A  .h  B ) )  =  ( A  .h  ( T `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5455  (class class class)co 6082   CCcc 8989   ~Hchil 22423    +h cva 22424    .h csm 22425   0hc0v 22428   LinOpclo 22451
This theorem is referenced by:  lnopmuli  23476  homco2  23481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-hilex 22503  ax-hfvadd 22504  ax-hvass 22506  ax-hv0cl 22507  ax-hvaddid 22508  ax-hfvmul 22509  ax-hvmulid 22510  ax-hvdistr2 22513  ax-hvmul0 22514
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-ltxr 9126  df-sub 9294  df-neg 9295  df-hvsub 22475  df-lnop 23345
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