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Theorem lnopmul 27179
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 26214 . . . 4  |-  0h  e.  ~H
2 lnopl 27126 . . . 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 682 . . 3  |-  ( ( ( T  e.  LinOp  /\  A  e.  CC )  /\  B  e.  ~H )  ->  ( T `  ( ( A  .h  B )  +h  0h ) )  =  ( ( A  .h  ( T `  B )
)  +h  ( T `
 0h ) ) )
433impa 1192 . 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 26224 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
6 ax-hvaddid 26215 . . . . 5  |-  ( ( A  .h  B )  e.  ~H  ->  (
( A  .h  B
)  +h  0h )  =  ( A  .h  B ) )
75, 6syl 17 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  .h  B )  +h  0h )  =  ( A  .h  B ) )
873adant1 1015 . . 3  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  (
( A  .h  B
)  +h  0h )  =  ( A  .h  B ) )
98fveq2d 5809 . 2  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  ( T `  ( ( A  .h  B )  +h  0h ) )  =  ( T `  ( A  .h  B )
) )
10 lnop0 27178 . . . . 5  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  0h )
1110oveq2d 6250 . . . 4  |-  ( T  e.  LinOp  ->  ( ( A  .h  ( T `  B ) )  +h  ( T `  0h ) )  =  ( ( A  .h  ( T `  B )
)  +h  0h )
)
12113ad2ant1 1018 . . 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 27071 . . . . . . . 8  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
1413ffvelrnda 5965 . . . . . . 7  |-  ( ( T  e.  LinOp  /\  B  e.  ~H )  ->  ( T `  B )  e.  ~H )
15 hvmulcl 26224 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( T `  B )  e.  ~H )  -> 
( A  .h  ( T `  B )
)  e.  ~H )
1614, 15sylan2 472 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T  e.  LinOp  /\  B  e.  ~H ) )  -> 
( A  .h  ( T `  B )
)  e.  ~H )
17163impb 1193 . . . . 5  |-  ( ( A  e.  CC  /\  T  e.  LinOp  /\  B  e.  ~H )  ->  ( A  .h  ( T `  B ) )  e. 
~H )
18173com12 1201 . . . 4  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  ( A  .h  ( T `  B ) )  e. 
~H )
19 ax-hvaddid 26215 . . . 4  |-  ( ( A  .h  ( T `
 B ) )  e.  ~H  ->  (
( A  .h  ( T `  B )
)  +h  0h )  =  ( A  .h  ( T `  B ) ) )
2018, 19syl 17 . . 3  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  (
( A  .h  ( T `  B )
)  +h  0h )  =  ( A  .h  ( T `  B ) ) )
2112, 20eqtrd 2443 . 2  |-  ( ( T  e.  LinOp  /\  A  e.  CC  /\  B  e. 
~H )  ->  (
( A  .h  ( T `  B )
)  +h  ( T `
 0h ) )  =  ( A  .h  ( T `  B ) ) )
224, 9, 213eqtr3d 2451 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5525  (class class class)co 6234   CCcc 9440   ~Hchil 26130    +h cva 26131    .h csm 26132   0hc0v 26135   LinOpclo 26158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-hilex 26210  ax-hfvadd 26211  ax-hvass 26213  ax-hv0cl 26214  ax-hvaddid 26215  ax-hfvmul 26216  ax-hvmulid 26217  ax-hvdistr2 26220  ax-hvmul0 26221
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-er 7268  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-ltxr 9583  df-sub 9763  df-neg 9764  df-hvsub 26182  df-lnop 27053
This theorem is referenced by:  lnopmuli  27184  homco2  27189
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