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Theorem hosubcli 23264
Description: Mapping of difference of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoeq.1  |-  S : ~H
--> ~H
hoeq.2  |-  T : ~H
--> ~H
Assertion
Ref Expression
hosubcli  |-  ( S  -op  T ) : ~H --> ~H

Proof of Theorem hosubcli
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hoeq.1 . . 3  |-  S : ~H
--> ~H
2 hoeq.2 . . 3  |-  T : ~H
--> ~H
3 hodmval 23232 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  -op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
41, 2, 3mp2an 654 . 2  |-  ( S  -op  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x )
) )
51ffvelrni 5861 . . 3  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
62ffvelrni 5861 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
7 hvsubcl 22512 . . 3  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( ( S `  x )  -h  ( T `  x )
)  e.  ~H )
85, 6, 7syl2anc 643 . 2  |-  ( x  e.  ~H  ->  (
( S `  x
)  -h  ( T `
 x ) )  e.  ~H )
94, 8fmpti 5884 1  |-  ( S  -op  T ) : ~H --> ~H
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073   ~Hchil 22414    -h cmv 22420    -op chod 22435
This theorem is referenced by:  hosubfni  23266  hosubcl  23268  hodsi  23270  hocsubdiri  23275  hodseqi  23289  ho0subi  23290  honegsubi  23291  hoaddsubi  23316  hosd1i  23317  honpncani  23322  hoddii  23484  unierri  23599  pjddii  23651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-hilex 22494  ax-hfvadd 22495  ax-hfvmul 22500
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285  df-neg 9286  df-hvsub 22466  df-hodif 23227
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