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Theorem hosubcli 23113
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 23081 . . 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 5801 . . 3  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
62ffvelrni 5801 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
7 hvsubcl 22361 . . 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 5824 1  |-  ( S  -op  T ) : ~H --> ~H
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717    e. cmpt 4200   -->wf 5383   ` cfv 5387  (class class class)co 6013   ~Hchil 22263    -h cmv 22269    -op chod 22284
This theorem is referenced by:  hosubfni  23115  hosubcl  23117  hodsi  23119  hocsubdiri  23124  hodseqi  23138  ho0subi  23139  honegsubi  23140  hoaddsubi  23165  hosd1i  23166  honpncani  23171  hoddii  23333  unierri  23448  pjddii  23500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-hilex 22343  ax-hfvadd 22344  ax-hfvmul 22349
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-ltxr 9051  df-sub 9218  df-neg 9219  df-hvsub 22315  df-hodif 23076
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