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Theorem hosubcli 25326
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 25294 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  -op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
41, 2, 3mp2an 672 . 2  |-  ( S  -op  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x )
) )
51ffvelrni 5952 . . 3  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
62ffvelrni 5952 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
7 hvsubcl 24572 . . 3  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( ( S `  x )  -h  ( T `  x )
)  e.  ~H )
85, 6, 7syl2anc 661 . 2  |-  ( x  e.  ~H  ->  (
( S `  x
)  -h  ( T `
 x ) )  e.  ~H )
94, 8fmpti 5976 1  |-  ( S  -op  T ) : ~H --> ~H
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758    |-> cmpt 4459   -->wf 5523   ` cfv 5527  (class class class)co 6201   ~Hchil 24474    -h cmv 24480    -op chod 24495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-hilex 24554  ax-hfvadd 24555  ax-hfvmul 24560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-ltxr 9535  df-sub 9709  df-neg 9710  df-hvsub 24526  df-hodif 25289
This theorem is referenced by:  hosubfni  25328  hosubcl  25330  hodsi  25332  hocsubdiri  25337  hodseqi  25351  ho0subi  25352  honegsubi  25353  hoaddsubi  25378  hosd1i  25379  honpncani  25384  hoddii  25546  unierri  25661  pjddii  25713
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