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Theorem honegsubi 23147
Description: Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
hodseq.2  |-  S : ~H
--> ~H
hodseq.3  |-  T : ~H
--> ~H
Assertion
Ref Expression
honegsubi  |-  ( S 
+op  ( -u 1  .op  T ) )  =  ( S  -op  T
)

Proof of Theorem honegsubi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hodseq.2 . . . . . 6  |-  S : ~H
--> ~H
2 neg1cn 9999 . . . . . . 7  |-  -u 1  e.  CC
3 hodseq.3 . . . . . . 7  |-  T : ~H
--> ~H
4 homulcl 23110 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  T : ~H --> ~H )  ->  ( -u 1  .op 
T ) : ~H --> ~H )
52, 3, 4mp2an 654 . . . . . 6  |-  ( -u
1  .op  T ) : ~H --> ~H
6 hosval 23091 . . . . . 6  |-  ( ( S : ~H --> ~H  /\  ( -u 1  .op  T
) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  +op  ( -u 1  .op  T
) ) `  x
)  =  ( ( S `  x )  +h  ( ( -u
1  .op  T ) `  x ) ) )
71, 5, 6mp3an12 1269 . . . . 5  |-  ( x  e.  ~H  ->  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S `
 x )  +h  ( ( -u 1  .op  T ) `  x
) ) )
81ffvelrni 5808 . . . . . . 7  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
93ffvelrni 5808 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
10 hvsubval 22367 . . . . . . 7  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( ( S `  x )  -h  ( T `  x )
)  =  ( ( S `  x )  +h  ( -u 1  .h  ( T `  x
) ) ) )
118, 9, 10syl2anc 643 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S `  x
)  -h  ( T `
 x ) )  =  ( ( S `
 x )  +h  ( -u 1  .h  ( T `  x
) ) ) )
12 homval 23092 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( -u 1  .op  T ) `  x
)  =  ( -u
1  .h  ( T `
 x ) ) )
132, 3, 12mp3an12 1269 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( -u 1  .op  T
) `  x )  =  ( -u 1  .h  ( T `  x
) ) )
1413oveq2d 6036 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S `  x
)  +h  ( (
-u 1  .op  T
) `  x )
)  =  ( ( S `  x )  +h  ( -u 1  .h  ( T `  x
) ) ) )
1511, 14eqtr4d 2422 . . . . 5  |-  ( x  e.  ~H  ->  (
( S `  x
)  -h  ( T `
 x ) )  =  ( ( S `
 x )  +h  ( ( -u 1  .op  T ) `  x
) ) )
167, 15eqtr4d 2422 . . . 4  |-  ( x  e.  ~H  ->  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
17 hodval 23093 . . . . 5  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  -op  T ) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
181, 3, 17mp3an12 1269 . . . 4  |-  ( x  e.  ~H  ->  (
( S  -op  T
) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
1916, 18eqtr4d 2422 . . 3  |-  ( x  e.  ~H  ->  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S  -op  T ) `  x ) )
2019rgen 2714 . 2  |-  A. x  e.  ~H  ( ( S 
+op  ( -u 1  .op  T ) ) `  x )  =  ( ( S  -op  T
) `  x )
211, 5hoaddcli 23119 . . 3  |-  ( S 
+op  ( -u 1  .op  T ) ) : ~H --> ~H
221, 3hosubcli 23120 . . 3  |-  ( S  -op  T ) : ~H --> ~H
2321, 22hoeqi 23112 . 2  |-  ( A. x  e.  ~H  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S  -op  T ) `  x )  <->  ( S  +op  ( -u 1  .op 
T ) )  =  ( S  -op  T
) )
2420, 23mpbi 200 1  |-  ( S 
+op  ( -u 1  .op  T ) )  =  ( S  -op  T
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   A.wral 2649   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   1c1 8924   -ucneg 9224   ~Hchil 22270    +h cva 22271    .h csm 22272    -h cmv 22276    +op chos 22289    .op chot 22290    -op chod 22291
This theorem is referenced by:  honegsub  23150  hosubeq0i  23177  lnophdi  23353  bdophdi  23448  nmoptri2i  23450
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-hilex 22350  ax-hfvadd 22351  ax-hfvmul 22356
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-sub 9225  df-neg 9226  df-hvsub 22322  df-hosum 23081  df-homul 23082  df-hodif 23083
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