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Theorem honegsubi 22392
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 9829 . . . . . . 7  |-  -u 1  e.  CC
3 hodseq.3 . . . . . . 7  |-  T : ~H
--> ~H
4 homulcl 22355 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  T : ~H --> ~H )  ->  ( -u 1  .op 
T ) : ~H --> ~H )
52, 3, 4mp2an 653 . . . . . 6  |-  ( -u
1  .op  T ) : ~H --> ~H
6 hosval 22336 . . . . . 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 1267 . . . . 5  |-  ( x  e.  ~H  ->  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S `
 x )  +h  ( ( -u 1  .op  T ) `  x
) ) )
81ffvelrni 5680 . . . . . . 7  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
93ffvelrni 5680 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
10 hvsubval 21612 . . . . . . 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 642 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S `  x
)  -h  ( T `
 x ) )  =  ( ( S `
 x )  +h  ( -u 1  .h  ( T `  x
) ) ) )
12 homval 22337 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( -u 1  .op  T ) `  x
)  =  ( -u
1  .h  ( T `
 x ) ) )
132, 3, 12mp3an12 1267 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( -u 1  .op  T
) `  x )  =  ( -u 1  .h  ( T `  x
) ) )
1413oveq2d 5890 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S `  x
)  +h  ( (
-u 1  .op  T
) `  x )
)  =  ( ( S `  x )  +h  ( -u 1  .h  ( T `  x
) ) ) )
1511, 14eqtr4d 2331 . . . . 5  |-  ( x  e.  ~H  ->  (
( S `  x
)  -h  ( T `
 x ) )  =  ( ( S `
 x )  +h  ( ( -u 1  .op  T ) `  x
) ) )
167, 15eqtr4d 2331 . . . 4  |-  ( x  e.  ~H  ->  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
17 hodval 22338 . . . . 5  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  -op  T ) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
181, 3, 17mp3an12 1267 . . . 4  |-  ( x  e.  ~H  ->  (
( S  -op  T
) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
1916, 18eqtr4d 2331 . . 3  |-  ( x  e.  ~H  ->  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S  -op  T ) `  x ) )
2019rgen 2621 . 2  |-  A. x  e.  ~H  ( ( S 
+op  ( -u 1  .op  T ) ) `  x )  =  ( ( S  -op  T
) `  x )
211, 5hoaddcli 22364 . . 3  |-  ( S 
+op  ( -u 1  .op  T ) ) : ~H --> ~H
221, 3hosubcli 22365 . . 3  |-  ( S  -op  T ) : ~H --> ~H
2321, 22hoeqi 22357 . 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 199 1  |-  ( S 
+op  ( -u 1  .op  T ) )  =  ( S  -op  T
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   A.wral 2556   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754   -ucneg 9054   ~Hchil 21515    +h cva 21516    .h csm 21517    -h cmv 21521    +op chos 21534    .op chot 21535    -op chod 21536
This theorem is referenced by:  honegsub  22395  hosubeq0i  22422  lnophdi  22598  bdophdi  22693  nmoptri2i  22695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-hilex 21595  ax-hfvadd 21596  ax-hfvmul 21601
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-hvsub 21567  df-hosum 22326  df-homul 22327  df-hodif 22328
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