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Theorem honegsubi 22378
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 9815 . . . . . . 7  |-  -u 1  e.  CC
3 hodseq.3 . . . . . . 7  |-  T : ~H
--> ~H
4 homulcl 22341 . . . . . . 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 22322 . . . . . 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 5666 . . . . . . 7  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
93ffvelrni 5666 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
10 hvsubval 21598 . . . . . . 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 22323 . . . . . . . 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 5876 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S `  x
)  +h  ( (
-u 1  .op  T
) `  x )
)  =  ( ( S `  x )  +h  ( -u 1  .h  ( T `  x
) ) ) )
1511, 14eqtr4d 2320 . . . . 5  |-  ( x  e.  ~H  ->  (
( S `  x
)  -h  ( T `
 x ) )  =  ( ( S `
 x )  +h  ( ( -u 1  .op  T ) `  x
) ) )
167, 15eqtr4d 2320 . . . 4  |-  ( x  e.  ~H  ->  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
17 hodval 22324 . . . . 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 2320 . . 3  |-  ( x  e.  ~H  ->  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S  -op  T ) `  x ) )
2019rgen 2610 . 2  |-  A. x  e.  ~H  ( ( S 
+op  ( -u 1  .op  T ) ) `  x )  =  ( ( S  -op  T
) `  x )
211, 5hoaddcli 22350 . . 3  |-  ( S 
+op  ( -u 1  .op  T ) ) : ~H --> ~H
221, 3hosubcli 22351 . . 3  |-  ( S  -op  T ) : ~H --> ~H
2321, 22hoeqi 22343 . 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 1625    e. wcel 1686   A.wral 2545   -->wf 5253   ` cfv 5257  (class class class)co 5860   CCcc 8737   1c1 8740   -ucneg 9040   ~Hchil 21501    +h cva 21502    .h csm 21503    -h cmv 21507    +op chos 21520    .op chot 21521    -op chod 21522
This theorem is referenced by:  honegsub  22381  hosubeq0i  22408  lnophdi  22584  bdophdi  22679  nmoptri2i  22681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-hilex 21581  ax-hfvadd 21582  ax-hfvmul 21587
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-ltxr 8874  df-sub 9041  df-neg 9042  df-hvsub 21553  df-hosum 22312  df-homul 22313  df-hodif 22314
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