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Theorem honegsubi 22372
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 9809 . . . . . . 7  |-  -u 1  e.  CC
3 hodseq.3 . . . . . . 7  |-  T : ~H
--> ~H
4 homulcl 22335 . . . . . . 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 22316 . . . . . 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 5626 . . . . . . 7  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
93ffvelrni 5626 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
10 hvsubval 21592 . . . . . . 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 22317 . . . . . . . 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 5836 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S `  x
)  +h  ( (
-u 1  .op  T
) `  x )
)  =  ( ( S `  x )  +h  ( -u 1  .h  ( T `  x
) ) ) )
1511, 14eqtr4d 2319 . . . . 5  |-  ( x  e.  ~H  ->  (
( S `  x
)  -h  ( T `
 x ) )  =  ( ( S `
 x )  +h  ( ( -u 1  .op  T ) `  x
) ) )
167, 15eqtr4d 2319 . . . 4  |-  ( x  e.  ~H  ->  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
17 hodval 22318 . . . . 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 2319 . . 3  |-  ( x  e.  ~H  ->  (
( S  +op  ( -u 1  .op  T ) ) `  x )  =  ( ( S  -op  T ) `  x ) )
2019rgen 2609 . 2  |-  A. x  e.  ~H  ( ( S 
+op  ( -u 1  .op  T ) ) `  x )  =  ( ( S  -op  T
) `  x )
211, 5hoaddcli 22344 . . 3  |-  ( S 
+op  ( -u 1  .op  T ) ) : ~H --> ~H
221, 3hosubcli 22345 . . 3  |-  ( S  -op  T ) : ~H --> ~H
2321, 22hoeqi 22337 . 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 1623    e. wcel 1685   A.wral 2544   -->wf 5217   ` cfv 5221  (class class class)co 5820   CCcc 8731   1c1 8734   -ucneg 9034   ~Hchil 21495    +h cva 21496    .h csm 21497    -h cmv 21501    +op chos 21514    .op chot 21515    -op chod 21516
This theorem is referenced by:  honegsub  22375  hosubeq0i  22402  lnophdi  22578  bdophdi  22673  nmoptri2i  22675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-hilex 21575  ax-hfvadd 21576  ax-hfvmul 21581
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-ltxr 8868  df-sub 9035  df-neg 9036  df-hvsub 21547  df-hosum 22306  df-homul 22307  df-hodif 22308
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