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Theorem counop 22501
Description: The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
counop  |-  ( ( S  e.  UniOp  /\  T  e.  UniOp )  ->  ( S  o.  T )  e.  UniOp )

Proof of Theorem counop
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopf1o 22496 . . . 4  |-  ( S  e.  UniOp  ->  S : ~H
-1-1-onto-> ~H )
2 unopf1o 22496 . . . 4  |-  ( T  e.  UniOp  ->  T : ~H
-1-1-onto-> ~H )
3 f1oco 5496 . . . 4  |-  ( ( S : ~H -1-1-onto-> ~H  /\  T : ~H
-1-1-onto-> ~H )  ->  ( S  o.  T ) : ~H -1-1-onto-> ~H )
41, 2, 3syl2an 463 . . 3  |-  ( ( S  e.  UniOp  /\  T  e.  UniOp )  ->  ( S  o.  T ) : ~H -1-1-onto-> ~H )
5 f1ofo 5479 . . 3  |-  ( ( S  o.  T ) : ~H -1-1-onto-> ~H  ->  ( S  o.  T ) : ~H -onto-> ~H )
64, 5syl 15 . 2  |-  ( ( S  e.  UniOp  /\  T  e.  UniOp )  ->  ( S  o.  T ) : ~H -onto-> ~H )
7 f1of 5472 . . . . . . . 8  |-  ( T : ~H -1-1-onto-> ~H  ->  T : ~H
--> ~H )
82, 7syl 15 . . . . . . 7  |-  ( T  e.  UniOp  ->  T : ~H
--> ~H )
98adantl 452 . . . . . 6  |-  ( ( S  e.  UniOp  /\  T  e.  UniOp )  ->  T : ~H --> ~H )
10 simpl 443 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  x  e.  ~H )
11 fvco3 5596 . . . . . 6  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  o.  T ) `  x
)  =  ( S `
 ( T `  x ) ) )
129, 10, 11syl2an 463 . . . . 5  |-  ( ( ( S  e.  UniOp  /\  T  e.  UniOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( S  o.  T ) `  x )  =  ( S `  ( T `
 x ) ) )
13 simpr 447 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  y  e.  ~H )
14 fvco3 5596 . . . . . 6  |-  ( ( T : ~H --> ~H  /\  y  e.  ~H )  ->  ( ( S  o.  T ) `  y
)  =  ( S `
 ( T `  y ) ) )
159, 13, 14syl2an 463 . . . . 5  |-  ( ( ( S  e.  UniOp  /\  T  e.  UniOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( S  o.  T ) `  y )  =  ( S `  ( T `
 y ) ) )
1612, 15oveq12d 5876 . . . 4  |-  ( ( ( S  e.  UniOp  /\  T  e.  UniOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
( S  o.  T
) `  x )  .ih  ( ( S  o.  T ) `  y
) )  =  ( ( S `  ( T `  x )
)  .ih  ( S `  ( T `  y
) ) ) )
17 ffvelrn 5663 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
18 ffvelrn 5663 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
1917, 18anim12dan 810 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( T `  x )  e.  ~H  /\  ( T `
 y )  e. 
~H ) )
208, 19sylan 457 . . . . . 6  |-  ( ( T  e.  UniOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( T `  x )  e.  ~H  /\  ( T `
 y )  e. 
~H ) )
21 unop 22495 . . . . . . 7  |-  ( ( S  e.  UniOp  /\  ( T `  x )  e.  ~H  /\  ( T `
 y )  e. 
~H )  ->  (
( S `  ( T `  x )
)  .ih  ( S `  ( T `  y
) ) )  =  ( ( T `  x )  .ih  ( T `  y )
) )
22213expb 1152 . . . . . 6  |-  ( ( S  e.  UniOp  /\  (
( T `  x
)  e.  ~H  /\  ( T `  y )  e.  ~H ) )  ->  ( ( S `
 ( T `  x ) )  .ih  ( S `  ( T `
 y ) ) )  =  ( ( T `  x ) 
.ih  ( T `  y ) ) )
2320, 22sylan2 460 . . . . 5  |-  ( ( S  e.  UniOp  /\  ( T  e.  UniOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
) )  ->  (
( S `  ( T `  x )
)  .ih  ( S `  ( T `  y
) ) )  =  ( ( T `  x )  .ih  ( T `  y )
) )
2423anassrs 629 . . . 4  |-  ( ( ( S  e.  UniOp  /\  T  e.  UniOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( S `  ( T `  x ) )  .ih  ( S `  ( T `
 y ) ) )  =  ( ( T `  x ) 
.ih  ( T `  y ) ) )
25 unop 22495 . . . . . 6  |-  ( ( T  e.  UniOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
( T `  x
)  .ih  ( T `  y ) )  =  ( x  .ih  y
) )
26253expb 1152 . . . . 5  |-  ( ( T  e.  UniOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( T `  x )  .ih  ( T `  y
) )  =  ( x  .ih  y ) )
2726adantll 694 . . . 4  |-  ( ( ( S  e.  UniOp  /\  T  e.  UniOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( T `  x )  .ih  ( T `  y
) )  =  ( x  .ih  y ) )
2816, 24, 273eqtrd 2319 . . 3  |-  ( ( ( S  e.  UniOp  /\  T  e.  UniOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
( S  o.  T
) `  x )  .ih  ( ( S  o.  T ) `  y
) )  =  ( x  .ih  y ) )
2928ralrimivva 2635 . 2  |-  ( ( S  e.  UniOp  /\  T  e.  UniOp )  ->  A. x  e.  ~H  A. y  e. 
~H  ( ( ( S  o.  T ) `
 x )  .ih  ( ( S  o.  T ) `  y
) )  =  ( x  .ih  y ) )
30 elunop 22452 . 2  |-  ( ( S  o.  T )  e.  UniOp 
<->  ( ( S  o.  T ) : ~H -onto-> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( ( S  o.  T ) `  x )  .ih  (
( S  o.  T
) `  y )
)  =  ( x 
.ih  y ) ) )
316, 29, 30sylanbrc 645 1  |-  ( ( S  e.  UniOp  /\  T  e.  UniOp )  ->  ( S  o.  T )  e.  UniOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    o. ccom 4693   -->wf 5251   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   ~Hchil 21499    .ih csp 21502   UniOpcuo 21529
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586  df-hvsub 21551  df-unop 22423
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