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Theorem adjadd 25644
Description: The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adjadd  |-  ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  ->  ( adjh `  ( S  +op  T ) )  =  ( ( adjh `  S )  +op  ( adjh `  T ) ) )

Proof of Theorem adjadd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmadjop 25439 . . 3  |-  ( S  e.  dom  adjh  ->  S : ~H --> ~H )
2 dmadjop 25439 . . 3  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
3 hoaddcl 25309 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
) : ~H --> ~H )
41, 2, 3syl2an 477 . 2  |-  ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  ->  ( S  +op  T ) : ~H --> ~H )
5 dmadjrn 25446 . . . 4  |-  ( S  e.  dom  adjh  ->  (
adjh `  S )  e.  dom  adjh )
6 dmadjop 25439 . . . 4  |-  ( (
adjh `  S )  e.  dom  adjh  ->  ( adjh `  S ) : ~H --> ~H )
75, 6syl 16 . . 3  |-  ( S  e.  dom  adjh  ->  (
adjh `  S ) : ~H --> ~H )
8 dmadjrn 25446 . . . 4  |-  ( T  e.  dom  adjh  ->  (
adjh `  T )  e.  dom  adjh )
9 dmadjop 25439 . . . 4  |-  ( (
adjh `  T )  e.  dom  adjh  ->  ( adjh `  T ) : ~H --> ~H )
108, 9syl 16 . . 3  |-  ( T  e.  dom  adjh  ->  (
adjh `  T ) : ~H --> ~H )
11 hoaddcl 25309 . . 3  |-  ( ( ( adjh `  S
) : ~H --> ~H  /\  ( adjh `  T ) : ~H --> ~H )  -> 
( ( adjh `  S
)  +op  ( adjh `  T ) ) : ~H --> ~H )
127, 10, 11syl2an 477 . 2  |-  ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  ->  ( ( adjh `  S )  +op  ( adjh `  T ) ) : ~H --> ~H )
13 adj2 25485 . . . . . . . 8  |-  ( ( S  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( ( S `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  S ) `  y
) ) )
14133expb 1189 . . . . . . 7  |-  ( ( S  e.  dom  adjh  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( S `  x )  .ih  y )  =  ( x  .ih  ( (
adjh `  S ) `  y ) ) )
1514adantlr 714 . . . . . 6  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( ( S `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  S ) `  y
) ) )
16 adj2 25485 . . . . . . . 8  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  T ) `  y
) ) )
17163expb 1189 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( (
adjh `  T ) `  y ) ) )
1817adantll 713 . . . . . 6  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  T ) `  y
) ) )
1915, 18oveq12d 6213 . . . . 5  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( ( ( S `
 x )  .ih  y )  +  ( ( T `  x
)  .ih  y )
)  =  ( ( x  .ih  ( (
adjh `  S ) `  y ) )  +  ( x  .ih  (
( adjh `  T ) `  y ) ) ) )
201ffvelrnda 5947 . . . . . . 7  |-  ( ( S  e.  dom  adjh  /\  x  e.  ~H )  ->  ( S `  x
)  e.  ~H )
2120ad2ant2r 746 . . . . . 6  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( S `  x
)  e.  ~H )
222ffvelrnda 5947 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
2322ad2ant2lr 747 . . . . . 6  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( T `  x
)  e.  ~H )
24 simprr 756 . . . . . 6  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
y  e.  ~H )
25 ax-his2 24632 . . . . . 6  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H  /\  y  e.  ~H )  ->  (
( ( S `  x )  +h  ( T `  x )
)  .ih  y )  =  ( ( ( S `  x ) 
.ih  y )  +  ( ( T `  x )  .ih  y
) ) )
2621, 23, 24, 25syl3anc 1219 . . . . 5  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( ( ( S `
 x )  +h  ( T `  x
) )  .ih  y
)  =  ( ( ( S `  x
)  .ih  y )  +  ( ( T `
 x )  .ih  y ) ) )
27 simprl 755 . . . . . 6  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  ->  x  e.  ~H )
28 adjcl 25483 . . . . . . 7  |-  ( ( S  e.  dom  adjh  /\  y  e.  ~H )  ->  ( ( adjh `  S
) `  y )  e.  ~H )
2928ad2ant2rl 748 . . . . . 6  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( ( adjh `  S
) `  y )  e.  ~H )
30 adjcl 25483 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  y  e.  ~H )  ->  ( ( adjh `  T
) `  y )  e.  ~H )
3130ad2ant2l 745 . . . . . 6  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( ( adjh `  T
) `  y )  e.  ~H )
32 his7 24639 . . . . . 6  |-  ( ( x  e.  ~H  /\  ( ( adjh `  S
) `  y )  e.  ~H  /\  ( (
adjh `  T ) `  y )  e.  ~H )  ->  ( x  .ih  ( ( ( adjh `  S ) `  y
)  +h  ( (
adjh `  T ) `  y ) ) )  =  ( ( x 
.ih  ( ( adjh `  S ) `  y
) )  +  ( x  .ih  ( (
adjh `  T ) `  y ) ) ) )
3327, 29, 31, 32syl3anc 1219 . . . . 5  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( x  .ih  (
( ( adjh `  S
) `  y )  +h  ( ( adjh `  T
) `  y )
) )  =  ( ( x  .ih  (
( adjh `  S ) `  y ) )  +  ( x  .ih  (
( adjh `  T ) `  y ) ) ) )
3419, 26, 333eqtr4rd 2504 . . . 4  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( x  .ih  (
( ( adjh `  S
) `  y )  +h  ( ( adjh `  T
) `  y )
) )  =  ( ( ( S `  x )  +h  ( T `  x )
)  .ih  y )
)
357, 10anim12i 566 . . . . . . 7  |-  ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  ->  ( ( adjh `  S ) : ~H --> ~H  /\  ( adjh `  T
) : ~H --> ~H )
)
36 hosval 25291 . . . . . . . 8  |-  ( ( ( adjh `  S
) : ~H --> ~H  /\  ( adjh `  T ) : ~H --> ~H  /\  y  e.  ~H )  ->  (
( ( adjh `  S
)  +op  ( adjh `  T ) ) `  y )  =  ( ( ( adjh `  S
) `  y )  +h  ( ( adjh `  T
) `  y )
) )
37363expa 1188 . . . . . . 7  |-  ( ( ( ( adjh `  S
) : ~H --> ~H  /\  ( adjh `  T ) : ~H --> ~H )  /\  y  e.  ~H )  ->  ( ( ( adjh `  S )  +op  ( adjh `  T ) ) `
 y )  =  ( ( ( adjh `  S ) `  y
)  +h  ( (
adjh `  T ) `  y ) ) )
3835, 37sylan 471 . . . . . 6  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  y  e. 
~H )  ->  (
( ( adjh `  S
)  +op  ( adjh `  T ) ) `  y )  =  ( ( ( adjh `  S
) `  y )  +h  ( ( adjh `  T
) `  y )
) )
3938adantrl 715 . . . . 5  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( ( ( adjh `  S )  +op  ( adjh `  T ) ) `
 y )  =  ( ( ( adjh `  S ) `  y
)  +h  ( (
adjh `  T ) `  y ) ) )
4039oveq2d 6211 . . . 4  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( x  .ih  (
( ( adjh `  S
)  +op  ( adjh `  T ) ) `  y ) )  =  ( x  .ih  (
( ( adjh `  S
) `  y )  +h  ( ( adjh `  T
) `  y )
) ) )
411, 2anim12i 566 . . . . . . 7  |-  ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  ->  ( S : ~H
--> ~H  /\  T : ~H
--> ~H ) )
42 hosval 25291 . . . . . . . 8  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  +op  T ) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
43423expa 1188 . . . . . . 7  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  (
( S  +op  T
) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
4441, 43sylan 471 . . . . . 6  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  x  e. 
~H )  ->  (
( S  +op  T
) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
4544adantrr 716 . . . . 5  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( ( S  +op  T ) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
4645oveq1d 6210 . . . 4  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( ( ( S 
+op  T ) `  x )  .ih  y
)  =  ( ( ( S `  x
)  +h  ( T `
 x ) ) 
.ih  y ) )
4734, 40, 463eqtr4rd 2504 . . 3  |-  ( ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  /\  ( x  e.  ~H  /\  y  e.  ~H ) )  -> 
( ( ( S 
+op  T ) `  x )  .ih  y
)  =  ( x 
.ih  ( ( (
adjh `  S )  +op  ( adjh `  T
) ) `  y
) ) )
4847ralrimivva 2908 . 2  |-  ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  ->  A. x  e.  ~H  A. y  e.  ~H  (
( ( S  +op  T ) `  x ) 
.ih  y )  =  ( x  .ih  (
( ( adjh `  S
)  +op  ( adjh `  T ) ) `  y ) ) )
49 adjeq 25486 . 2  |-  ( ( ( S  +op  T
) : ~H --> ~H  /\  ( ( adjh `  S
)  +op  ( adjh `  T ) ) : ~H --> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( ( ( S  +op  T ) `
 x )  .ih  y )  =  ( x  .ih  ( ( ( adjh `  S
)  +op  ( adjh `  T ) ) `  y ) ) )  ->  ( adjh `  ( S  +op  T ) )  =  ( ( adjh `  S )  +op  ( adjh `  T ) ) )
504, 12, 48, 49syl3anc 1219 1  |-  ( ( S  e.  dom  adjh  /\  T  e.  dom  adjh )  ->  ( adjh `  ( S  +op  T ) )  =  ( ( adjh `  S )  +op  ( adjh `  T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796   dom cdm 4943   -->wf 5517   ` cfv 5521  (class class class)co 6195    + caddc 9391   ~Hchil 24468    +h cva 24469    .ih csp 24471    +op chos 24487   adjhcado 24504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-hilex 24548  ax-hfvadd 24549  ax-hvcom 24550  ax-hvass 24551  ax-hv0cl 24552  ax-hvaddid 24553  ax-hfvmul 24554  ax-hvmulid 24555  ax-hvdistr2 24558  ax-hvmul0 24559  ax-hfi 24628  ax-his1 24631  ax-his2 24632  ax-his3 24633  ax-his4 24634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-2 10486  df-cj 12701  df-re 12702  df-im 12703  df-hvsub 24520  df-hosum 25281  df-adjh 25400
This theorem is referenced by: (None)
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