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Theorem lnopunilem2 23516
Description: Lemma for lnopunii 23517. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnopunilem.1  |-  T  e. 
LinOp
lnopunilem.2  |-  A. x  e.  ~H  ( normh `  ( T `  x )
)  =  ( normh `  x )
lnopunilem.3  |-  A  e. 
~H
lnopunilem.4  |-  B  e. 
~H
Assertion
Ref Expression
lnopunilem2  |-  ( ( T `  A ) 
.ih  ( T `  B ) )  =  ( A  .ih  B
)
Distinct variable group:    x, T
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem lnopunilem2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 oveq1 6090 . . . . . 6  |-  ( y  =  if ( y  e.  CC ,  y ,  0 )  -> 
( y  x.  (
( T `  A
)  .ih  ( T `  B ) ) )  =  ( if ( y  e.  CC , 
y ,  0 )  x.  ( ( T `
 A )  .ih  ( T `  B ) ) ) )
21fveq2d 5734 . . . . 5  |-  ( y  =  if ( y  e.  CC ,  y ,  0 )  -> 
( Re `  (
y  x.  ( ( T `  A ) 
.ih  ( T `  B ) ) ) )  =  ( Re
`  ( if ( y  e.  CC , 
y ,  0 )  x.  ( ( T `
 A )  .ih  ( T `  B ) ) ) ) )
3 oveq1 6090 . . . . . 6  |-  ( y  =  if ( y  e.  CC ,  y ,  0 )  -> 
( y  x.  ( A  .ih  B ) )  =  ( if ( y  e.  CC , 
y ,  0 )  x.  ( A  .ih  B ) ) )
43fveq2d 5734 . . . . 5  |-  ( y  =  if ( y  e.  CC ,  y ,  0 )  -> 
( Re `  (
y  x.  ( A 
.ih  B ) ) )  =  ( Re
`  ( if ( y  e.  CC , 
y ,  0 )  x.  ( A  .ih  B ) ) ) )
52, 4eqeq12d 2452 . . . 4  |-  ( y  =  if ( y  e.  CC ,  y ,  0 )  -> 
( ( Re `  ( y  x.  (
( T `  A
)  .ih  ( T `  B ) ) ) )  =  ( Re
`  ( y  x.  ( A  .ih  B
) ) )  <->  ( Re `  ( if ( y  e.  CC ,  y ,  0 )  x.  ( ( T `  A )  .ih  ( T `  B )
) ) )  =  ( Re `  ( if ( y  e.  CC ,  y ,  0 )  x.  ( A 
.ih  B ) ) ) ) )
6 lnopunilem.1 . . . . 5  |-  T  e. 
LinOp
7 lnopunilem.2 . . . . 5  |-  A. x  e.  ~H  ( normh `  ( T `  x )
)  =  ( normh `  x )
8 lnopunilem.3 . . . . 5  |-  A  e. 
~H
9 lnopunilem.4 . . . . 5  |-  B  e. 
~H
10 0cn 9086 . . . . . 6  |-  0  e.  CC
1110elimel 3793 . . . . 5  |-  if ( y  e.  CC , 
y ,  0 )  e.  CC
126, 7, 8, 9, 11lnopunilem1 23515 . . . 4  |-  ( Re
`  ( if ( y  e.  CC , 
y ,  0 )  x.  ( ( T `
 A )  .ih  ( T `  B ) ) ) )  =  ( Re `  ( if ( y  e.  CC ,  y ,  0 )  x.  ( A 
.ih  B ) ) )
135, 12dedth 3782 . . 3  |-  ( y  e.  CC  ->  (
Re `  ( y  x.  ( ( T `  A )  .ih  ( T `  B )
) ) )  =  ( Re `  (
y  x.  ( A 
.ih  B ) ) ) )
1413rgen 2773 . 2  |-  A. y  e.  CC  ( Re `  ( y  x.  (
( T `  A
)  .ih  ( T `  B ) ) ) )  =  ( Re
`  ( y  x.  ( A  .ih  B
) ) )
156lnopfi 23474 . . . . . 6  |-  T : ~H
--> ~H
1615ffvelrni 5871 . . . . 5  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
178, 16ax-mp 8 . . . 4  |-  ( T `
 A )  e. 
~H
1815ffvelrni 5871 . . . . 5  |-  ( B  e.  ~H  ->  ( T `  B )  e.  ~H )
199, 18ax-mp 8 . . . 4  |-  ( T `
 B )  e. 
~H
2017, 19hicli 22585 . . 3  |-  ( ( T `  A ) 
.ih  ( T `  B ) )  e.  CC
218, 9hicli 22585 . . 3  |-  ( A 
.ih  B )  e.  CC
22 recan 12142 . . 3  |-  ( ( ( ( T `  A )  .ih  ( T `  B )
)  e.  CC  /\  ( A  .ih  B )  e.  CC )  -> 
( A. y  e.  CC  ( Re `  ( y  x.  (
( T `  A
)  .ih  ( T `  B ) ) ) )  =  ( Re
`  ( y  x.  ( A  .ih  B
) ) )  <->  ( ( T `  A )  .ih  ( T `  B
) )  =  ( A  .ih  B ) ) )
2320, 21, 22mp2an 655 . 2  |-  ( A. y  e.  CC  (
Re `  ( y  x.  ( ( T `  A )  .ih  ( T `  B )
) ) )  =  ( Re `  (
y  x.  ( A 
.ih  B ) ) )  <->  ( ( T `
 A )  .ih  ( T `  B ) )  =  ( A 
.ih  B ) )
2414, 23mpbi 201 1  |-  ( ( T `  A ) 
.ih  ( T `  B ) )  =  ( A  .ih  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   A.wral 2707   ifcif 3741   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992    x. cmul 8997   Recre 11904   ~Hchil 22424    .ih csp 22427   normhcno 22428   LinOpclo 22452
This theorem is referenced by:  lnopunii  23517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-hilex 22504  ax-hfvadd 22505  ax-hv0cl 22508  ax-hfvmul 22510  ax-hvmul0 22515  ax-hfi 22583  ax-his1 22586  ax-his2 22587  ax-his3 22588  ax-his4 22589
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-hnorm 22473  df-lnop 23346
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