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Theorem lnolin 26378
Description: Basic linearity property of a linear operator. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnoval.1  |-  X  =  ( BaseSet `  U )
lnoval.2  |-  Y  =  ( BaseSet `  W )
lnoval.3  |-  G  =  ( +v `  U
)
lnoval.4  |-  H  =  ( +v `  W
)
lnoval.5  |-  R  =  ( .sOLD `  U )
lnoval.6  |-  S  =  ( .sOLD `  W )
lnoval.7  |-  L  =  ( U  LnOp  W
)
Assertion
Ref Expression
lnolin  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( T `  (
( A R B ) G C ) )  =  ( ( A S ( T `
 B ) ) H ( T `  C ) ) )

Proof of Theorem lnolin
Dummy variables  u  t  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnoval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
2 lnoval.2 . . . . 5  |-  Y  =  ( BaseSet `  W )
3 lnoval.3 . . . . 5  |-  G  =  ( +v `  U
)
4 lnoval.4 . . . . 5  |-  H  =  ( +v `  W
)
5 lnoval.5 . . . . 5  |-  R  =  ( .sOLD `  U )
6 lnoval.6 . . . . 5  |-  S  =  ( .sOLD `  W )
7 lnoval.7 . . . . 5  |-  L  =  ( U  LnOp  W
)
81, 2, 3, 4, 5, 6, 7islno 26377 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  L  <->  ( T : X --> Y  /\  A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( ( u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `
 t ) ) ) ) )
98biimp3a 1364 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T : X --> Y  /\  A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( (
u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `  t
) ) ) )
109simprd 464 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( ( u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `
 t ) ) )
11 oveq1 6308 . . . . . 6  |-  ( u  =  A  ->  (
u R w )  =  ( A R w ) )
1211oveq1d 6316 . . . . 5  |-  ( u  =  A  ->  (
( u R w ) G t )  =  ( ( A R w ) G t ) )
1312fveq2d 5881 . . . 4  |-  ( u  =  A  ->  ( T `  ( (
u R w ) G t ) )  =  ( T `  ( ( A R w ) G t ) ) )
14 oveq1 6308 . . . . 5  |-  ( u  =  A  ->  (
u S ( T `
 w ) )  =  ( A S ( T `  w
) ) )
1514oveq1d 6316 . . . 4  |-  ( u  =  A  ->  (
( u S ( T `  w ) ) H ( T `
 t ) )  =  ( ( A S ( T `  w ) ) H ( T `  t
) ) )
1613, 15eqeq12d 2444 . . 3  |-  ( u  =  A  ->  (
( T `  (
( u R w ) G t ) )  =  ( ( u S ( T `
 w ) ) H ( T `  t ) )  <->  ( T `  ( ( A R w ) G t ) )  =  ( ( A S ( T `  w ) ) H ( T `
 t ) ) ) )
17 oveq2 6309 . . . . . 6  |-  ( w  =  B  ->  ( A R w )  =  ( A R B ) )
1817oveq1d 6316 . . . . 5  |-  ( w  =  B  ->  (
( A R w ) G t )  =  ( ( A R B ) G t ) )
1918fveq2d 5881 . . . 4  |-  ( w  =  B  ->  ( T `  ( ( A R w ) G t ) )  =  ( T `  (
( A R B ) G t ) ) )
20 fveq2 5877 . . . . . 6  |-  ( w  =  B  ->  ( T `  w )  =  ( T `  B ) )
2120oveq2d 6317 . . . . 5  |-  ( w  =  B  ->  ( A S ( T `  w ) )  =  ( A S ( T `  B ) ) )
2221oveq1d 6316 . . . 4  |-  ( w  =  B  ->  (
( A S ( T `  w ) ) H ( T `
 t ) )  =  ( ( A S ( T `  B ) ) H ( T `  t
) ) )
2319, 22eqeq12d 2444 . . 3  |-  ( w  =  B  ->  (
( T `  (
( A R w ) G t ) )  =  ( ( A S ( T `
 w ) ) H ( T `  t ) )  <->  ( T `  ( ( A R B ) G t ) )  =  ( ( A S ( T `  B ) ) H ( T `
 t ) ) ) )
24 oveq2 6309 . . . . 5  |-  ( t  =  C  ->  (
( A R B ) G t )  =  ( ( A R B ) G C ) )
2524fveq2d 5881 . . . 4  |-  ( t  =  C  ->  ( T `  ( ( A R B ) G t ) )  =  ( T `  (
( A R B ) G C ) ) )
26 fveq2 5877 . . . . 5  |-  ( t  =  C  ->  ( T `  t )  =  ( T `  C ) )
2726oveq2d 6317 . . . 4  |-  ( t  =  C  ->  (
( A S ( T `  B ) ) H ( T `
 t ) )  =  ( ( A S ( T `  B ) ) H ( T `  C
) ) )
2825, 27eqeq12d 2444 . . 3  |-  ( t  =  C  ->  (
( T `  (
( A R B ) G t ) )  =  ( ( A S ( T `
 B ) ) H ( T `  t ) )  <->  ( T `  ( ( A R B ) G C ) )  =  ( ( A S ( T `  B ) ) H ( T `
 C ) ) ) )
2916, 23, 28rspc3v 3194 . 2  |-  ( ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )  ->  ( A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( ( u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `
 t ) )  ->  ( T `  ( ( A R B ) G C ) )  =  ( ( A S ( T `  B ) ) H ( T `
 C ) ) ) )
3010, 29mpan9 471 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( T `  (
( A R B ) G C ) )  =  ( ( A S ( T `
 B ) ) H ( T `  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   -->wf 5593   ` cfv 5597  (class class class)co 6301   CCcc 9537   NrmCVeccnv 26186   +vcpv 26187   BaseSetcba 26188   .sOLDcns 26189    LnOp clno 26364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-map 7478  df-lno 26368
This theorem is referenced by:  lno0  26380  lnocoi  26381  lnoadd  26382  lnosub  26383  lnomul  26384
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