Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oddcomabszz Structured version   Unicode version

Theorem oddcomabszz 35546
Description: An odd function which takes nonnegative values on nonnegative arguments commutes with  abs. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Hypotheses
Ref Expression
oddcomabszz.1  |-  ( (
ph  /\  x  e.  ZZ )  ->  A  e.  RR )
oddcomabszz.2  |-  ( (
ph  /\  x  e.  ZZ  /\  0  <_  x
)  ->  0  <_  A )
oddcomabszz.3  |-  ( (
ph  /\  y  e.  ZZ )  ->  C  = 
-u B )
oddcomabszz.4  |-  ( x  =  y  ->  A  =  B )
oddcomabszz.5  |-  ( x  =  -u y  ->  A  =  C )
oddcomabszz.6  |-  ( x  =  D  ->  A  =  E )
oddcomabszz.7  |-  ( x  =  ( abs `  D
)  ->  A  =  F )
Assertion
Ref Expression
oddcomabszz  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  E )  =  F )
Distinct variable groups:    x, B    x, C    x, D, y   
x, E    x, F    y, A    ph, x, y
Allowed substitution hints:    A( x)    B( y)    C( y)    E( y)    F( y)

Proof of Theorem oddcomabszz
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eleq1 2492 . . . . . 6  |-  ( a  =  D  ->  (
a  e.  ZZ  <->  D  e.  ZZ ) )
21anbi2d 708 . . . . 5  |-  ( a  =  D  ->  (
( ph  /\  a  e.  ZZ )  <->  ( ph  /\  D  e.  ZZ ) ) )
3 csbeq1 3395 . . . . . . 7  |-  ( a  =  D  ->  [_ a  /  x ]_ A  = 
[_ D  /  x ]_ A )
43fveq2d 5876 . . . . . 6  |-  ( a  =  D  ->  ( abs `  [_ a  /  x ]_ A )  =  ( abs `  [_ D  /  x ]_ A ) )
5 fveq2 5872 . . . . . . 7  |-  ( a  =  D  ->  ( abs `  a )  =  ( abs `  D
) )
65csbeq1d 3399 . . . . . 6  |-  ( a  =  D  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ ( abs `  D )  /  x ]_ A
)
74, 6eqeq12d 2442 . . . . 5  |-  ( a  =  D  ->  (
( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A  <->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A ) )
82, 7imbi12d 321 . . . 4  |-  ( a  =  D  ->  (
( ( ph  /\  a  e.  ZZ )  ->  ( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A )  <->  ( ( ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A ) ) )
9 nfv 1751 . . . . . . . . . 10  |-  F/ x
( ph  /\  a  e.  ZZ )
10 nfcsb1v 3408 . . . . . . . . . . 11  |-  F/_ x [_ a  /  x ]_ A
1110nfel1 2598 . . . . . . . . . 10  |-  F/ x [_ a  /  x ]_ A  e.  RR
129, 11nfim 1975 . . . . . . . . 9  |-  F/ x
( ( ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR )
13 eleq1 2492 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
x  e.  ZZ  <->  a  e.  ZZ ) )
1413anbi2d 708 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( ph  /\  x  e.  ZZ )  <->  ( ph  /\  a  e.  ZZ ) ) )
15 csbeq1a 3401 . . . . . . . . . . 11  |-  ( x  =  a  ->  A  =  [_ a  /  x ]_ A )
1615eleq1d 2489 . . . . . . . . . 10  |-  ( x  =  a  ->  ( A  e.  RR  <->  [_ a  /  x ]_ A  e.  RR ) )
1714, 16imbi12d 321 . . . . . . . . 9  |-  ( x  =  a  ->  (
( ( ph  /\  x  e.  ZZ )  ->  A  e.  RR )  <-> 
( ( ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR ) ) )
18 oddcomabszz.1 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ZZ )  ->  A  e.  RR )
1912, 17, 18chvar 2066 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR )
2019adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  [_ a  /  x ]_ A  e.  RR )
21 nfv 1751 . . . . . . . . . 10  |-  F/ x
( ph  /\  a  e.  ZZ  /\  0  <_ 
a )
22 nfcv 2582 . . . . . . . . . . 11  |-  F/_ x
0
23 nfcv 2582 . . . . . . . . . . 11  |-  F/_ x  <_
2422, 23, 10nfbr 4461 . . . . . . . . . 10  |-  F/ x
0  <_  [_ a  /  x ]_ A
2521, 24nfim 1975 . . . . . . . . 9  |-  F/ x
( ( ph  /\  a  e.  ZZ  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A )
26 breq2 4421 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
0  <_  x  <->  0  <_  a ) )
2713, 263anbi23d 1338 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( ph  /\  x  e.  ZZ  /\  0  <_  x )  <->  ( ph  /\  a  e.  ZZ  /\  0  <_  a ) ) )
2815breq2d 4429 . . . . . . . . . 10  |-  ( x  =  a  ->  (
0  <_  A  <->  0  <_  [_ a  /  x ]_ A ) )
2927, 28imbi12d 321 . . . . . . . . 9  |-  ( x  =  a  ->  (
( ( ph  /\  x  e.  ZZ  /\  0  <_  x )  ->  0  <_  A )  <->  ( ( ph  /\  a  e.  ZZ  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A ) ) )
30 oddcomabszz.2 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ZZ  /\  0  <_  x
)  ->  0  <_  A )
3125, 29, 30chvar 2066 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ  /\  0  <_  a
)  ->  0  <_  [_ a  /  x ]_ A )
32313expa 1205 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A )
3320, 32absidd 13452 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ a  /  x ]_ A )
34 zre 10930 . . . . . . . . 9  |-  ( a  e.  ZZ  ->  a  e.  RR )
3534ad2antlr 731 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  a  e.  RR )
36 absid 13327 . . . . . . . 8  |-  ( ( a  e.  RR  /\  0  <_  a )  -> 
( abs `  a
)  =  a )
3735, 36sylancom 671 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  a )  =  a )
3837csbeq1d 3399 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ a  /  x ]_ A
)
3933, 38eqtr4d 2464 . . . . 5  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ ( abs `  a
)  /  x ]_ A )
40 nfv 1751 . . . . . . . 8  |-  F/ y ( ( ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  =  -u [_ a  /  x ]_ A )
41 eleq1 2492 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  e.  ZZ  <->  a  e.  ZZ ) )
4241anbi2d 708 . . . . . . . . 9  |-  ( y  =  a  ->  (
( ph  /\  y  e.  ZZ )  <->  ( ph  /\  a  e.  ZZ ) ) )
43 negex 9862 . . . . . . . . . . . 12  |-  -u y  e.  _V
44 oddcomabszz.5 . . . . . . . . . . . 12  |-  ( x  =  -u y  ->  A  =  C )
4543, 44csbie 3418 . . . . . . . . . . 11  |-  [_ -u y  /  x ]_ A  =  C
46 negeq 9856 . . . . . . . . . . . 12  |-  ( y  =  a  ->  -u y  =  -u a )
4746csbeq1d 3399 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ -u y  /  x ]_ A  = 
[_ -u a  /  x ]_ A )
4845, 47syl5eqr 2475 . . . . . . . . . 10  |-  ( y  =  a  ->  C  =  [_ -u a  /  x ]_ A )
49 vex 3081 . . . . . . . . . . . . 13  |-  y  e. 
_V
50 oddcomabszz.4 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  A  =  B )
5149, 50csbie 3418 . . . . . . . . . . . 12  |-  [_ y  /  x ]_ A  =  B
52 csbeq1 3395 . . . . . . . . . . . 12  |-  ( y  =  a  ->  [_ y  /  x ]_ A  = 
[_ a  /  x ]_ A )
5351, 52syl5eqr 2475 . . . . . . . . . . 11  |-  ( y  =  a  ->  B  =  [_ a  /  x ]_ A )
5453negeqd 9858 . . . . . . . . . 10  |-  ( y  =  a  ->  -u B  =  -u [_ a  /  x ]_ A )
5548, 54eqeq12d 2442 . . . . . . . . 9  |-  ( y  =  a  ->  ( C  =  -u B  <->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A ) )
5642, 55imbi12d 321 . . . . . . . 8  |-  ( y  =  a  ->  (
( ( ph  /\  y  e.  ZZ )  ->  C  =  -u B
)  <->  ( ( ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  =  -u [_ a  /  x ]_ A ) ) )
57 oddcomabszz.3 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ZZ )  ->  C  = 
-u B )
5840, 56, 57chvar 2066 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A )
5958adantr 466 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A )
6034ad2antlr 731 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  a  e.  RR )
61 absnid 13329 . . . . . . . 8  |-  ( ( a  e.  RR  /\  a  <_  0 )  -> 
( abs `  a
)  =  -u a
)
6260, 61sylancom 671 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  a )  = 
-u a )
6362csbeq1d 3399 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ -u a  /  x ]_ A )
6419adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ a  /  x ]_ A  e.  RR )
65 znegcl 10961 . . . . . . . . . . 11  |-  ( a  e.  ZZ  ->  -u a  e.  ZZ )
66 nfv 1751 . . . . . . . . . . . . . 14  |-  F/ x
( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )
67 nfcsb1v 3408 . . . . . . . . . . . . . . 15  |-  F/_ x [_ -u a  /  x ]_ A
6822, 23, 67nfbr 4461 . . . . . . . . . . . . . 14  |-  F/ x
0  <_  [_ -u a  /  x ]_ A
6966, 68nfim 1975 . . . . . . . . . . . . 13  |-  F/ x
( ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  -> 
0  <_  [_ -u a  /  x ]_ A )
70 negex 9862 . . . . . . . . . . . . 13  |-  -u a  e.  _V
71 eleq1 2492 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  (
x  e.  ZZ  <->  -u a  e.  ZZ ) )
72 breq2 4421 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  (
0  <_  x  <->  0  <_  -u a ) )
7371, 723anbi23d 1338 . . . . . . . . . . . . . 14  |-  ( x  =  -u a  ->  (
( ph  /\  x  e.  ZZ  /\  0  <_  x )  <->  ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a ) ) )
74 csbeq1a 3401 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  A  =  [_ -u a  /  x ]_ A )
7574breq2d 4429 . . . . . . . . . . . . . 14  |-  ( x  =  -u a  ->  (
0  <_  A  <->  0  <_  [_ -u a  /  x ]_ A ) )
7673, 75imbi12d 321 . . . . . . . . . . . . 13  |-  ( x  =  -u a  ->  (
( ( ph  /\  x  e.  ZZ  /\  0  <_  x )  ->  0  <_  A )  <->  ( ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  ->  0  <_  [_ -u a  /  x ]_ A ) ) )
7769, 70, 76, 30vtoclf 3129 . . . . . . . . . . . 12  |-  ( (
ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  ->  0  <_  [_ -u a  /  x ]_ A )
78773expia 1207 . . . . . . . . . . 11  |-  ( (
ph  /\  -u a  e.  ZZ )  ->  (
0  <_  -u a  -> 
0  <_  [_ -u a  /  x ]_ A ) )
7965, 78sylan2 476 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  -u a  ->  0  <_  [_ -u a  /  x ]_ A ) )
8058breq2d 4429 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  [_ -u a  /  x ]_ A  <->  0  <_  -u [_ a  /  x ]_ A ) )
8179, 80sylibd 217 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  -u a  ->  0  <_ 
-u [_ a  /  x ]_ A ) )
8234adantl 467 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  a  e.  RR )
8382le0neg1d 10174 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( a  <_  0  <->  0  <_  -u a ) )
8419le0neg1d 10174 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( [_ a  /  x ]_ A  <_  0  <->  0  <_  -u [_ a  /  x ]_ A ) )
8581, 83, 843imtr4d 271 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( a  <_  0  ->  [_ a  /  x ]_ A  <_ 
0 ) )
8685imp 430 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ a  /  x ]_ A  <_ 
0 )
8764, 86absnidd 13443 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  [_ a  /  x ]_ A )  = 
-u [_ a  /  x ]_ A )
8859, 63, 873eqtr4rd 2472 . . . . 5  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ ( abs `  a
)  /  x ]_ A )
89 0re 9632 . . . . . . 7  |-  0  e.  RR
90 letric 9723 . . . . . . 7  |-  ( ( 0  e.  RR  /\  a  e.  RR )  ->  ( 0  <_  a  \/  a  <_  0 ) )
9189, 34, 90sylancr 667 . . . . . 6  |-  ( a  e.  ZZ  ->  (
0  <_  a  \/  a  <_  0 ) )
9291adantl 467 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  a  \/  a  <_  0 ) )
9339, 88, 92mpjaodan 793 . . . 4  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A
)
948, 93vtoclg 3136 . . 3  |-  ( D  e.  ZZ  ->  (
( ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  = 
[_ ( abs `  D
)  /  x ]_ A ) )
9594anabsi7 826 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A
)
96 nfcvd 2583 . . . . 5  |-  ( D  e.  ZZ  ->  F/_ x E )
97 oddcomabszz.6 . . . . 5  |-  ( x  =  D  ->  A  =  E )
9896, 97csbiegf 3416 . . . 4  |-  ( D  e.  ZZ  ->  [_ D  /  x ]_ A  =  E )
9998fveq2d 5876 . . 3  |-  ( D  e.  ZZ  ->  ( abs `  [_ D  /  x ]_ A )  =  ( abs `  E
) )
10099adantl 467 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  ( abs `  E ) )
101 fvex 5882 . . . 4  |-  ( abs `  D )  e.  _V
102 oddcomabszz.7 . . . 4  |-  ( x  =  ( abs `  D
)  ->  A  =  F )
103101, 102csbie 3418 . . 3  |-  [_ ( abs `  D )  /  x ]_ A  =  F
104103a1i 11 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  [_ ( abs `  D )  /  x ]_ A  =  F )
10595, 100, 1043eqtr3d 2469 1  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  E )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   [_csb 3392   class class class wbr 4417   ` cfv 5592   RRcr 9527   0cc0 9528    <_ cle 9665   -ucneg 9850   ZZcz 10926   abscabs 13265
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 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267
This theorem is referenced by:  rmyabs  35562
  Copyright terms: Public domain W3C validator