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Theorem vdwmc2 14929
Description: Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdwmc.1  |-  X  e. 
_V
vdwmc.2  |-  ( ph  ->  K  e.  NN0 )
vdwmc.3  |-  ( ph  ->  F : X --> R )
vdwmc2.4  |-  ( ph  ->  A  e.  X )
Assertion
Ref Expression
vdwmc2  |-  ( ph  ->  ( K MonoAP  F  <->  E. c  e.  R  E. a  e.  NN  E. d  e.  NN  A. m  e.  ( 0 ... ( K  -  1 ) ) ( a  +  ( m  x.  d
) )  e.  ( `' F " { c } ) ) )
Distinct variable groups:    a, c,
d, m, F    K, a, c, d, m    ph, c    R, a, c, d    ph, a,
d
Allowed substitution hints:    ph( m)    A( m, a, c, d)    R( m)    X( m, a, c, d)

Proof of Theorem vdwmc2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vdwmc.1 . . 3  |-  X  e. 
_V
2 vdwmc.2 . . 3  |-  ( ph  ->  K  e.  NN0 )
3 vdwmc.3 . . 3  |-  ( ph  ->  F : X --> R )
41, 2, 3vdwmc 14928 . 2  |-  ( ph  ->  ( K MonoAP  F  <->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
5 vdwapid1 14925 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  a  e.  NN  /\  d  e.  NN )  ->  a  e.  ( a (AP `  K ) d ) )
6 ne0i 3737 . . . . . . . . . . . 12  |-  ( a  e.  ( a (AP
`  K ) d )  ->  ( a
(AP `  K )
d )  =/=  (/) )
75, 6syl 17 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  a  e.  NN  /\  d  e.  NN )  ->  (
a (AP `  K
) d )  =/=  (/) )
873expb 1209 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  ( a  e.  NN  /\  d  e.  NN ) )  ->  ( a
(AP `  K )
d )  =/=  (/) )
98adantll 720 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( a
(AP `  K )
d )  =/=  (/) )
10 ssn0 3767 . . . . . . . . . 10  |-  ( ( ( a (AP `  K ) d ) 
C_  ( `' F " { c } )  /\  ( a (AP
`  K ) d )  =/=  (/) )  -> 
( `' F " { c } )  =/=  (/) )
1110expcom 437 . . . . . . . . 9  |-  ( ( a (AP `  K
) d )  =/=  (/)  ->  ( ( a (AP `  K ) d )  C_  ( `' F " { c } )  ->  ( `' F " { c } )  =/=  (/) ) )
129, 11syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( (
a (AP `  K
) d )  C_  ( `' F " { c } )  ->  ( `' F " { c } )  =/=  (/) ) )
13 disjsn 4032 . . . . . . . . . 10  |-  ( ( R  i^i  { c } )  =  (/)  <->  -.  c  e.  R )
143adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  K  e.  NN )  ->  F : X
--> R )
15 fimacnvdisj 5761 . . . . . . . . . . . . 13  |-  ( ( F : X --> R  /\  ( R  i^i  { c } )  =  (/) )  ->  ( `' F " { c } )  =  (/) )
1615ex 436 . . . . . . . . . . . 12  |-  ( F : X --> R  -> 
( ( R  i^i  { c } )  =  (/)  ->  ( `' F " { c } )  =  (/) ) )
1714, 16syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  K  e.  NN )  ->  ( ( R  i^i  { c } )  =  (/)  ->  ( `' F " { c } )  =  (/) ) )
1817adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( ( R  i^i  { c } )  =  (/)  ->  ( `' F " { c } )  =  (/) ) )
1913, 18syl5bir 222 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( -.  c  e.  R  ->  ( `' F " { c } )  =  (/) ) )
2019necon1ad 2641 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( ( `' F " { c } )  =/=  (/)  ->  c  e.  R ) )
2112, 20syld 45 . . . . . . 7  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( (
a (AP `  K
) d )  C_  ( `' F " { c } )  ->  c  e.  R ) )
2221rexlimdvva 2886 . . . . . 6  |-  ( (
ph  /\  K  e.  NN )  ->  ( E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } )  ->  c  e.  R ) )
2322pm4.71rd 641 . . . . 5  |-  ( (
ph  /\  K  e.  NN )  ->  ( E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } )  <->  ( c  e.  R  /\  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) ) ) )
2423exbidv 1768 . . . 4  |-  ( (
ph  /\  K  e.  NN )  ->  ( E. c E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } )  <->  E. c ( c  e.  R  /\  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) ) ) )
25 df-rex 2743 . . . 4  |-  ( E. c  e.  R  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } )  <->  E. c ( c  e.  R  /\  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) ) )
2624, 25syl6bbr 267 . . 3  |-  ( (
ph  /\  K  e.  NN )  ->  ( E. c E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } )  <->  E. c  e.  R  E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
27 vdwmc2.4 . . . . . . . . 9  |-  ( ph  ->  A  e.  X )
283, 27ffvelrnd 6023 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  R )
29 ne0i 3737 . . . . . . . 8  |-  ( ( F `  A )  e.  R  ->  R  =/=  (/) )
3028, 29syl 17 . . . . . . 7  |-  ( ph  ->  R  =/=  (/) )
3130adantr 467 . . . . . 6  |-  ( (
ph  /\  K  = 
0 )  ->  R  =/=  (/) )
32 1nn 10620 . . . . . . . . 9  |-  1  e.  NN
3332ne0ii 3738 . . . . . . . 8  |-  NN  =/=  (/)
34 simpllr 769 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  K  =  0 )
3534fveq2d 5869 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  (AP `  K
)  =  (AP ` 
0 ) )
3635oveqd 6307 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  =  ( a (AP `  0 ) d ) )
37 vdwap0 14926 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ( a (AP ` 
0 ) d )  =  (/) )
3837adantll 720 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  0 ) d )  =  (/) )
3936, 38eqtrd 2485 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  =  (/) )
40 0ss 3763 . . . . . . . . . . . 12  |-  (/)  C_  ( `' F " { c } )
4139, 40syl6eqss 3482 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
4241ralrimiva 2802 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  ->  A. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
43 r19.2z 3858 . . . . . . . . . 10  |-  ( ( NN  =/=  (/)  /\  A. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) )  ->  E. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
4433, 42, 43sylancr 669 . . . . . . . . 9  |-  ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  ->  E. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
4544ralrimiva 2802 . . . . . . . 8  |-  ( (
ph  /\  K  = 
0 )  ->  A. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
46 r19.2z 3858 . . . . . . . 8  |-  ( ( NN  =/=  (/)  /\  A. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )  ->  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
4733, 45, 46sylancr 669 . . . . . . 7  |-  ( (
ph  /\  K  = 
0 )  ->  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
4847ralrimivw 2803 . . . . . 6  |-  ( (
ph  /\  K  = 
0 )  ->  A. c  e.  R  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
49 r19.2z 3858 . . . . . 6  |-  ( ( R  =/=  (/)  /\  A. c  e.  R  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )  ->  E. c  e.  R  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
5031, 48, 49syl2anc 667 . . . . 5  |-  ( (
ph  /\  K  = 
0 )  ->  E. c  e.  R  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
51 rexex 2844 . . . . 5  |-  ( E. c  e.  R  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } )  ->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) )
5250, 51syl 17 . . . 4  |-  ( (
ph  /\  K  = 
0 )  ->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) )
5352, 502thd 244 . . 3  |-  ( (
ph  /\  K  = 
0 )  ->  ( E. c E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } )  <->  E. c  e.  R  E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
54 elnn0 10871 . . . 4  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
552, 54sylib 200 . . 3  |-  ( ph  ->  ( K  e.  NN  \/  K  =  0
) )
5626, 53, 55mpjaodan 795 . 2  |-  ( ph  ->  ( E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } )  <->  E. c  e.  R  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) ) )
57 vdwapval 14923 . . . . . . . . 9  |-  ( ( K  e.  NN0  /\  a  e.  NN  /\  d  e.  NN )  ->  (
x  e.  ( a (AP `  K ) d )  <->  E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( a  +  ( m  x.  d ) ) ) )
58573expb 1209 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  ( a  e.  NN  /\  d  e.  NN ) )  ->  ( x  e.  ( a (AP `  K ) d )  <->  E. m  e.  (
0 ... ( K  - 
1 ) ) x  =  ( a  +  ( m  x.  d
) ) ) )
592, 58sylan 474 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  NN  /\  d  e.  NN ) )  -> 
( x  e.  ( a (AP `  K
) d )  <->  E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( a  +  ( m  x.  d ) ) ) )
6059imbi1d 319 . . . . . 6  |-  ( (
ph  /\  ( a  e.  NN  /\  d  e.  NN ) )  -> 
( ( x  e.  ( a (AP `  K ) d )  ->  x  e.  ( `' F " { c } ) )  <->  ( E. m  e.  ( 0 ... ( K  - 
1 ) ) x  =  ( a  +  ( m  x.  d
) )  ->  x  e.  ( `' F " { c } ) ) ) )
6160albidv 1767 . . . . 5  |-  ( (
ph  /\  ( a  e.  NN  /\  d  e.  NN ) )  -> 
( A. x ( x  e.  ( a (AP `  K ) d )  ->  x  e.  ( `' F " { c } ) )  <->  A. x ( E. m  e.  ( 0 ... ( K  - 
1 ) ) x  =  ( a  +  ( m  x.  d
) )  ->  x  e.  ( `' F " { c } ) ) ) )
62 dfss2 3421 . . . . 5  |-  ( ( a (AP `  K
) d )  C_  ( `' F " { c } )  <->  A. x
( x  e.  ( a (AP `  K
) d )  ->  x  e.  ( `' F " { c } ) ) )
63 ralcom4 3066 . . . . . 6  |-  ( A. m  e.  ( 0 ... ( K  - 
1 ) ) A. x ( x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) )  <->  A. x A. m  e.  (
0 ... ( K  - 
1 ) ) ( x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) ) )
64 ovex 6318 . . . . . . . 8  |-  ( a  +  ( m  x.  d ) )  e. 
_V
65 eleq1 2517 . . . . . . . 8  |-  ( x  =  ( a  +  ( m  x.  d
) )  ->  (
x  e.  ( `' F " { c } )  <->  ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) ) )
6664, 65ceqsalv 3075 . . . . . . 7  |-  ( A. x ( x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) )  <->  ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) )
6766ralbii 2819 . . . . . 6  |-  ( A. m  e.  ( 0 ... ( K  - 
1 ) ) A. x ( x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) )  <->  A. m  e.  ( 0 ... ( K  -  1 ) ) ( a  +  ( m  x.  d
) )  e.  ( `' F " { c } ) )
68 r19.23v 2867 . . . . . . 7  |-  ( A. m  e.  ( 0 ... ( K  - 
1 ) ) ( x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) )  <->  ( E. m  e.  ( 0 ... ( K  - 
1 ) ) x  =  ( a  +  ( m  x.  d
) )  ->  x  e.  ( `' F " { c } ) ) )
6968albii 1691 . . . . . 6  |-  ( A. x A. m  e.  ( 0 ... ( K  -  1 ) ) ( x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) )  <->  A. x
( E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) ) )
7063, 67, 693bitr3i 279 . . . . 5  |-  ( A. m  e.  ( 0 ... ( K  - 
1 ) ) ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } )  <->  A. x ( E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) ) )
7161, 62, 703bitr4g 292 . . . 4  |-  ( (
ph  /\  ( a  e.  NN  /\  d  e.  NN ) )  -> 
( ( a (AP
`  K ) d )  C_  ( `' F " { c } )  <->  A. m  e.  ( 0 ... ( K  -  1 ) ) ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) ) )
72712rexbidva 2907 . . 3  |-  ( ph  ->  ( E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } )  <->  E. a  e.  NN  E. d  e.  NN  A. m  e.  ( 0 ... ( K  - 
1 ) ) ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) ) )
7372rexbidv 2901 . 2  |-  ( ph  ->  ( E. c  e.  R  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } )  <->  E. c  e.  R  E. a  e.  NN  E. d  e.  NN  A. m  e.  ( 0 ... ( K  - 
1 ) ) ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) ) )
744, 56, 733bitrd 283 1  |-  ( ph  ->  ( K MonoAP  F  <->  E. c  e.  R  E. a  e.  NN  E. d  e.  NN  A. m  e.  ( 0 ... ( K  -  1 ) ) ( a  +  ( m  x.  d
) )  e.  ( `' F " { c } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3045    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968   class class class wbr 4402   `'ccnv 4833   "cima 4837   -->wf 5578   ` cfv 5582  (class class class)co 6290   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    - cmin 9860   NNcn 10609   NN0cn0 10869   ...cfz 11784  APcvdwa 14915   MonoAP cvdwm 14916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-vdwap 14918  df-vdwmc 14919
This theorem is referenced by:  vdw  14944
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