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Theorem vdwmc2 15008
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 15007 . 2  |-  ( ph  ->  ( K MonoAP  F  <->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
5 vdwapid1 15004 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  a  e.  NN  /\  d  e.  NN )  ->  a  e.  ( a (AP `  K ) d ) )
6 ne0i 3728 . . . . . . . . . . . 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 1232 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  ( a  e.  NN  /\  d  e.  NN ) )  ->  ( a
(AP `  K )
d )  =/=  (/) )
98adantll 728 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( a
(AP `  K )
d )  =/=  (/) )
10 ssn0 3770 . . . . . . . . . 10  |-  ( ( ( a (AP `  K ) d ) 
C_  ( `' F " { c } )  /\  ( a (AP
`  K ) d )  =/=  (/) )  -> 
( `' F " { c } )  =/=  (/) )
1110expcom 442 . . . . . . . . 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 4023 . . . . . . . . . 10  |-  ( ( R  i^i  { c } )  =  (/)  <->  -.  c  e.  R )
143adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  K  e.  NN )  ->  F : X
--> R )
15 fimacnvdisj 5774 . . . . . . . . . . . . 13  |-  ( ( F : X --> R  /\  ( R  i^i  { c } )  =  (/) )  ->  ( `' F " { c } )  =  (/) )
1615ex 441 . . . . . . . . . . . 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 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( ( R  i^i  { c } )  =  (/)  ->  ( `' F " { c } )  =  (/) ) )
1913, 18syl5bir 226 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( -.  c  e.  R  ->  ( `' F " { c } )  =  (/) ) )
2019necon1ad 2660 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( ( `' F " { c } )  =/=  (/)  ->  c  e.  R ) )
2112, 20syld 44 . . . . . . 7  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( (
a (AP `  K
) d )  C_  ( `' F " { c } )  ->  c  e.  R ) )
2221rexlimdvva 2878 . . . . . 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 647 . . . . 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 1776 . . . 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 2762 . . . 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 271 . . 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 6038 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  R )
29 ne0i 3728 . . . . . . . 8  |-  ( ( F `  A )  e.  R  ->  R  =/=  (/) )
3028, 29syl 17 . . . . . . 7  |-  ( ph  ->  R  =/=  (/) )
3130adantr 472 . . . . . 6  |-  ( (
ph  /\  K  = 
0 )  ->  R  =/=  (/) )
32 1nn 10642 . . . . . . . . 9  |-  1  e.  NN
3332ne0ii 3729 . . . . . . . 8  |-  NN  =/=  (/)
34 simpllr 777 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  K  =  0 )
3534fveq2d 5883 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  (AP `  K
)  =  (AP ` 
0 ) )
3635oveqd 6325 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  =  ( a (AP `  0 ) d ) )
37 vdwap0 15005 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ( a (AP ` 
0 ) d )  =  (/) )
3837adantll 728 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  0 ) d )  =  (/) )
3936, 38eqtrd 2505 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  =  (/) )
40 0ss 3766 . . . . . . . . . . . 12  |-  (/)  C_  ( `' F " { c } )
4139, 40syl6eqss 3468 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
4241ralrimiva 2809 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  ->  A. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
43 r19.2z 3849 . . . . . . . . . 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 676 . . . . . . . . 9  |-  ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  ->  E. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
4544ralrimiva 2809 . . . . . . . 8  |-  ( (
ph  /\  K  = 
0 )  ->  A. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
46 r19.2z 3849 . . . . . . . 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 676 . . . . . . 7  |-  ( (
ph  /\  K  = 
0 )  ->  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
4847ralrimivw 2810 . . . . . 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 3849 . . . . . 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 673 . . . . 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 2843 . . . . 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 248 . . 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 10895 . . . 4  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
552, 54sylib 201 . . 3  |-  ( ph  ->  ( K  e.  NN  \/  K  =  0
) )
5626, 53, 55mpjaodan 803 . 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 15002 . . . . . . . . 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 1232 . . . . . . . 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 479 . . . . . . 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 324 . . . . . 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 1775 . . . . 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 3407 . . . . 5  |-  ( ( a (AP `  K
) d )  C_  ( `' F " { c } )  <->  A. x
( x  e.  ( a (AP `  K
) d )  ->  x  e.  ( `' F " { c } ) ) )
63 ralcom4 3052 . . . . . 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 6336 . . . . . . . 8  |-  ( a  +  ( m  x.  d ) )  e. 
_V
65 eleq1 2537 . . . . . . . 8  |-  ( x  =  ( a  +  ( m  x.  d
) )  ->  (
x  e.  ( `' F " { c } )  <->  ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) ) )
6664, 65ceqsalv 3061 . . . . . . 7  |-  ( A. x ( x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) )  <->  ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) )
6766ralbii 2823 . . . . . 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 2863 . . . . . . 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 1699 . . . . . 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 283 . . . . 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 296 . . . 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 2896 . . 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 2892 . 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 287 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 189    \/ wo 375    /\ wa 376    /\ w3a 1007   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   {csn 3959   class class class wbr 4395   `'ccnv 4838   "cima 4842   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    - cmin 9880   NNcn 10631   NN0cn0 10893   ...cfz 11810  APcvdwa 14994   MonoAP cvdwm 14995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-vdwap 14997  df-vdwmc 14998
This theorem is referenced by:  vdw  15023
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