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Theorem vdwmc2 14509
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 14508 . 2  |-  ( ph  ->  ( K MonoAP  F  <->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
5 vdwapid1 14505 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  a  e.  NN  /\  d  e.  NN )  ->  a  e.  ( a (AP `  K ) d ) )
6 ne0i 3799 . . . . . . . . . . . 12  |-  ( a  e.  ( a (AP
`  K ) d )  ->  ( a
(AP `  K )
d )  =/=  (/) )
75, 6syl 16 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  a  e.  NN  /\  d  e.  NN )  ->  (
a (AP `  K
) d )  =/=  (/) )
873expb 1197 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  ( a  e.  NN  /\  d  e.  NN ) )  ->  ( a
(AP `  K )
d )  =/=  (/) )
98adantll 713 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( a
(AP `  K )
d )  =/=  (/) )
10 ssn0 3827 . . . . . . . . . 10  |-  ( ( ( a (AP `  K ) d ) 
C_  ( `' F " { c } )  /\  ( a (AP
`  K ) d )  =/=  (/) )  -> 
( `' F " { c } )  =/=  (/) )
1110expcom 435 . . . . . . . . 9  |-  ( ( a (AP `  K
) d )  =/=  (/)  ->  ( ( a (AP `  K ) d )  C_  ( `' F " { c } )  ->  ( `' F " { c } )  =/=  (/) ) )
129, 11syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( (
a (AP `  K
) d )  C_  ( `' F " { c } )  ->  ( `' F " { c } )  =/=  (/) ) )
13 disjsn 4092 . . . . . . . . . 10  |-  ( ( R  i^i  { c } )  =  (/)  <->  -.  c  e.  R )
143adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  K  e.  NN )  ->  F : X
--> R )
15 fimacnvdisj 5769 . . . . . . . . . . . . 13  |-  ( ( F : X --> R  /\  ( R  i^i  { c } )  =  (/) )  ->  ( `' F " { c } )  =  (/) )
1615ex 434 . . . . . . . . . . . 12  |-  ( F : X --> R  -> 
( ( R  i^i  { c } )  =  (/)  ->  ( `' F " { c } )  =  (/) ) )
1714, 16syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  K  e.  NN )  ->  ( ( R  i^i  { c } )  =  (/)  ->  ( `' F " { c } )  =  (/) ) )
1817adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( ( R  i^i  { c } )  =  (/)  ->  ( `' F " { c } )  =  (/) ) )
1913, 18syl5bir 218 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  NN )  /\  (
a  e.  NN  /\  d  e.  NN )
)  ->  ( -.  c  e.  R  ->  ( `' F " { c } )  =  (/) ) )
2019necon1ad 2673 . . . . . . . 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 2956 . . . . . 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 635 . . . . 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 1715 . . . 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 2813 . . . 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 263 . . 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 6033 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  R )
29 ne0i 3799 . . . . . . . 8  |-  ( ( F `  A )  e.  R  ->  R  =/=  (/) )
3028, 29syl 16 . . . . . . 7  |-  ( ph  ->  R  =/=  (/) )
3130adantr 465 . . . . . 6  |-  ( (
ph  /\  K  = 
0 )  ->  R  =/=  (/) )
32 1nn 10567 . . . . . . . . 9  |-  1  e.  NN
3332ne0ii 3800 . . . . . . . 8  |-  NN  =/=  (/)
34 simpllr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  K  =  0 )
3534fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  (AP `  K
)  =  (AP ` 
0 ) )
3635oveqd 6313 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  =  ( a (AP `  0 ) d ) )
37 vdwap0 14506 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ( a (AP ` 
0 ) d )  =  (/) )
3837adantll 713 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  0 ) d )  =  (/) )
3936, 38eqtrd 2498 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  =  (/) )
40 0ss 3823 . . . . . . . . . . . 12  |-  (/)  C_  ( `' F " { c } )
4139, 40syl6eqss 3549 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
4241ralrimiva 2871 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  ->  A. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
43 r19.2z 3921 . . . . . . . . . 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 663 . . . . . . . . 9  |-  ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  ->  E. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
4544ralrimiva 2871 . . . . . . . 8  |-  ( (
ph  /\  K  = 
0 )  ->  A. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
46 r19.2z 3921 . . . . . . . 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 663 . . . . . . 7  |-  ( (
ph  /\  K  = 
0 )  ->  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
4847ralrimivw 2872 . . . . . 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 3921 . . . . . 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 661 . . . . 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 2914 . . . . 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 16 . . . 4  |-  ( (
ph  /\  K  = 
0 )  ->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) )
5352, 502thd 240 . . 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 10818 . . . 4  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
552, 54sylib 196 . . 3  |-  ( ph  ->  ( K  e.  NN  \/  K  =  0
) )
5626, 53, 55mpjaodan 786 . 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 14503 . . . . . . . . 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 1197 . . . . . . . 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 471 . . . . . . 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 317 . . . . . 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 1714 . . . . 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 3488 . . . . 5  |-  ( ( a (AP `  K
) d )  C_  ( `' F " { c } )  <->  A. x
( x  e.  ( a (AP `  K
) d )  ->  x  e.  ( `' F " { c } ) ) )
63 ralcom4 3128 . . . . . 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 6324 . . . . . . . 8  |-  ( a  +  ( m  x.  d ) )  e. 
_V
65 eleq1 2529 . . . . . . . 8  |-  ( x  =  ( a  +  ( m  x.  d
) )  ->  (
x  e.  ( `' F " { c } )  <->  ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) ) )
6664, 65ceqsalv 3137 . . . . . . 7  |-  ( A. x ( x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) )  <->  ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) )
6766ralbii 2888 . . . . . 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 2937 . . . . . . 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 1641 . . . . . 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 275 . . . . 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 288 . . . 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 2974 . . 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 2968 . 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 279 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 184    \/ wo 368    /\ wa 369    /\ w3a 973   A.wal 1393    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   class class class wbr 4456   `'ccnv 5007   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824   NNcn 10556   NN0cn0 10816   ...cfz 11697  APcvdwa 14495   MonoAP cvdwm 14496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-vdwap 14498  df-vdwmc 14499
This theorem is referenced by:  vdw  14524
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