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Theorem vdwmc2 14159
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 14158 . 2  |-  ( ph  ->  ( K MonoAP  F  <->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
5 vdwapid1 14155 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  a  e.  NN  /\  d  e.  NN )  ->  a  e.  ( a (AP `  K ) d ) )
6 ne0i 3752 . . . . . . . . . . . 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 1189 . . . . . . . . . 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 3779 . . . . . . . . . 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 4045 . . . . . . . . . 10  |-  ( ( R  i^i  { c } )  =  (/)  <->  -.  c  e.  R )
143adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  K  e.  NN )  ->  F : X
--> R )
15 fimacnvdisj 5698 . . . . . . . . . . . . 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 2668 . . . . . . . 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 2954 . . . . . 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 1681 . . . 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 2805 . . . 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 5954 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  R )
29 ne0i 3752 . . . . . . . 8  |-  ( ( F `  A )  e.  R  ->  R  =/=  (/) )
3028, 29syl 16 . . . . . . 7  |-  ( ph  ->  R  =/=  (/) )
3130adantr 465 . . . . . 6  |-  ( (
ph  /\  K  = 
0 )  ->  R  =/=  (/) )
32 1nn 10445 . . . . . . . . 9  |-  1  e.  NN
33 ne0i 3752 . . . . . . . . 9  |-  ( 1  e.  NN  ->  NN  =/=  (/) )
3432, 33ax-mp 5 . . . . . . . 8  |-  NN  =/=  (/)
35 simpllr 758 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  K  =  0 )
3635fveq2d 5804 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  (AP `  K
)  =  (AP ` 
0 ) )
3736oveqd 6218 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  =  ( a (AP `  0 ) d ) )
38 vdwap0 14156 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ( a (AP ` 
0 ) d )  =  (/) )
3938adantll 713 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  0 ) d )  =  (/) )
4037, 39eqtrd 2495 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  =  (/) )
41 0ss 3775 . . . . . . . . . . . 12  |-  (/)  C_  ( `' F " { c } )
4240, 41syl6eqss 3515 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
4342ralrimiva 2830 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  ->  A. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
44 r19.2z 3878 . . . . . . . . . 10  |-  ( ( NN  =/=  (/)  /\  A. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) )  ->  E. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
4534, 43, 44sylancr 663 . . . . . . . . 9  |-  ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  ->  E. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
4645ralrimiva 2830 . . . . . . . 8  |-  ( (
ph  /\  K  = 
0 )  ->  A. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
47 r19.2z 3878 . . . . . . . 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 } ) )
4834, 46, 47sylancr 663 . . . . . . 7  |-  ( (
ph  /\  K  = 
0 )  ->  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
4948ralrimivw 2831 . . . . . 6  |-  ( (
ph  /\  K  = 
0 )  ->  A. c  e.  R  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
50 r19.2z 3878 . . . . . 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 } ) )
5131, 49, 50syl2anc 661 . . . . 5  |-  ( (
ph  /\  K  = 
0 )  ->  E. c  e.  R  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
52 rexex 2893 . . . . 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 } ) )
5351, 52syl 16 . . . 4  |-  ( (
ph  /\  K  = 
0 )  ->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) )
5453, 512thd 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 } ) ) )
55 elnn0 10693 . . . 4  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
562, 55sylib 196 . . 3  |-  ( ph  ->  ( K  e.  NN  \/  K  =  0
) )
5726, 54, 56mpjaodan 784 . 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 } ) ) )
58 vdwapval 14153 . . . . . . . . 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 ) ) ) )
59583expb 1189 . . . . . . . 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
) ) ) )
602, 59sylan 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 ) ) ) )
6160imbi1d 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 } ) ) ) )
6261albidv 1680 . . . . 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 } ) ) ) )
63 dfss2 3454 . . . . 5  |-  ( ( a (AP `  K
) d )  C_  ( `' F " { c } )  <->  A. x
( x  e.  ( a (AP `  K
) d )  ->  x  e.  ( `' F " { c } ) ) )
64 ralcom4 3097 . . . . . 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 } ) ) )
65 ovex 6226 . . . . . . . 8  |-  ( a  +  ( m  x.  d ) )  e. 
_V
66 eleq1 2526 . . . . . . . 8  |-  ( x  =  ( a  +  ( m  x.  d
) )  ->  (
x  e.  ( `' F " { c } )  <->  ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) ) )
6765, 66ceqsalv 3106 . . . . . . 7  |-  ( A. x ( x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) )  <->  ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) )
6867ralbii 2839 . . . . . 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 } ) )
69 r19.23v 2939 . . . . . . 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 } ) ) )
7069albii 1611 . . . . . 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 } ) ) )
7164, 68, 703bitr3i 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 } ) ) )
7262, 63, 713bitr4g 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 } ) ) )
73722rexbidva 2874 . . 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 } ) ) )
7473rexbidv 2868 . 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 } ) ) )
754, 57, 743bitrd 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 965   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   _Vcvv 3078    i^i cin 3436    C_ wss 3437   (/)c0 3746   {csn 3986   class class class wbr 4401   `'ccnv 4948   "cima 4952   -->wf 5523   ` cfv 5527  (class class class)co 6201   0cc0 9394   1c1 9395    + caddc 9397    x. cmul 9399    - cmin 9707   NNcn 10434   NN0cn0 10691   ...cfz 11555  APcvdwa 14145   MonoAP cvdwm 14146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-vdwap 14148  df-vdwmc 14149
This theorem is referenced by:  vdw  14174
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