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Theorem vdwmc2 14345
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 14344 . 2  |-  ( ph  ->  ( K MonoAP  F  <->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
5 vdwapid1 14341 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  a  e.  NN  /\  d  e.  NN )  ->  a  e.  ( a (AP `  K ) d ) )
6 ne0i 3784 . . . . . . . . . . . 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 1192 . . . . . . . . . 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 3811 . . . . . . . . . 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 4081 . . . . . . . . . 10  |-  ( ( R  i^i  { c } )  =  (/)  <->  -.  c  e.  R )
143adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  K  e.  NN )  ->  F : X
--> R )
15 fimacnvdisj 5754 . . . . . . . . . . . . 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 2676 . . . . . . . 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 2955 . . . . . 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 1685 . . . 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 6013 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  R )
29 ne0i 3784 . . . . . . . 8  |-  ( ( F `  A )  e.  R  ->  R  =/=  (/) )
3028, 29syl 16 . . . . . . 7  |-  ( ph  ->  R  =/=  (/) )
3130adantr 465 . . . . . 6  |-  ( (
ph  /\  K  = 
0 )  ->  R  =/=  (/) )
32 1nn 10536 . . . . . . . . 9  |-  1  e.  NN
33 ne0i 3784 . . . . . . . . 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 5861 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  (AP `  K
)  =  (AP ` 
0 ) )
3736oveqd 6292 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  =  ( a (AP `  0 ) d ) )
38 vdwap0 14342 . . . . . . . . . . . . . 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 2501 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  =  (/) )
41 0ss 3807 . . . . . . . . . . . 12  |-  (/)  C_  ( `' F " { c } )
4240, 41syl6eqss 3547 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  /\  d  e.  NN )  ->  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
4342ralrimiva 2871 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  =  0 )  /\  a  e.  NN )  ->  A. d  e.  NN  ( a (AP `  K ) d ) 
C_  ( `' F " { c } ) )
44 r19.2z 3910 . . . . . . . . . 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 2871 . . . . . . . 8  |-  ( (
ph  /\  K  = 
0 )  ->  A. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
47 r19.2z 3910 . . . . . . . 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 2872 . . . . . 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 3910 . . . . . 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 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 } ) )
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 10786 . . . 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 14339 . . . . . . . . 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 1192 . . . . . . . 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 1684 . . . . 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 3486 . . . . 5  |-  ( ( a (AP `  K
) d )  C_  ( `' F " { c } )  <->  A. x
( x  e.  ( a (AP `  K
) d )  ->  x  e.  ( `' F " { c } ) ) )
64 ralcom4 3125 . . . . . 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 6300 . . . . . . . 8  |-  ( a  +  ( m  x.  d ) )  e. 
_V
66 eleq1 2532 . . . . . . . 8  |-  ( x  =  ( a  +  ( m  x.  d
) )  ->  (
x  e.  ( `' F " { c } )  <->  ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) ) )
6765, 66ceqsalv 3134 . . . . . . 7  |-  ( A. x ( x  =  ( a  +  ( m  x.  d ) )  ->  x  e.  ( `' F " { c } ) )  <->  ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } ) )
6867ralbii 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 } ) )
69 r19.23v 2936 . . . . . . 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 1615 . . . . . 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 2972 . . 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 2966 . 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 968   A.wal 1372    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   _Vcvv 3106    i^i cin 3468    C_ wss 3469   (/)c0 3778   {csn 4020   class class class wbr 4440   `'ccnv 4991   "cima 4995   -->wf 5575   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9794   NNcn 10525   NN0cn0 10784   ...cfz 11661  APcvdwa 14331   MonoAP cvdwm 14332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-vdwap 14334  df-vdwmc 14335
This theorem is referenced by:  vdw  14360
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