MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vdwmc Structured version   Visualization version   Unicode version

Theorem vdwmc 14928
Description: The predicate " The  <. R ,  N >.-coloring  F contains a monochromatic AP of length 
K". (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 )
Assertion
Ref Expression
vdwmc  |-  ( ph  ->  ( K MonoAP  F  <->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
Distinct variable groups:    a, c,
d, F    K, a,
c, d    ph, c
Allowed substitution hints:    ph( a, d)    R( a, c, d)    X( a, c, d)

Proof of Theorem vdwmc
Dummy variables  f 
k  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdwmc.2 . . 3  |-  ( ph  ->  K  e.  NN0 )
2 vdwmc.3 . . . 4  |-  ( ph  ->  F : X --> R )
3 vdwmc.1 . . . 4  |-  X  e. 
_V
4 fex 6138 . . . 4  |-  ( ( F : X --> R  /\  X  e.  _V )  ->  F  e.  _V )
52, 3, 4sylancl 668 . . 3  |-  ( ph  ->  F  e.  _V )
6 fveq2 5865 . . . . . . . 8  |-  ( k  =  K  ->  (AP `  k )  =  (AP
`  K ) )
76rneqd 5062 . . . . . . 7  |-  ( k  =  K  ->  ran  (AP `  k )  =  ran  (AP `  K
) )
8 cnveq 5008 . . . . . . . . 9  |-  ( f  =  F  ->  `' f  =  `' F
)
98imaeq1d 5167 . . . . . . . 8  |-  ( f  =  F  ->  ( `' f " {
c } )  =  ( `' F " { c } ) )
109pweqd 3956 . . . . . . 7  |-  ( f  =  F  ->  ~P ( `' f " {
c } )  =  ~P ( `' F " { c } ) )
117, 10ineqan12d 3636 . . . . . 6  |-  ( ( k  =  K  /\  f  =  F )  ->  ( ran  (AP `  k )  i^i  ~P ( `' f " {
c } ) )  =  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) ) )
1211neeq1d 2683 . . . . 5  |-  ( ( k  =  K  /\  f  =  F )  ->  ( ( ran  (AP `  k )  i^i  ~P ( `' f " {
c } ) )  =/=  (/)  <->  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
1312exbidv 1768 . . . 4  |-  ( ( k  =  K  /\  f  =  F )  ->  ( E. c ( ran  (AP `  k
)  i^i  ~P ( `' f " {
c } ) )  =/=  (/)  <->  E. c ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
14 df-vdwmc 14919 . . . 4  |- MonoAP  =  { <. k ,  f >.  |  E. c ( ran  (AP `  k )  i^i  ~P ( `' f " { c } ) )  =/=  (/) }
1513, 14brabga 4715 . . 3  |-  ( ( K  e.  NN0  /\  F  e.  _V )  ->  ( K MonoAP  F  <->  E. c
( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
161, 5, 15syl2anc 667 . 2  |-  ( ph  ->  ( K MonoAP  F  <->  E. c
( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
17 vdwapf 14922 . . . . 5  |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN ) --> ~P NN )
18 ffn 5728 . . . . 5  |-  ( (AP
`  K ) : ( NN  X.  NN )
--> ~P NN  ->  (AP `  K )  Fn  ( NN  X.  NN ) )
19 selpw 3958 . . . . . . 7  |-  ( z  e.  ~P ( `' F " { c } )  <->  z  C_  ( `' F " { c } ) )
20 sseq1 3453 . . . . . . 7  |-  ( z  =  ( (AP `  K ) `  w
)  ->  ( z  C_  ( `' F " { c } )  <-> 
( (AP `  K
) `  w )  C_  ( `' F " { c } ) ) )
2119, 20syl5bb 261 . . . . . 6  |-  ( z  =  ( (AP `  K ) `  w
)  ->  ( z  e.  ~P ( `' F " { c } )  <-> 
( (AP `  K
) `  w )  C_  ( `' F " { c } ) ) )
2221rexrn 6024 . . . . 5  |-  ( (AP
`  K )  Fn  ( NN  X.  NN )  ->  ( E. z  e.  ran  (AP `  K
) z  e.  ~P ( `' F " { c } )  <->  E. w  e.  ( NN  X.  NN ) ( (AP `  K ) `  w
)  C_  ( `' F " { c } ) ) )
231, 17, 18, 224syl 19 . . . 4  |-  ( ph  ->  ( E. z  e. 
ran  (AP `  K
) z  e.  ~P ( `' F " { c } )  <->  E. w  e.  ( NN  X.  NN ) ( (AP `  K ) `  w
)  C_  ( `' F " { c } ) ) )
24 elin 3617 . . . . . 6  |-  ( z  e.  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  <->  ( z  e.  ran  (AP `  K
)  /\  z  e.  ~P ( `' F " { c } ) ) )
2524exbii 1718 . . . . 5  |-  ( E. z  z  e.  ( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  <->  E. z
( z  e.  ran  (AP `  K )  /\  z  e.  ~P ( `' F " { c } ) ) )
26 n0 3741 . . . . 5  |-  ( ( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) 
<->  E. z  z  e.  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) ) )
27 df-rex 2743 . . . . 5  |-  ( E. z  e.  ran  (AP `  K ) z  e. 
~P ( `' F " { c } )  <->  E. z ( z  e. 
ran  (AP `  K
)  /\  z  e.  ~P ( `' F " { c } ) ) )
2825, 26, 273bitr4ri 282 . . . 4  |-  ( E. z  e.  ran  (AP `  K ) z  e. 
~P ( `' F " { c } )  <-> 
( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) )
29 fveq2 5865 . . . . . . 7  |-  ( w  =  <. a ,  d
>.  ->  ( (AP `  K ) `  w
)  =  ( (AP
`  K ) `  <. a ,  d >.
) )
30 df-ov 6293 . . . . . . 7  |-  ( a (AP `  K ) d )  =  ( (AP `  K ) `
 <. a ,  d
>. )
3129, 30syl6eqr 2503 . . . . . 6  |-  ( w  =  <. a ,  d
>.  ->  ( (AP `  K ) `  w
)  =  ( a (AP `  K ) d ) )
3231sseq1d 3459 . . . . 5  |-  ( w  =  <. a ,  d
>.  ->  ( ( (AP
`  K ) `  w )  C_  ( `' F " { c } )  <->  ( a
(AP `  K )
d )  C_  ( `' F " { c } ) ) )
3332rexxp 4977 . . . 4  |-  ( E. w  e.  ( NN 
X.  NN ) ( (AP `  K ) `
 w )  C_  ( `' F " { c } )  <->  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
3423, 28, 333bitr3g 291 . . 3  |-  ( ph  ->  ( ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  =/=  (/) 
<->  E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
3534exbidv 1768 . 2  |-  ( ph  ->  ( E. c ( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) 
<->  E. c E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) ) )
3616, 35bitrd 257 1  |-  ( ph  ->  ( K MonoAP  F  <->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   E.wrex 2738   _Vcvv 3045    i^i cin 3403    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   {csn 3968   <.cop 3974   class class class wbr 4402    X. cxp 4832   `'ccnv 4833   ran crn 4835   "cima 4837    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   NNcn 10609   NN0cn0 10869  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:  vdwmc2  14929  vdwlem1  14931  vdwlem2  14932  vdwlem9  14939  vdwlem10  14940  vdwlem12  14942  vdwlem13  14943
  Copyright terms: Public domain W3C validator