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Theorem nconstwlpolem 15085
Description: Lemma for nconstwlpo 15086. (Contributed by Jim Kingdon, 23-Jul-2024.)
Hypotheses
Ref Expression
nconstwlpo.f  |-  ( ph  ->  F : RR --> ZZ )
nconstwlpo.0  |-  ( ph  ->  ( F `  0
)  =  0 )
nconstwlpo.rp  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( F `  x )  =/=  0
)
nconstwlpo.g  |-  ( ph  ->  G : NN --> { 0 ,  1 } )
nconstwlpo.a  |-  A  = 
sum_ i  e.  NN  ( ( 1  / 
( 2 ^ i
) )  x.  ( G `  i )
)
Assertion
Ref Expression
nconstwlpolem  |-  ( ph  ->  ( A. y  e.  NN  ( G `  y )  =  0  \/  -.  A. y  e.  NN  ( G `  y )  =  0 ) )
Distinct variable groups:    x, A    y, A    x, F    y, F    i, G, y    ph, x    ph, y, i
Allowed substitution hints:    A( i)    F( i)    G( x)

Proof of Theorem nconstwlpolem
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 breq2 4019 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
2 fveq2 5527 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
32neeq1d 2375 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
( F `  x
)  =/=  0  <->  ( F `  A )  =/=  0 ) )
41, 3imbi12d 234 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
( 0  <  x  ->  ( F `  x
)  =/=  0 )  <-> 
( 0  <  A  ->  ( F `  A
)  =/=  0 ) ) )
5 elrp 9668 . . . . . . . . . . . . . 14  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
6 nconstwlpo.rp . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( F `  x )  =/=  0
)
75, 6sylan2br 288 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  RR  /\  0  < 
x ) )  -> 
( F `  x
)  =/=  0 )
87expr 375 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  RR )  ->  ( 0  <  x  ->  ( F `  x )  =/=  0 ) )
98ralrimiva 2560 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  RR  ( 0  <  x  ->  ( F `  x
)  =/=  0 ) )
10 nconstwlpo.g . . . . . . . . . . . 12  |-  ( ph  ->  G : NN --> { 0 ,  1 } )
11 nconstwlpo.a . . . . . . . . . . . 12  |-  A  = 
sum_ i  e.  NN  ( ( 1  / 
( 2 ^ i
) )  x.  ( G `  i )
)
1210, 11trilpolemcl 15057 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
134, 9, 12rspcdva 2858 . . . . . . . . . 10  |-  ( ph  ->  ( 0  <  A  ->  ( F `  A
)  =/=  0 ) )
1413necon2bd 2415 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  A )  =  0  ->  -.  0  <  A ) )
1514imp 124 . . . . . . . 8  |-  ( (
ph  /\  ( F `  A )  =  0 )  ->  -.  0  <  A )
1610adantr 276 . . . . . . . . . . . 12  |-  ( (
ph  /\  E. y  e.  NN  ( G `  y )  =  1 )  ->  G : NN
--> { 0 ,  1 } )
17 simpr 110 . . . . . . . . . . . . 13  |-  ( (
ph  /\  E. y  e.  NN  ( G `  y )  =  1 )  ->  E. y  e.  NN  ( G `  y )  =  1 )
18 fveqeq2 5536 . . . . . . . . . . . . . 14  |-  ( y  =  a  ->  (
( G `  y
)  =  1  <->  ( G `  a )  =  1 ) )
1918cbvrexv 2716 . . . . . . . . . . . . 13  |-  ( E. y  e.  NN  ( G `  y )  =  1  <->  E. a  e.  NN  ( G `  a )  =  1 )
2017, 19sylib 122 . . . . . . . . . . . 12  |-  ( (
ph  /\  E. y  e.  NN  ( G `  y )  =  1 )  ->  E. a  e.  NN  ( G `  a )  =  1 )
2116, 11, 20nconstwlpolemgt0 15084 . . . . . . . . . . 11  |-  ( (
ph  /\  E. y  e.  NN  ( G `  y )  =  1 )  ->  0  <  A )
2221ex 115 . . . . . . . . . 10  |-  ( ph  ->  ( E. y  e.  NN  ( G `  y )  =  1  ->  0  <  A
) )
2322con3d 632 . . . . . . . . 9  |-  ( ph  ->  ( -.  0  < 
A  ->  -.  E. y  e.  NN  ( G `  y )  =  1 ) )
2423adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( F `  A )  =  0 )  ->  ( -.  0  <  A  ->  -.  E. y  e.  NN  ( G `  y )  =  1 ) )
2515, 24mpd 13 . . . . . . 7  |-  ( (
ph  /\  ( F `  A )  =  0 )  ->  -.  E. y  e.  NN  ( G `  y )  =  1 )
26 ralnex 2475 . . . . . . 7  |-  ( A. y  e.  NN  -.  ( G `  y )  =  1  <->  -.  E. y  e.  NN  ( G `  y )  =  1 )
2725, 26sylibr 134 . . . . . 6  |-  ( (
ph  /\  ( F `  A )  =  0 )  ->  A. y  e.  NN  -.  ( G `
 y )  =  1 )
2827r19.21bi 2575 . . . . 5  |-  ( ( ( ph  /\  ( F `  A )  =  0 )  /\  y  e.  NN )  ->  -.  ( G `  y )  =  1 )
2910ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  ( F `  A )  =  0 )  /\  y  e.  NN )  ->  G : NN --> { 0 ,  1 } )
30 simpr 110 . . . . . . 7  |-  ( ( ( ph  /\  ( F `  A )  =  0 )  /\  y  e.  NN )  ->  y  e.  NN )
3129, 30ffvelcdmd 5665 . . . . . 6  |-  ( ( ( ph  /\  ( F `  A )  =  0 )  /\  y  e.  NN )  ->  ( G `  y
)  e.  { 0 ,  1 } )
32 elpri 3627 . . . . . 6  |-  ( ( G `  y )  e.  { 0 ,  1 }  ->  (
( G `  y
)  =  0  \/  ( G `  y
)  =  1 ) )
3331, 32syl 14 . . . . 5  |-  ( ( ( ph  /\  ( F `  A )  =  0 )  /\  y  e.  NN )  ->  ( ( G `  y )  =  0  \/  ( G `  y )  =  1 ) )
3428, 33ecased 1359 . . . 4  |-  ( ( ( ph  /\  ( F `  A )  =  0 )  /\  y  e.  NN )  ->  ( G `  y
)  =  0 )
3534ralrimiva 2560 . . 3  |-  ( (
ph  /\  ( F `  A )  =  0 )  ->  A. y  e.  NN  ( G `  y )  =  0 )
3635orcd 734 . 2  |-  ( (
ph  /\  ( F `  A )  =  0 )  ->  ( A. y  e.  NN  ( G `  y )  =  0  \/  -.  A. y  e.  NN  ( G `  y )  =  0 ) )
3710adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  A. y  e.  NN  ( G `  y )  =  0 )  ->  G : NN
--> { 0 ,  1 } )
38 simpr 110 . . . . . . . . 9  |-  ( (
ph  /\  A. y  e.  NN  ( G `  y )  =  0 )  ->  A. y  e.  NN  ( G `  y )  =  0 )
3937, 11, 38nconstwlpolem0 15083 . . . . . . . 8  |-  ( (
ph  /\  A. y  e.  NN  ( G `  y )  =  0 )  ->  A  = 
0 )
4039fveq2d 5531 . . . . . . 7  |-  ( (
ph  /\  A. y  e.  NN  ( G `  y )  =  0 )  ->  ( F `  A )  =  ( F `  0 ) )
41 nconstwlpo.0 . . . . . . . 8  |-  ( ph  ->  ( F `  0
)  =  0 )
4241adantr 276 . . . . . . 7  |-  ( (
ph  /\  A. y  e.  NN  ( G `  y )  =  0 )  ->  ( F `  0 )  =  0 )
4340, 42eqtrd 2220 . . . . . 6  |-  ( (
ph  /\  A. y  e.  NN  ( G `  y )  =  0 )  ->  ( F `  A )  =  0 )
4443ex 115 . . . . 5  |-  ( ph  ->  ( A. y  e.  NN  ( G `  y )  =  0  ->  ( F `  A )  =  0 ) )
4544con3d 632 . . . 4  |-  ( ph  ->  ( -.  ( F `
 A )  =  0  ->  -.  A. y  e.  NN  ( G `  y )  =  0 ) )
4645imp 124 . . 3  |-  ( (
ph  /\  -.  ( F `  A )  =  0 )  ->  -.  A. y  e.  NN  ( G `  y )  =  0 )
4746olcd 735 . 2  |-  ( (
ph  /\  -.  ( F `  A )  =  0 )  -> 
( A. y  e.  NN  ( G `  y )  =  0  \/  -.  A. y  e.  NN  ( G `  y )  =  0 ) )
48 nconstwlpo.f . . . . 5  |-  ( ph  ->  F : RR --> ZZ )
4948, 12ffvelcdmd 5665 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  ZZ )
50 0z 9277 . . . 4  |-  0  e.  ZZ
51 zdceq 9341 . . . 4  |-  ( ( ( F `  A
)  e.  ZZ  /\  0  e.  ZZ )  -> DECID  ( F `  A )  =  0 )
5249, 50, 51sylancl 413 . . 3  |-  ( ph  -> DECID  ( F `  A )  =  0 )
53 exmiddc 837 . . 3  |-  (DECID  ( F `
 A )  =  0  ->  ( ( F `  A )  =  0  \/  -.  ( F `  A )  =  0 ) )
5452, 53syl 14 . 2  |-  ( ph  ->  ( ( F `  A )  =  0  \/  -.  ( F `
 A )  =  0 ) )
5536, 47, 54mpjaodan 799 1  |-  ( ph  ->  ( A. y  e.  NN  ( G `  y )  =  0  \/  -.  A. y  e.  NN  ( G `  y )  =  0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709  DECID wdc 835    = wceq 1363    e. wcel 2158    =/= wne 2357   A.wral 2465   E.wrex 2466   {cpr 3605   class class class wbr 4015   -->wf 5224   ` cfv 5228  (class class class)co 5888   RRcr 7823   0cc0 7824   1c1 7825    x. cmul 7829    < clt 8005    / cdiv 8642   NNcn 8932   2c2 8983   ZZcz 9266   RR+crp 9666   ^cexp 10532   sum_csu 11374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-frec 6405  df-1o 6430  df-oadd 6434  df-er 6548  df-en 6754  df-dom 6755  df-fin 6756  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-ico 9907  df-fz 10022  df-fzo 10156  df-seqfrec 10459  df-exp 10533  df-ihash 10769  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-clim 11300  df-sumdc 11375
This theorem is referenced by:  nconstwlpo  15086
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