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Theorem fiblem 27963
Description: Lemma for fib0 27964, fib1 27965 and fibp1 27966. (Contributed by Thierry Arnoux, 25-Apr-2019.)
Assertion
Ref Expression
fiblem  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= ` 
2 ) ) ) 
|->  ( ( w `  ( ( # `  w
)  -  2 ) )  +  ( w `
 ( ( # `  w )  -  1 ) ) ) ) : (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) ) --> NN0

Proof of Theorem fiblem
StepHypRef Expression
1 s2len 12802 . . . . . . 7  |-  ( # `  <" 0 1 "> )  =  2
21eqcomi 2473 . . . . . 6  |-  2  =  ( # `  <" 0 1 "> )
32fveq2i 5860 . . . . 5  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( # `  <" 0 1 "> ) )
43imaeq2i 5326 . . . 4  |-  ( `' # " ( ZZ>= `  2
) )  =  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) )
54ineq2i 3690 . . 3  |-  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  2 ) ) )  =  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) ) )
6 eqid 2460 . . 3  |-  ( ( w `  ( (
# `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) )  =  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) )
75, 6mpteq12i 4524 . 2  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= ` 
2 ) ) ) 
|->  ( ( w `  ( ( # `  w
)  -  2 ) )  +  ( w `
 ( ( # `  w )  -  1 ) ) ) )  =  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) )
8 elin 3680 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  <->  ( w  e. Word  NN0  /\  w  e.  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) ) ) )
98simplbi 460 . . . . 5  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  ->  w  e. Word  NN0 )
10 wrdf 12506 . . . . 5  |-  ( w  e. Word  NN0  ->  w :
( 0..^ ( # `  w ) ) --> NN0 )
119, 10syl 16 . . . 4  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  ->  w : ( 0..^ (
# `  w )
) --> NN0 )
128simprbi 464 . . . . . . . . . 10  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  ->  w  e.  ( `' #
" ( ZZ>= `  ( # `
 <" 0 1 "> ) ) ) )
13 hashf 12367 . . . . . . . . . . 11  |-  # : _V
--> ( NN0  u.  { +oo } )
14 ffn 5722 . . . . . . . . . . 11  |-  ( # : _V --> ( NN0  u.  { +oo } )  ->  #  Fn  _V )
15 elpreima 5992 . . . . . . . . . . 11  |-  ( #  Fn  _V  ->  ( w  e.  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) )  <->  ( w  e.  _V  /\  ( # `  w )  e.  (
ZZ>= `  ( # `  <" 0 1 "> ) ) ) ) )
1613, 14, 15mp2b 10 . . . . . . . . . 10  |-  ( w  e.  ( `' # " ( ZZ>= `  ( # `  <" 0 1 "> ) ) )  <->  ( w  e.  _V  /\  ( # `  w )  e.  (
ZZ>= `  ( # `  <" 0 1 "> ) ) ) )
1712, 16sylib 196 . . . . . . . . 9  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( w  e.  _V  /\  ( # `  w
)  e.  ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )
1817simprd 463 . . . . . . . 8  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  ( # `
 <" 0 1 "> ) ) )
1918, 3syl6eleqr 2559 . . . . . . 7  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  2
) )
20 uznn0sub 11102 . . . . . . 7  |-  ( (
# `  w )  e.  ( ZZ>= `  2 )  ->  ( ( # `  w
)  -  2 )  e.  NN0 )
2119, 20syl 16 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  2 )  e.  NN0 )
22 1zzd 10884 . . . . . . . 8  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
1  e.  ZZ )
23 1p1e2 10638 . . . . . . . . . 10  |-  ( 1  +  1 )  =  2
2423fveq2i 5860 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  +  1 ) )  =  (
ZZ>= `  2 )
2519, 24syl6eleqr 2559 . . . . . . . 8  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  (
1  +  1 ) ) )
26 peano2uzr 11125 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  ( # `  w )  e.  ( ZZ>= `  (
1  +  1 ) ) )  ->  ( # `
 w )  e.  ( ZZ>= `  1 )
)
2722, 25, 26syl2anc 661 . . . . . . 7  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  1
) )
28 nnuz 11106 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2927, 28syl6eleqr 2559 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  NN )
3029nnred 10540 . . . . . . 7  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  RR )
31 2rp 11214 . . . . . . . 8  |-  2  e.  RR+
3231a1i 11 . . . . . . 7  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
2  e.  RR+ )
3330, 32ltsubrpd 11273 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  2 )  <  ( # `  w
) )
3421, 29, 333jca 1171 . . . . 5  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( ( # `  w )  -  2 )  e.  NN0  /\  ( # `  w )  e.  NN  /\  (
( # `  w )  -  2 )  < 
( # `  w ) ) )
35 elfzo0 11820 . . . . 5  |-  ( ( ( # `  w
)  -  2 )  e.  ( 0..^ (
# `  w )
)  <->  ( ( (
# `  w )  -  2 )  e. 
NN0  /\  ( # `  w
)  e.  NN  /\  ( ( # `  w
)  -  2 )  <  ( # `  w
) ) )
3634, 35sylibr 212 . . . 4  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  2 )  e.  ( 0..^ (
# `  w )
) )
3711, 36ffvelrnd 6013 . . 3  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( w `  (
( # `  w )  -  2 ) )  e.  NN0 )
38 fzo0end 11861 . . . . 5  |-  ( (
# `  w )  e.  NN  ->  ( ( # `
 w )  - 
1 )  e.  ( 0..^ ( # `  w
) ) )
3929, 38syl 16 . . . 4  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  1 )  e.  ( 0..^ (
# `  w )
) )
4011, 39ffvelrnd 6013 . . 3  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( w `  (
( # `  w )  -  1 ) )  e.  NN0 )
4137, 40nn0addcld 10845 . 2  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( w `  ( ( # `  w
)  -  2 ) )  +  ( w `
 ( ( # `  w )  -  1 ) ) )  e. 
NN0 )
427, 41fmpti 6035 1  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= ` 
2 ) ) ) 
|->  ( ( w `  ( ( # `  w
)  -  2 ) )  +  ( w `
 ( ( # `  w )  -  1 ) ) ) ) : (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) ) --> NN0
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 968    e. wcel 1762   _Vcvv 3106    u. cun 3467    i^i cin 3468   {csn 4020   class class class wbr 4440    |-> cmpt 4498   `'ccnv 4991   "cima 4995    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484   +oocpnf 9614    < clt 9617    - cmin 9794   NNcn 10525   2c2 10574   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   RR+crp 11209  ..^cfzo 11781   #chash 12360  Word cword 12487   <"cs2 12756
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-int 4276  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-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  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-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-concat 12497  df-s1 12498  df-s2 12763
This theorem is referenced by:  fib0  27964  fib1  27965  fibp1  27966
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