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Theorem fiblem 29224
Description: Lemma for fib0 29225, fib1 29226 and fibp1 29227. (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 12978 . . . . . . 7  |-  ( # `  <" 0 1 "> )  =  2
21eqcomi 2459 . . . . . 6  |-  2  =  ( # `  <" 0 1 "> )
32fveq2i 5866 . . . . 5  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( # `  <" 0 1 "> ) )
43imaeq2i 5165 . . . 4  |-  ( `' # " ( ZZ>= `  2
) )  =  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) )
54ineq2i 3630 . . 3  |-  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  2 ) ) )  =  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) ) )
6 eqid 2450 . . 3  |-  ( ( w `  ( (
# `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) )  =  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) )
75, 6mpteq12i 4486 . 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 3616 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  <->  ( w  e. Word  NN0  /\  w  e.  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) ) ) )
98simplbi 462 . . . . 5  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  ->  w  e. Word  NN0 )
10 wrdf 12673 . . . . 5  |-  ( w  e. Word  NN0  ->  w :
( 0..^ ( # `  w ) ) --> NN0 )
119, 10syl 17 . . . 4  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  ->  w : ( 0..^ (
# `  w )
) --> NN0 )
128simprbi 466 . . . . . . . . 9  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  ->  w  e.  ( `' #
" ( ZZ>= `  ( # `
 <" 0 1 "> ) ) ) )
13 hashf 12519 . . . . . . . . . 10  |-  # : _V
--> ( NN0  u.  { +oo } )
14 ffn 5726 . . . . . . . . . 10  |-  ( # : _V --> ( NN0  u.  { +oo } )  ->  #  Fn  _V )
15 elpreima 6000 . . . . . . . . . 10  |-  ( #  Fn  _V  ->  ( w  e.  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) )  <->  ( w  e.  _V  /\  ( # `  w )  e.  (
ZZ>= `  ( # `  <" 0 1 "> ) ) ) ) )
1613, 14, 15mp2b 10 . . . . . . . . 9  |-  ( w  e.  ( `' # " ( ZZ>= `  ( # `  <" 0 1 "> ) ) )  <->  ( w  e.  _V  /\  ( # `  w )  e.  (
ZZ>= `  ( # `  <" 0 1 "> ) ) ) )
1712, 16sylib 200 . . . . . . . 8  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( w  e.  _V  /\  ( # `  w
)  e.  ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )
1817simprd 465 . . . . . . 7  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  ( # `
 <" 0 1 "> ) ) )
1918, 3syl6eleqr 2539 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  2
) )
20 uznn0sub 11187 . . . . . 6  |-  ( (
# `  w )  e.  ( ZZ>= `  2 )  ->  ( ( # `  w
)  -  2 )  e.  NN0 )
2119, 20syl 17 . . . . 5  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  2 )  e.  NN0 )
22 1zzd 10965 . . . . . . 7  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
1  e.  ZZ )
23 1p1e2 10720 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
2423fveq2i 5866 . . . . . . . 8  |-  ( ZZ>= `  ( 1  +  1 ) )  =  (
ZZ>= `  2 )
2519, 24syl6eleqr 2539 . . . . . . 7  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  (
1  +  1 ) ) )
26 peano2uzr 11211 . . . . . . 7  |-  ( ( 1  e.  ZZ  /\  ( # `  w )  e.  ( ZZ>= `  (
1  +  1 ) ) )  ->  ( # `
 w )  e.  ( ZZ>= `  1 )
)
2722, 25, 26syl2anc 666 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  1
) )
28 nnuz 11191 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
2927, 28syl6eleqr 2539 . . . . 5  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  NN )
3029nnred 10621 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  RR )
31 2rp 11304 . . . . . . 7  |-  2  e.  RR+
3231a1i 11 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
2  e.  RR+ )
3330, 32ltsubrpd 11367 . . . . 5  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  2 )  <  ( # `  w
) )
34 elfzo0 11953 . . . . 5  |-  ( ( ( # `  w
)  -  2 )  e.  ( 0..^ (
# `  w )
)  <->  ( ( (
# `  w )  -  2 )  e. 
NN0  /\  ( # `  w
)  e.  NN  /\  ( ( # `  w
)  -  2 )  <  ( # `  w
) ) )
3521, 29, 33, 34syl3anbrc 1191 . . . 4  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  2 )  e.  ( 0..^ (
# `  w )
) )
3611, 35ffvelrnd 6021 . . 3  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( w `  (
( # `  w )  -  2 ) )  e.  NN0 )
37 fzo0end 12000 . . . . 5  |-  ( (
# `  w )  e.  NN  ->  ( ( # `
 w )  - 
1 )  e.  ( 0..^ ( # `  w
) ) )
3829, 37syl 17 . . . 4  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  1 )  e.  ( 0..^ (
# `  w )
) )
3911, 38ffvelrnd 6021 . . 3  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( w `  (
( # `  w )  -  1 ) )  e.  NN0 )
4036, 39nn0addcld 10926 . 2  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( w `  ( ( # `  w
)  -  2 ) )  +  ( w `
 ( ( # `  w )  -  1 ) ) )  e. 
NN0 )
417, 40fmpti 6043 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 188    /\ wa 371    e. wcel 1886   _Vcvv 3044    u. cun 3401    i^i cin 3402   {csn 3967   class class class wbr 4401    |-> cmpt 4460   `'ccnv 4832   "cima 4836    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6288   0cc0 9536   1c1 9537    + caddc 9539   +oocpnf 9669    < clt 9672    - cmin 9857   NNcn 10606   2c2 10656   NN0cn0 10866   ZZcz 10934   ZZ>=cuz 11156   RR+crp 11299  ..^cfzo 11912   #chash 12512  Word cword 12653   <"cs2 12932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-fz 11782  df-fzo 11913  df-hash 12513  df-word 12661  df-concat 12663  df-s1 12664  df-s2 12939
This theorem is referenced by:  fib0  29225  fib1  29226  fibp1  29227
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