Theorem fiblem 28534

Description: Lemma for fib0 28535, fib1 28536 and fibp1 28537. (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

Step	Hyp	Ref	Expression
1		s2len 12864	. . . . . . 7 |- ( # ` <" 0 1 "> ) = 2
2	1	eqcomi 2470	. . . . . 6 |- 2 = ( # ` <" 0 1 "> )
3	2	fveq2i 5875	. . . . 5 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( # ` <" 0 1 "> ) )
4	3	imaeq2i 5345	. . . 4 |- ( `' # " ( ZZ>= ` 2 ) ) = ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) )
5	4	ineq2i 3693	. . 3 |- (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) = (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
6		eqid 2457	. . 3 |- ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) = ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) )
7	5, 6	mpteq12i 4541	. 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 3683	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) <-> ( w e. Word NN0 /\ w e. ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) )
9	8	simplbi 460	. . . . 5 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> w e. Word NN0 )
10		wrdf 12558	. . . . 5 |- ( w e. Word NN0 -> w : ( 0..^ ( # ` w ) ) --> NN0 )
11	9, 10	syl 16	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> w : ( 0..^ ( # ` w ) ) --> NN0 )
12	8	simprbi 464	. . . . . . . . 9 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> w e. ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
13		hashf 12415	. . . . . . . . . 10 |- # : _V --> ( NN0 u. { +oo } )
14		ffn 5737	. . . . . . . . . 10 |- ( # : _V --> ( NN0 u. { +oo } ) -> # Fn _V )
15		elpreima 6008	. . . . . . . . . 10 |- ( # Fn _V -> ( w e. ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) <-> ( w e. _V /\ ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) )
16	13, 14, 15	mp2b 10	. . . . . . . . 9 |- ( w e. ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) <-> ( w e. _V /\ ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
17	12, 16	sylib 196	. . . . . . . 8 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( w e. _V /\ ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
18	17	simprd 463	. . . . . . 7 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) )
19	18, 3	syl6eleqr 2556	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` 2 ) )
20		uznn0sub 11137	. . . . . 6 |- ( ( # ` w ) e. ( ZZ>= ` 2 ) -> ( ( # ` w ) - 2 ) e. NN0 )
21	19, 20	syl 16	. . . . 5 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 2 ) e. NN0 )
22		1zzd 10916	. . . . . . 7 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> 1 e. ZZ )
23		1p1e2 10670	. . . . . . . . 9 |- ( 1 + 1 ) = 2
24	23	fveq2i 5875	. . . . . . . 8 |- ( ZZ>= ` ( 1 + 1 ) ) = ( ZZ>= ` 2 )
25	19, 24	syl6eleqr 2556	. . . . . . 7 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` ( 1 + 1 ) ) )
26		peano2uzr 11161	. . . . . . 7 |- ( ( 1 e. ZZ /\ ( # ` w ) e. ( ZZ>= ` ( 1 + 1 ) ) ) -> ( # ` w ) e. ( ZZ>= ` 1 ) )
27	22, 25, 26	syl2anc 661	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` 1 ) )
28		nnuz 11141	. . . . . 6 |- NN = ( ZZ>= ` 1 )
29	27, 28	syl6eleqr 2556	. . . . 5 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. NN )
30	29	nnred 10571	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. RR )
31		2rp 11250	. . . . . . 7 |- 2 e. RR+
32	31	a1i 11	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> 2 e. RR+ )
33	30, 32	ltsubrpd 11309	. . . . 5 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 2 ) < ( # ` w ) )
34		elfzo0 11862	. . . . 5 |- ( ( ( # ` w ) - 2 ) e. ( 0..^ ( # ` w ) ) <-> ( ( ( # ` w ) - 2 ) e. NN0 /\ ( # ` w ) e. NN /\ ( ( # ` w ) - 2 ) < ( # ` w ) ) )
35	21, 29, 33, 34	syl3anbrc 1180	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 2 ) e. ( 0..^ ( # ` w ) ) )
36	11, 35	ffvelrnd 6033	. . 3 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( w ` ( ( # ` w ) - 2 ) ) e. NN0 )
37		fzo0end 11907	. . . . 5 |- ( ( # ` w ) e. NN -> ( ( # ` w ) - 1 ) e. ( 0..^ ( # ` w ) ) )
38	29, 37	syl 16	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 1 ) e. ( 0..^ ( # ` w ) ) )
39	11, 38	ffvelrnd 6033	. . 3 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( w ` ( ( # ` w ) - 1 ) ) e. NN0 )
40	36, 39	nn0addcld 10877	. 2 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) e. NN0 )
41	7, 40	fmpti 6055	1 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) : (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) --> NN0

Syntax hints:   <-> wb 184   /\ wa 369   e. wcel 1819   _Vcvv 3109   u. cun 3469   i^i cin 3470   {csn 4032   class class class wbr 4456   |-> cmpt 4515   `'ccnv 5007   "cima 5011   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510   + caddc 9512   +oocpnf 9642   < clt 9645   - cmin 9824   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245  ..^cfzo 11821   #chash 12408  Word cword 12538   <"cs2 12818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825

This theorem is referenced by:  fib0  28535  fib1  28536  fibp1  28537