Theorem fiblem 30460

Description: Lemma for fib0 30461, fib1 30462 and fibp1 30463. (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 13634	. . . . . . 7 |- ( # ` <" 0 1 "> ) = 2
2	1	eqcomi 2631	. . . . . 6 |- 2 = ( # ` <" 0 1 "> )
3	2	fveq2i 6194	. . . . 5 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( # ` <" 0 1 "> ) )
4	3	imaeq2i 5464	. . . 4 |- ( `' # " ( ZZ>= ` 2 ) ) = ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) )
5	4	ineq2i 3811	. . 3 |- (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) = (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
6		eqid 2622	. . 3 |- ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) = ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) )
7	5, 6	mpteq12i 4742	. 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 3796	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) <-> ( w e. Word NN0 /\ w e. ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) )
9	8	simplbi 476	. . . . 5 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> w e. Word NN0 )
10		wrdf 13310	. . . . 5 |- ( w e. Word NN0 -> w : ( 0..^ ( # ` w ) ) --> NN0 )
11	9, 10	syl 17	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> w : ( 0..^ ( # ` w ) ) --> NN0 )
12	8	simprbi 480	. . . . . . . . 9 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> w e. ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
13		hashf 13125	. . . . . . . . . 10 |- # : _V --> ( NN0 u. { +oo } )
14		ffn 6045	. . . . . . . . . 10 |- ( # : _V --> ( NN0 u. { +oo } ) -> # Fn _V )
15		elpreima 6337	. . . . . . . . . 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 208	. . . . . . . 8 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( w e. _V /\ ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
18	17	simprd 479	. . . . . . 7 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) )
19	18, 3	syl6eleqr 2712	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` 2 ) )
20		uznn0sub 11719	. . . . . 6 |- ( ( # ` w ) e. ( ZZ>= ` 2 ) -> ( ( # ` w ) - 2 ) e. NN0 )
21	19, 20	syl 17	. . . . 5 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 2 ) e. NN0 )
22		1zzd 11408	. . . . . . 7 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> 1 e. ZZ )
23		1p1e2 11134	. . . . . . . . 9 |- ( 1 + 1 ) = 2
24	23	fveq2i 6194	. . . . . . . 8 |- ( ZZ>= ` ( 1 + 1 ) ) = ( ZZ>= ` 2 )
25	19, 24	syl6eleqr 2712	. . . . . . 7 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` ( 1 + 1 ) ) )
26		peano2uzr 11743	. . . . . . 7 |- ( ( 1 e. ZZ /\ ( # ` w ) e. ( ZZ>= ` ( 1 + 1 ) ) ) -> ( # ` w ) e. ( ZZ>= ` 1 ) )
27	22, 25, 26	syl2anc 693	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` 1 ) )
28		nnuz 11723	. . . . . 6 |- NN = ( ZZ>= ` 1 )
29	27, 28	syl6eleqr 2712	. . . . 5 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. NN )
30	29	nnred 11035	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. RR )
31		2rp 11837	. . . . . . 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 11904	. . . . 5 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 2 ) < ( # ` w ) )
34		elfzo0 12508	. . . . 5 |- ( ( ( # ` w ) - 2 ) e. ( 0..^ ( # ` w ) ) <-> ( ( ( # ` w ) - 2 ) e. NN0 /\ ( # ` w ) e. NN /\ ( ( # ` w ) - 2 ) < ( # ` w ) ) )
35	21, 29, 33, 34	syl3anbrc 1246	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 2 ) e. ( 0..^ ( # ` w ) ) )
36	11, 35	ffvelrnd 6360	. . 3 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( w ` ( ( # ` w ) - 2 ) ) e. NN0 )
37		fzo0end 12560	. . . . 5 |- ( ( # ` w ) e. NN -> ( ( # ` w ) - 1 ) e. ( 0..^ ( # ` w ) ) )
38	29, 37	syl 17	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 1 ) e. ( 0..^ ( # ` w ) ) )
39	11, 38	ffvelrnd 6360	. . 3 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( w ` ( ( # ` w ) - 1 ) ) e. NN0 )
40	36, 39	nn0addcld 11355	. 2 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) e. NN0 )
41	7, 40	fmpti 6383	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 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   < clt 10074   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs2 13586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593

This theorem is referenced by:  fib0  30461  fib1  30462  fibp1  30463