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

Step	Hyp	Ref	Expression
1		s2len 12802	. . . . . . 7 |- ( # ` <" 0 1 "> ) = 2
2	1	eqcomi 2473	. . . . . 6 |- 2 = ( # ` <" 0 1 "> )
3	2	fveq2i 5860	. . . . 5 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( # ` <" 0 1 "> ) )
4	3	imaeq2i 5326	. . . 4 |- ( `' # " ( ZZ>= ` 2 ) ) = ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) )
5	4	ineq2i 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 ) ) )
7	5, 6	mpteq12i 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 "> ) ) ) ) )
9	8	simplbi 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 )
11	9, 10	syl 16	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> w : ( 0..^ ( # ` w ) ) --> NN0 )
12	8	simprbi 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 "> ) ) ) ) )
16	13, 14, 15	mp2b 10	. . . . . . . . . 10 |- ( w e. ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) <-> ( w e. _V /\ ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
17	12, 16	sylib 196	. . . . . . . . 9 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( w e. _V /\ ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
18	17	simprd 463	. . . . . . . 8 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) )
19	18, 3	syl6eleqr 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 )
21	19, 20	syl 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
24	23	fveq2i 5860	. . . . . . . . 9 |- ( ZZ>= ` ( 1 + 1 ) ) = ( ZZ>= ` 2 )
25	19, 24	syl6eleqr 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 ) )
27	22, 25, 26	syl2anc 661	. . . . . . 7 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` 1 ) )
28		nnuz 11106	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
29	27, 28	syl6eleqr 2559	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. NN )
30	29	nnred 10540	. . . . . . 7 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. RR )
31		2rp 11214	. . . . . . . 8 |- 2 e. RR+
32	31	a1i 11	. . . . . . 7 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> 2 e. RR+ )
33	30, 32	ltsubrpd 11273	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 2 ) < ( # ` w ) )
34	21, 29, 33	3jca 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 ) ) )
36	34, 35	sylibr 212	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 2 ) e. ( 0..^ ( # ` w ) ) )
37	11, 36	ffvelrnd 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 ) ) )
39	29, 38	syl 16	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 1 ) e. ( 0..^ ( # ` w ) ) )
40	11, 39	ffvelrnd 6013	. . 3 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( w ` ( ( # ` w ) - 1 ) ) e. NN0 )
41	37, 40	nn0addcld 10845	. 2 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) e. NN0 )
42	7, 41	fmpti 6035	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   /\ 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