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

Step	Hyp	Ref	Expression
1		s2len 12978	. . . . . . 7 |- ( # ` <" 0 1 "> ) = 2
2	1	eqcomi 2459	. . . . . 6 |- 2 = ( # ` <" 0 1 "> )
3	2	fveq2i 5866	. . . . 5 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( # ` <" 0 1 "> ) )
4	3	imaeq2i 5165	. . . 4 |- ( `' # " ( ZZ>= ` 2 ) ) = ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) )
5	4	ineq2i 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 ) ) )
7	5, 6	mpteq12i 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 "> ) ) ) ) )
9	8	simplbi 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 )
11	9, 10	syl 17	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> w : ( 0..^ ( # ` w ) ) --> NN0 )
12	8	simprbi 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 "> ) ) ) ) )
16	13, 14, 15	mp2b 10	. . . . . . . . 9 |- ( w e. ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) <-> ( w e. _V /\ ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
17	12, 16	sylib 200	. . . . . . . 8 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( w e. _V /\ ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
18	17	simprd 465	. . . . . . 7 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) )
19	18, 3	syl6eleqr 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 )
21	19, 20	syl 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
24	23	fveq2i 5866	. . . . . . . 8 |- ( ZZ>= ` ( 1 + 1 ) ) = ( ZZ>= ` 2 )
25	19, 24	syl6eleqr 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 ) )
27	22, 25, 26	syl2anc 666	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. ( ZZ>= ` 1 ) )
28		nnuz 11191	. . . . . 6 |- NN = ( ZZ>= ` 1 )
29	27, 28	syl6eleqr 2539	. . . . 5 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. NN )
30	29	nnred 10621	. . . . . 6 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( # ` w ) e. RR )
31		2rp 11304	. . . . . . 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 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 ) ) )
35	21, 29, 33, 34	syl3anbrc 1191	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 2 ) e. ( 0..^ ( # ` w ) ) )
36	11, 35	ffvelrnd 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 ) ) )
38	29, 37	syl 17	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( # ` w ) - 1 ) e. ( 0..^ ( # ` w ) ) )
39	11, 38	ffvelrnd 6021	. . 3 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( w ` ( ( # ` w ) - 1 ) ) e. NN0 )
40	36, 39	nn0addcld 10926	. 2 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) -> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) e. NN0 )
41	7, 40	fmpti 6043	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 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