Theorem signstfveq0a 30653

Description: Lemma for signstfveq0 30654. (Contributed by Thierry Arnoux, 11-Oct-2018.)

Hypotheses

Ref	Expression
signsv.p	|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
signsv.w	|- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
signsv.t	|- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
signsv.v	|- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
signstfveq0.1	|- N = ( # ` F )

Assertion

Ref	Expression
signstfveq0a	|- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N e. ( ZZ>= ` 2 ) )

Distinct variable groups:   a, b, .+^   f, i, n, F   f, W, i, n   F, a, b, f, i, n   N, a   f, b, i, n, N   T, a, b

Allowed substitution hints:   .+^ ( f, i, j, n)   T( f, i, j, n)   F( j)   N( j)   V( f, i, j, n, a, b)   W( j, a, b)

Proof of Theorem signstfveq0a

Step	Hyp	Ref	Expression
1		simpll 790	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> F e. (Word RR \ { (/) } ) )
2	1	eldifad 3586	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> F e. Word RR )
3		signstfveq0.1	. . . . 5 |- N = ( # ` F )
4		lencl 13324	. . . . 5 |- ( F e. Word RR -> ( # ` F ) e. NN0 )
5	3, 4	syl5eqel 2705	. . . 4 |- ( F e. Word RR -> N e. NN0 )
6	2, 5	syl 17	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N e. NN0 )
7		eldifsn 4317	. . . . 5 |- ( F e. (Word RR \ { (/) } ) <-> ( F e. Word RR /\ F =/= (/) ) )
8	1, 7	sylib 208	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F e. Word RR /\ F =/= (/) ) )
9		hasheq0 13154	. . . . . . 7 |- ( F e. Word RR -> ( ( # ` F ) = 0 <-> F = (/) ) )
10	9	necon3bid 2838	. . . . . 6 |- ( F e. Word RR -> ( ( # ` F ) =/= 0 <-> F =/= (/) ) )
11	10	biimpar 502	. . . . 5 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( # ` F ) =/= 0 )
12	3	neeq1i 2858	. . . . 5 |- ( N =/= 0 <-> ( # ` F ) =/= 0 )
13	11, 12	sylibr 224	. . . 4 |- ( ( F e. Word RR /\ F =/= (/) ) -> N =/= 0 )
14	8, 13	syl 17	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N =/= 0 )
15		elnnne0 11306	. . 3 |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
16	6, 14, 15	sylanbrc 698	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N e. NN )
17		simplr 792	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F ` 0 ) =/= 0 )
18		simpr 477	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F ` ( N - 1 ) ) = 0 )
19	17, 18	neeqtrrd 2868	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F ` 0 ) =/= ( F ` ( N - 1 ) ) )
20	19	necomd 2849	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F ` ( N - 1 ) ) =/= ( F ` 0 ) )
21		oveq1 6657	. . . . . 6 |- ( N = 1 -> ( N - 1 ) = ( 1 - 1 ) )
22		1m1e0 11089	. . . . . 6 |- ( 1 - 1 ) = 0
23	21, 22	syl6eq 2672	. . . . 5 |- ( N = 1 -> ( N - 1 ) = 0 )
24	23	fveq2d 6195	. . . 4 |- ( N = 1 -> ( F ` ( N - 1 ) ) = ( F ` 0 ) )
25	24	necon3i 2826	. . 3 |- ( ( F ` ( N - 1 ) ) =/= ( F ` 0 ) -> N =/= 1 )
26	20, 25	syl 17	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N =/= 1 )
27		eluz2b3 11762	. 2 |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
28	16, 26, 27	sylanbrc 698	1 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N e. ( ZZ>= ` 2 ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   (/)c0 3915   ifcif 4086   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  sgncsgn 13826   sum_csu 14416   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   gsum_g cgsu 16101

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-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299

This theorem is referenced by:  signstfveq0  30654