Theorem signsvfpn 30662

Description: Adding a letter of the same sign as the highest coefficient does not change the sign. (Contributed by Thierry Arnoux, 12-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 ) )
signsvf.e	|- ( ph -> E e. (Word RR \ { (/) } ) )
signsvf.0	|- ( ph -> ( E ` 0 ) =/= 0 )
signsvf.f	|- ( ph -> F = ( E ++ <" A "> ) )
signsvf.a	|- ( ph -> A e. RR )
signsvf.n	|- N = ( # ` E )
signsvf.b	|- B = ( E ` ( N - 1 ) )

Assertion

Ref	Expression
signsvfpn	|- ( ( ph /\ 0 < ( B x. A ) ) -> ( V ` F ) = ( V ` E ) )

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

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

Proof of Theorem signsvfpn

Step	Hyp	Ref	Expression
1		signsvf.a	. . . . . . . . 9 |- ( ph -> A e. RR )
2	1	recnd 10068	. . . . . . . 8 |- ( ph -> A e. CC )
3		signsvf.b	. . . . . . . . 9 |- B = ( E ` ( N - 1 ) )
4		signsvf.e	. . . . . . . . . . . . 13 |- ( ph -> E e. (Word RR \ { (/) } ) )
5	4	eldifad 3586	. . . . . . . . . . . 12 |- ( ph -> E e. Word RR )
6		wrdf 13310	. . . . . . . . . . . 12 |- ( E e. Word RR -> E : ( 0..^ ( # ` E ) ) --> RR )
7	5, 6	syl 17	. . . . . . . . . . 11 |- ( ph -> E : ( 0..^ ( # ` E ) ) --> RR )
8		signsvf.n	. . . . . . . . . . . . 13 |- N = ( # ` E )
9	8	oveq1i 6660	. . . . . . . . . . . 12 |- ( N - 1 ) = ( ( # ` E ) - 1 )
10		eldifsn 4317	. . . . . . . . . . . . . 14 |- ( E e. (Word RR \ { (/) } ) <-> ( E e. Word RR /\ E =/= (/) ) )
11	4, 10	sylib 208	. . . . . . . . . . . . 13 |- ( ph -> ( E e. Word RR /\ E =/= (/) ) )
12		lennncl 13325	. . . . . . . . . . . . 13 |- ( ( E e. Word RR /\ E =/= (/) ) -> ( # ` E ) e. NN )
13		fzo0end 12560	. . . . . . . . . . . . 13 |- ( ( # ` E ) e. NN -> ( ( # ` E ) - 1 ) e. ( 0..^ ( # ` E ) ) )
14	11, 12, 13	3syl 18	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` E ) - 1 ) e. ( 0..^ ( # ` E ) ) )
15	9, 14	syl5eqel 2705	. . . . . . . . . . 11 |- ( ph -> ( N - 1 ) e. ( 0..^ ( # ` E ) ) )
16	7, 15	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( E ` ( N - 1 ) ) e. RR )
17	16	recnd 10068	. . . . . . . . 9 |- ( ph -> ( E ` ( N - 1 ) ) e. CC )
18	3, 17	syl5eqel 2705	. . . . . . . 8 |- ( ph -> B e. CC )
19	2, 18	mulcomd 10061	. . . . . . 7 |- ( ph -> ( A x. B ) = ( B x. A ) )
20	19	breq2d 4665	. . . . . 6 |- ( ph -> ( 0 < ( A x. B ) <-> 0 < ( B x. A ) ) )
21	3, 16	syl5eqel 2705	. . . . . . 7 |- ( ph -> B e. RR )
22		sgnmulsgp 30612	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( A x. B ) <-> 0 < ( (sgn ` A ) x. (sgn ` B ) ) ) )
23	1, 21, 22	syl2anc 693	. . . . . 6 |- ( ph -> ( 0 < ( A x. B ) <-> 0 < ( (sgn ` A ) x. (sgn ` B ) ) ) )
24	20, 23	bitr3d 270	. . . . 5 |- ( ph -> ( 0 < ( B x. A ) <-> 0 < ( (sgn ` A ) x. (sgn ` B ) ) ) )
25	24	biimpa 501	. . . 4 |- ( ( ph /\ 0 < ( B x. A ) ) -> 0 < ( (sgn ` A ) x. (sgn ` B ) ) )
26	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ 0 < ( B x. A ) ) -> E e. (Word RR \ { (/) } ) )
27	18	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ 0 < ( B x. A ) ) -> B e. CC )
28	2	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ 0 < ( B x. A ) ) -> A e. CC )
29		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ 0 < ( B x. A ) ) -> 0 < ( B x. A ) )
30	29	gt0ne0d 10592	. . . . . . . . . 10 |- ( ( ph /\ 0 < ( B x. A ) ) -> ( B x. A ) =/= 0 )
31	27, 28, 30	mulne0bad 10682	. . . . . . . . 9 |- ( ( ph /\ 0 < ( B x. A ) ) -> B =/= 0 )
32	3, 31	syl5eqner 2869	. . . . . . . 8 |- ( ( ph /\ 0 < ( B x. A ) ) -> ( E ` ( N - 1 ) ) =/= 0 )
33		signsv.p	. . . . . . . . . 10 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
34		signsv.w	. . . . . . . . . 10 |- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
35		signsv.t	. . . . . . . . . 10 |- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
36		signsv.v	. . . . . . . . . 10 |- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
37	33, 34, 35, 36, 8	signsvtn0 30647	. . . . . . . . 9 |- ( ( E e. (Word RR \ { (/) } ) /\ ( E ` ( N - 1 ) ) =/= 0 ) -> ( ( T ` E ) ` ( N - 1 ) ) = (sgn ` ( E ` ( N - 1 ) ) ) )
38	3	fveq2i 6194	. . . . . . . . 9 |- (sgn ` B ) = (sgn ` ( E ` ( N - 1 ) ) )
39	37, 38	syl6eqr 2674	. . . . . . . 8 |- ( ( E e. (Word RR \ { (/) } ) /\ ( E ` ( N - 1 ) ) =/= 0 ) -> ( ( T ` E ) ` ( N - 1 ) ) = (sgn ` B ) )
40	26, 32, 39	syl2anc 693	. . . . . . 7 |- ( ( ph /\ 0 < ( B x. A ) ) -> ( ( T ` E ) ` ( N - 1 ) ) = (sgn ` B ) )
41	40	fveq2d 6195	. . . . . 6 |- ( ( ph /\ 0 < ( B x. A ) ) -> (sgn ` ( ( T ` E ) ` ( N - 1 ) ) ) = (sgn ` (sgn ` B ) ) )
42	21	adantr 481	. . . . . . . 8 |- ( ( ph /\ 0 < ( B x. A ) ) -> B e. RR )
43	42	rexrd 10089	. . . . . . 7 |- ( ( ph /\ 0 < ( B x. A ) ) -> B e. RR* )
44		sgnsgn 30610	. . . . . . 7 |- ( B e. RR* -> (sgn ` (sgn ` B ) ) = (sgn ` B ) )
45	43, 44	syl 17	. . . . . 6 |- ( ( ph /\ 0 < ( B x. A ) ) -> (sgn ` (sgn ` B ) ) = (sgn ` B ) )
46	41, 45	eqtrd 2656	. . . . 5 |- ( ( ph /\ 0 < ( B x. A ) ) -> (sgn ` ( ( T ` E ) ` ( N - 1 ) ) ) = (sgn ` B ) )
47	46	oveq2d 6666	. . . 4 |- ( ( ph /\ 0 < ( B x. A ) ) -> ( (sgn ` A ) x. (sgn ` ( ( T ` E ) ` ( N - 1 ) ) ) ) = ( (sgn ` A ) x. (sgn ` B ) ) )
48	25, 47	breqtrrd 4681	. . 3 |- ( ( ph /\ 0 < ( B x. A ) ) -> 0 < ( (sgn ` A ) x. (sgn ` ( ( T ` E ) ` ( N - 1 ) ) ) ) )
49	1	adantr 481	. . . 4 |- ( ( ph /\ 0 < ( B x. A ) ) -> A e. RR )
50		sgnclre 30601	. . . . . 6 |- ( B e. RR -> (sgn ` B ) e. RR )
51	42, 50	syl 17	. . . . 5 |- ( ( ph /\ 0 < ( B x. A ) ) -> (sgn ` B ) e. RR )
52	40, 51	eqeltrd 2701	. . . 4 |- ( ( ph /\ 0 < ( B x. A ) ) -> ( ( T ` E ) ` ( N - 1 ) ) e. RR )
53		sgnmulsgp 30612	. . . 4 |- ( ( A e. RR /\ ( ( T ` E ) ` ( N - 1 ) ) e. RR ) -> ( 0 < ( A x. ( ( T ` E ) ` ( N - 1 ) ) ) <-> 0 < ( (sgn ` A ) x. (sgn ` ( ( T ` E ) ` ( N - 1 ) ) ) ) ) )
54	49, 52, 53	syl2anc 693	. . 3 |- ( ( ph /\ 0 < ( B x. A ) ) -> ( 0 < ( A x. ( ( T ` E ) ` ( N - 1 ) ) ) <-> 0 < ( (sgn ` A ) x. (sgn ` ( ( T ` E ) ` ( N - 1 ) ) ) ) ) )
55	48, 54	mpbird 247	. 2 |- ( ( ph /\ 0 < ( B x. A ) ) -> 0 < ( A x. ( ( T ` E ) ` ( N - 1 ) ) ) )
56		signsvf.0	. . 3 |- ( ph -> ( E ` 0 ) =/= 0 )
57		signsvf.f	. . 3 |- ( ph -> F = ( E ++ <" A "> ) )
58		eqid 2622	. . 3 |- ( ( T ` E ) ` ( N - 1 ) ) = ( ( T ` E ) ` ( N - 1 ) )
59	33, 34, 35, 36, 4, 56, 57, 1, 8, 58	signsvtp 30660	. 2 |- ( ( ph /\ 0 < ( A x. ( ( T ` E ) ` ( N - 1 ) ) ) ) -> ( V ` F ) = ( V ` E ) )
60	55, 59	syldan 487	1 |- ( ( ph /\ 0 < ( B x. A ) ) -> ( V ` F ) = ( V ` E ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   (/)c0 3915   ifcif 4086   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   NNcn 11020   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294  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-inf2 8538  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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-rmo 2920  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  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-sup 8348  df-oi 8415  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-sgn 13827  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mulg 17541  df-cntz 17750

This theorem is referenced by: (None)