Theorem ftalem7 24805

Description: Lemma for fta 24806. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)

Hypotheses

Ref	Expression
ftalem.1	|- A = (coeff ` F )
ftalem.2	|- N = (deg ` F )
ftalem.3	|- ( ph -> F e. (Poly ` S ) )
ftalem.4	|- ( ph -> N e. NN )
ftalem7.5	|- ( ph -> X e. CC )
ftalem7.6	|- ( ph -> ( F ` X ) =/= 0 )

Assertion

Ref	Expression
ftalem7	|- ( ph -> -. A. x e. CC ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` x ) ) )

Distinct variable groups:   x, A   x, N   x, F   ph, x   x, X

Allowed substitution hint:   S( x)

Proof of Theorem ftalem7

Dummy variables z w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- (coeff ` ( z e. CC |-> ( F ` ( z + X ) ) ) ) = (coeff ` ( z e. CC |-> ( F ` ( z + X ) ) ) )
2		eqid 2622	. . . 4 |- (deg ` ( z e. CC |-> ( F ` ( z + X ) ) ) ) = (deg ` ( z e. CC |-> ( F ` ( z + X ) ) ) )
3		simpr 477	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> z e. CC )
4		ftalem7.5	. . . . . . . 8 |- ( ph -> X e. CC )
5	4	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> X e. CC )
6	3, 5	addcld 10059	. . . . . 6 |- ( ( ph /\ z e. CC ) -> ( z + X ) e. CC )
7		cnex 10017	. . . . . . . . 9 |- CC e. _V
8	7	a1i 11	. . . . . . . 8 |- ( ph -> CC e. _V )
9	4	negcld 10379	. . . . . . . . 9 |- ( ph -> -u X e. CC )
10	9	adantr 481	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> -u X e. CC )
11		df-idp 23945	. . . . . . . . . 10 |- Xp = ( _I |` CC )
12		mptresid 5456	. . . . . . . . . 10 |- ( z e. CC |-> z ) = ( _I |` CC )
13	11, 12	eqtr4i 2647	. . . . . . . . 9 |- Xp = ( z e. CC |-> z )
14	13	a1i 11	. . . . . . . 8 |- ( ph -> Xp = ( z e. CC |-> z ) )
15		fconstmpt 5163	. . . . . . . . 9 |- ( CC X. { -u X } ) = ( z e. CC |-> -u X )
16	15	a1i 11	. . . . . . . 8 |- ( ph -> ( CC X. { -u X } ) = ( z e. CC |-> -u X ) )
17	8, 3, 10, 14, 16	offval2 6914	. . . . . . 7 |- ( ph -> ( Xp oF - ( CC X. { -u X } ) ) = ( z e. CC |-> ( z - -u X ) ) )
18		id 22	. . . . . . . . 9 |- ( z e. CC -> z e. CC )
19		subneg 10330	. . . . . . . . 9 |- ( ( z e. CC /\ X e. CC ) -> ( z - -u X ) = ( z + X ) )
20	18, 4, 19	syl2anr 495	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> ( z - -u X ) = ( z + X ) )
21	20	mpteq2dva 4744	. . . . . . 7 |- ( ph -> ( z e. CC |-> ( z - -u X ) ) = ( z e. CC |-> ( z + X ) ) )
22	17, 21	eqtrd 2656	. . . . . 6 |- ( ph -> ( Xp oF - ( CC X. { -u X } ) ) = ( z e. CC |-> ( z + X ) ) )
23		ftalem.3	. . . . . . . 8 |- ( ph -> F e. (Poly ` S ) )
24		plyf 23954	. . . . . . . 8 |- ( F e. (Poly ` S ) -> F : CC --> CC )
25	23, 24	syl 17	. . . . . . 7 |- ( ph -> F : CC --> CC )
26	25	feqmptd 6249	. . . . . 6 |- ( ph -> F = ( y e. CC |-> ( F ` y ) ) )
27		fveq2 6191	. . . . . 6 |- ( y = ( z + X ) -> ( F ` y ) = ( F ` ( z + X ) ) )
28	6, 22, 26, 27	fmptco 6396	. . . . 5 |- ( ph -> ( F o. ( Xp oF - ( CC X. { -u X } ) ) ) = ( z e. CC |-> ( F ` ( z + X ) ) ) )
29		plyssc 23956	. . . . . . 7 |- (Poly ` S ) C_ (Poly ` CC )
30	29, 23	sseldi 3601	. . . . . 6 |- ( ph -> F e. (Poly ` CC ) )
31		eqid 2622	. . . . . . . . 9 |- ( Xp oF - ( CC X. { -u X } ) ) = ( Xp oF - ( CC X. { -u X } ) )
32	31	plyremlem 24059	. . . . . . . 8 |- ( -u X e. CC -> ( ( Xp oF - ( CC X. { -u X } ) ) e. (Poly ` CC ) /\ (deg ` ( Xp oF - ( CC X. { -u X } ) ) ) = 1 /\ ( `' ( Xp oF - ( CC X. { -u X } ) ) " { 0 } ) = { -u X } ) )
33	9, 32	syl 17	. . . . . . 7 |- ( ph -> ( ( Xp oF - ( CC X. { -u X } ) ) e. (Poly ` CC ) /\ (deg ` ( Xp oF - ( CC X. { -u X } ) ) ) = 1 /\ ( `' ( Xp oF - ( CC X. { -u X } ) ) " { 0 } ) = { -u X } ) )
34	33	simp1d 1073	. . . . . 6 |- ( ph -> ( Xp oF - ( CC X. { -u X } ) ) e. (Poly ` CC ) )
35		addcl 10018	. . . . . . 7 |- ( ( z e. CC /\ w e. CC ) -> ( z + w ) e. CC )
36	35	adantl 482	. . . . . 6 |- ( ( ph /\ ( z e. CC /\ w e. CC ) ) -> ( z + w ) e. CC )
37		mulcl 10020	. . . . . . 7 |- ( ( z e. CC /\ w e. CC ) -> ( z x. w ) e. CC )
38	37	adantl 482	. . . . . 6 |- ( ( ph /\ ( z e. CC /\ w e. CC ) ) -> ( z x. w ) e. CC )
39	30, 34, 36, 38	plyco 23997	. . . . 5 |- ( ph -> ( F o. ( Xp oF - ( CC X. { -u X } ) ) ) e. (Poly ` CC ) )
40	28, 39	eqeltrrd 2702	. . . 4 |- ( ph -> ( z e. CC |-> ( F ` ( z + X ) ) ) e. (Poly ` CC ) )
41	28	fveq2d 6195	. . . . 5 |- ( ph -> (deg ` ( F o. ( Xp oF - ( CC X. { -u X } ) ) ) ) = (deg ` ( z e. CC |-> ( F ` ( z + X ) ) ) ) )
42		ftalem.2	. . . . . . 7 |- N = (deg ` F )
43		eqid 2622	. . . . . . 7 |- (deg ` ( Xp oF - ( CC X. { -u X } ) ) ) = (deg ` ( Xp oF - ( CC X. { -u X } ) ) )
44	42, 43, 30, 34	dgrco 24031	. . . . . 6 |- ( ph -> (deg ` ( F o. ( Xp oF - ( CC X. { -u X } ) ) ) ) = ( N x. (deg ` ( Xp oF - ( CC X. { -u X } ) ) ) ) )
45		ftalem.4	. . . . . . 7 |- ( ph -> N e. NN )
46	33	simp2d 1074	. . . . . . . 8 |- ( ph -> (deg ` ( Xp oF - ( CC X. { -u X } ) ) ) = 1 )
47		1nn 11031	. . . . . . . 8 |- 1 e. NN
48	46, 47	syl6eqel 2709	. . . . . . 7 |- ( ph -> (deg ` ( Xp oF - ( CC X. { -u X } ) ) ) e. NN )
49	45, 48	nnmulcld 11068	. . . . . 6 |- ( ph -> ( N x. (deg ` ( Xp oF - ( CC X. { -u X } ) ) ) ) e. NN )
50	44, 49	eqeltrd 2701	. . . . 5 |- ( ph -> (deg ` ( F o. ( Xp oF - ( CC X. { -u X } ) ) ) ) e. NN )
51	41, 50	eqeltrrd 2702	. . . 4 |- ( ph -> (deg ` ( z e. CC |-> ( F ` ( z + X ) ) ) ) e. NN )
52		0cn 10032	. . . . . . 7 |- 0 e. CC
53		oveq1 6657	. . . . . . . . 9 |- ( z = 0 -> ( z + X ) = ( 0 + X ) )
54	53	fveq2d 6195	. . . . . . . 8 |- ( z = 0 -> ( F ` ( z + X ) ) = ( F ` ( 0 + X ) ) )
55		eqid 2622	. . . . . . . 8 |- ( z e. CC |-> ( F ` ( z + X ) ) ) = ( z e. CC |-> ( F ` ( z + X ) ) )
56		fvex 6201	. . . . . . . 8 |- ( F ` ( 0 + X ) ) e. _V
57	54, 55, 56	fvmpt 6282	. . . . . . 7 |- ( 0 e. CC -> ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) = ( F ` ( 0 + X ) ) )
58	52, 57	ax-mp 5	. . . . . 6 |- ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) = ( F ` ( 0 + X ) )
59	4	addid2d 10237	. . . . . . 7 |- ( ph -> ( 0 + X ) = X )
60	59	fveq2d 6195	. . . . . 6 |- ( ph -> ( F ` ( 0 + X ) ) = ( F ` X ) )
61	58, 60	syl5eq 2668	. . . . 5 |- ( ph -> ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) = ( F ` X ) )
62		ftalem7.6	. . . . 5 |- ( ph -> ( F ` X ) =/= 0 )
63	61, 62	eqnetrd 2861	. . . 4 |- ( ph -> ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) =/= 0 )
64	1, 2, 40, 51, 63	ftalem6 24804	. . 3 |- ( ph -> E. y e. CC ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` y ) ) < ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) ) )
65		id 22	. . . . . 6 |- ( y e. CC -> y e. CC )
66		addcl 10018	. . . . . 6 |- ( ( y e. CC /\ X e. CC ) -> ( y + X ) e. CC )
67	65, 4, 66	syl2anr 495	. . . . 5 |- ( ( ph /\ y e. CC ) -> ( y + X ) e. CC )
68		oveq1 6657	. . . . . . . . . . . 12 |- ( z = y -> ( z + X ) = ( y + X ) )
69	68	fveq2d 6195	. . . . . . . . . . 11 |- ( z = y -> ( F ` ( z + X ) ) = ( F ` ( y + X ) ) )
70		fvex 6201	. . . . . . . . . . 11 |- ( F ` ( y + X ) ) e. _V
71	69, 55, 70	fvmpt 6282	. . . . . . . . . 10 |- ( y e. CC -> ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` y ) = ( F ` ( y + X ) ) )
72	71	adantl 482	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` y ) = ( F ` ( y + X ) ) )
73	72	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ y e. CC ) -> ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` y ) ) = ( abs ` ( F ` ( y + X ) ) ) )
74	61	adantr 481	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) = ( F ` X ) )
75	74	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ y e. CC ) -> ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) ) = ( abs ` ( F ` X ) ) )
76	73, 75	breq12d 4666	. . . . . . 7 |- ( ( ph /\ y e. CC ) -> ( ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` y ) ) < ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) ) <-> ( abs ` ( F ` ( y + X ) ) ) < ( abs ` ( F ` X ) ) ) )
77	25	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ y e. CC ) -> F : CC --> CC )
78	77, 67	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> ( F ` ( y + X ) ) e. CC )
79	78	abscld 14175	. . . . . . . 8 |- ( ( ph /\ y e. CC ) -> ( abs ` ( F ` ( y + X ) ) ) e. RR )
80	25, 4	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( F ` X ) e. CC )
81	80	abscld 14175	. . . . . . . . 9 |- ( ph -> ( abs ` ( F ` X ) ) e. RR )
82	81	adantr 481	. . . . . . . 8 |- ( ( ph /\ y e. CC ) -> ( abs ` ( F ` X ) ) e. RR )
83	79, 82	ltnled 10184	. . . . . . 7 |- ( ( ph /\ y e. CC ) -> ( ( abs ` ( F ` ( y + X ) ) ) < ( abs ` ( F ` X ) ) <-> -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` ( y + X ) ) ) ) )
84	76, 83	bitrd 268	. . . . . 6 |- ( ( ph /\ y e. CC ) -> ( ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` y ) ) < ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) ) <-> -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` ( y + X ) ) ) ) )
85	84	biimpd 219	. . . . 5 |- ( ( ph /\ y e. CC ) -> ( ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` y ) ) < ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) ) -> -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` ( y + X ) ) ) ) )
86		fveq2 6191	. . . . . . . . 9 |- ( x = ( y + X ) -> ( F ` x ) = ( F ` ( y + X ) ) )
87	86	fveq2d 6195	. . . . . . . 8 |- ( x = ( y + X ) -> ( abs ` ( F ` x ) ) = ( abs ` ( F ` ( y + X ) ) ) )
88	87	breq2d 4665	. . . . . . 7 |- ( x = ( y + X ) -> ( ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` x ) ) <-> ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` ( y + X ) ) ) ) )
89	88	notbid 308	. . . . . 6 |- ( x = ( y + X ) -> ( -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` x ) ) <-> -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` ( y + X ) ) ) ) )
90	89	rspcev 3309	. . . . 5 |- ( ( ( y + X ) e. CC /\ -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` ( y + X ) ) ) ) -> E. x e. CC -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` x ) ) )
91	67, 85, 90	syl6an 568	. . . 4 |- ( ( ph /\ y e. CC ) -> ( ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` y ) ) < ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) ) -> E. x e. CC -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` x ) ) ) )
92	91	rexlimdva 3031	. . 3 |- ( ph -> ( E. y e. CC ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` y ) ) < ( abs ` ( ( z e. CC |-> ( F ` ( z + X ) ) ) ` 0 ) ) -> E. x e. CC -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` x ) ) ) )
93	64, 92	mpd 15	. 2 |- ( ph -> E. x e. CC -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` x ) ) )
94		rexnal 2995	. 2 |- ( E. x e. CC -. ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` x ) ) <-> -. A. x e. CC ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` x ) ) )
95	93, 94	sylib 208	1 |- ( ph -> -. A. x e. CC ( abs ` ( F ` X ) ) <_ ( abs ` ( F ` x ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   {csn 4177   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   NNcn 11020   abscabs 13974  Polycply 23940   Xpcidp 23941  coeffccoe 23942  degcdgr 23943

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  ax-addf 10015  ax-mulf 10016

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-iin 4523  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-of 6897  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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-0p 23437  df-limc 23630  df-dv 23631  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-log 24303  df-cxp 24304

This theorem is referenced by:  fta  24806