Theorem cncficcgt0 40101

Description: A the absolute value of a continuous function on a closed interval, that is never 0, has a strictly positive lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
cncficcgt0.f	|- F = ( x e. ( A [,] B ) |-> C )
cncficcgt0.a	|- ( ph -> A e. RR )
cncficcgt0.b	|- ( ph -> B e. RR )
cncficcgt0.aleb	|- ( ph -> A <_ B )
cncficcgt0.fcn	|- ( ph -> F e. ( ( A [,] B ) -cn-> ( RR \ { 0 } ) ) )

Assertion

Ref	Expression
cncficcgt0	|- ( ph -> E. y e. RR+ A. x e. ( A [,] B ) y <_ ( abs ` C ) )

Distinct variable groups:   x, A, y   x, B, y   y, C   y, F   ph, x

Allowed substitution hints:   ph( y)   C( x)   F( x)

Proof of Theorem cncficcgt0

Dummy variables a b c d z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cncficcgt0.fcn	. . . . . . . 8 |- ( ph -> F e. ( ( A [,] B ) -cn-> ( RR \ { 0 } ) ) )
2		cncff 22696	. . . . . . . 8 |- ( F e. ( ( A [,] B ) -cn-> ( RR \ { 0 } ) ) -> F : ( A [,] B ) --> ( RR \ { 0 } ) )
3		ffun 6048	. . . . . . . 8 |- ( F : ( A [,] B ) --> ( RR \ { 0 } ) -> Fun F )
4	1, 2, 3	3syl 18	. . . . . . 7 |- ( ph -> Fun F )
5	4	adantr 481	. . . . . 6 |- ( ( ph /\ c e. ( A [,] B ) ) -> Fun F )
6		simpr 477	. . . . . . 7 |- ( ( ph /\ c e. ( A [,] B ) ) -> c e. ( A [,] B ) )
7	1, 2	syl 17	. . . . . . . . . 10 |- ( ph -> F : ( A [,] B ) --> ( RR \ { 0 } ) )
8		fdm 6051	. . . . . . . . . 10 |- ( F : ( A [,] B ) --> ( RR \ { 0 } ) -> dom F = ( A [,] B ) )
9	7, 8	syl 17	. . . . . . . . 9 |- ( ph -> dom F = ( A [,] B ) )
10	9	eqcomd 2628	. . . . . . . 8 |- ( ph -> ( A [,] B ) = dom F )
11	10	adantr 481	. . . . . . 7 |- ( ( ph /\ c e. ( A [,] B ) ) -> ( A [,] B ) = dom F )
12	6, 11	eleqtrd 2703	. . . . . 6 |- ( ( ph /\ c e. ( A [,] B ) ) -> c e. dom F )
13		fvco 6274	. . . . . 6 |- ( ( Fun F /\ c e. dom F ) -> ( ( abs o. F ) ` c ) = ( abs ` ( F ` c ) ) )
14	5, 12, 13	syl2anc 693	. . . . 5 |- ( ( ph /\ c e. ( A [,] B ) ) -> ( ( abs o. F ) ` c ) = ( abs ` ( F ` c ) ) )
15	7	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ c e. ( A [,] B ) ) -> ( F ` c ) e. ( RR \ { 0 } ) )
16	15	eldifad 3586	. . . . . . 7 |- ( ( ph /\ c e. ( A [,] B ) ) -> ( F ` c ) e. RR )
17	16	recnd 10068	. . . . . 6 |- ( ( ph /\ c e. ( A [,] B ) ) -> ( F ` c ) e. CC )
18		eldifsni 4320	. . . . . . 7 |- ( ( F ` c ) e. ( RR \ { 0 } ) -> ( F ` c ) =/= 0 )
19	15, 18	syl 17	. . . . . 6 |- ( ( ph /\ c e. ( A [,] B ) ) -> ( F ` c ) =/= 0 )
20	17, 19	absrpcld 14187	. . . . 5 |- ( ( ph /\ c e. ( A [,] B ) ) -> ( abs ` ( F ` c ) ) e. RR+ )
21	14, 20	eqeltrd 2701	. . . 4 |- ( ( ph /\ c e. ( A [,] B ) ) -> ( ( abs o. F ) ` c ) e. RR+ )
22	21	adantr 481	. . 3 |- ( ( ( ph /\ c e. ( A [,] B ) ) /\ A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) ) -> ( ( abs o. F ) ` c ) e. RR+ )
23		nfv 1843	. . . . 5 |- F/ x ( ph /\ c e. ( A [,] B ) )
24		nfcv 2764	. . . . . 6 |- F/_ x ( A [,] B )
25		nfcv 2764	. . . . . . . . 9 |- F/_ x abs
26		cncficcgt0.f	. . . . . . . . . 10 |- F = ( x e. ( A [,] B ) |-> C )
27		nfmpt1 4747	. . . . . . . . . 10 |- F/_ x ( x e. ( A [,] B ) |-> C )
28	26, 27	nfcxfr 2762	. . . . . . . . 9 |- F/_ x F
29	25, 28	nfco 5287	. . . . . . . 8 |- F/_ x ( abs o. F )
30		nfcv 2764	. . . . . . . 8 |- F/_ x c
31	29, 30	nffv 6198	. . . . . . 7 |- F/_ x ( ( abs o. F ) ` c )
32		nfcv 2764	. . . . . . 7 |- F/_ x <_
33		nfcv 2764	. . . . . . . 8 |- F/_ x d
34	29, 33	nffv 6198	. . . . . . 7 |- F/_ x ( ( abs o. F ) ` d )
35	31, 32, 34	nfbr 4699	. . . . . 6 |- F/ x ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d )
36	24, 35	nfral 2945	. . . . 5 |- F/ x A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d )
37	23, 36	nfan 1828	. . . 4 |- F/ x ( ( ph /\ c e. ( A [,] B ) ) /\ A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) )
38		fveq2 6191	. . . . . . . . 9 |- ( d = x -> ( ( abs o. F ) ` d ) = ( ( abs o. F ) ` x ) )
39	38	breq2d 4665	. . . . . . . 8 |- ( d = x -> ( ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) <-> ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` x ) ) )
40	39	rspccva 3308	. . . . . . 7 |- ( ( A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) /\ x e. ( A [,] B ) ) -> ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` x ) )
41	40	adantll 750	. . . . . 6 |- ( ( ( ( ph /\ c e. ( A [,] B ) ) /\ A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) ) /\ x e. ( A [,] B ) ) -> ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` x ) )
42		absf 14077	. . . . . . . . . . 11 |- abs : CC --> RR
43	42	a1i 11	. . . . . . . . . 10 |- ( ph -> abs : CC --> RR )
44		difss 3737	. . . . . . . . . . . . 13 |- ( RR \ { 0 } ) C_ RR
45		ax-resscn 9993	. . . . . . . . . . . . 13 |- RR C_ CC
46	44, 45	sstri 3612	. . . . . . . . . . . 12 |- ( RR \ { 0 } ) C_ CC
47	46	a1i 11	. . . . . . . . . . 11 |- ( ph -> ( RR \ { 0 } ) C_ CC )
48	7, 47	fssd 6057	. . . . . . . . . 10 |- ( ph -> F : ( A [,] B ) --> CC )
49		fcompt 6400	. . . . . . . . . 10 |- ( ( abs : CC --> RR /\ F : ( A [,] B ) --> CC ) -> ( abs o. F ) = ( z e. ( A [,] B ) |-> ( abs ` ( F ` z ) ) ) )
50	43, 48, 49	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( abs o. F ) = ( z e. ( A [,] B ) |-> ( abs ` ( F ` z ) ) ) )
51		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ x z
52	28, 51	nffv 6198	. . . . . . . . . . . 12 |- F/_ x ( F ` z )
53	25, 52	nffv 6198	. . . . . . . . . . 11 |- F/_ x ( abs ` ( F ` z ) )
54		nfcv 2764	. . . . . . . . . . 11 |- F/_ z ( abs ` ( F ` x ) )
55		fveq2 6191	. . . . . . . . . . . 12 |- ( z = x -> ( F ` z ) = ( F ` x ) )
56	55	fveq2d 6195	. . . . . . . . . . 11 |- ( z = x -> ( abs ` ( F ` z ) ) = ( abs ` ( F ` x ) ) )
57	53, 54, 56	cbvmpt 4749	. . . . . . . . . 10 |- ( z e. ( A [,] B ) |-> ( abs ` ( F ` z ) ) ) = ( x e. ( A [,] B ) |-> ( abs ` ( F ` x ) ) )
58	57	a1i 11	. . . . . . . . 9 |- ( ph -> ( z e. ( A [,] B ) |-> ( abs ` ( F ` z ) ) ) = ( x e. ( A [,] B ) |-> ( abs ` ( F ` x ) ) ) )
59	26	a1i 11	. . . . . . . . . . . 12 |- ( ph -> F = ( x e. ( A [,] B ) |-> C ) )
60	59, 7	feq1dd 39347	. . . . . . . . . . . . 13 |- ( ph -> ( x e. ( A [,] B ) |-> C ) : ( A [,] B ) --> ( RR \ { 0 } ) )
61	60	mptex2 6384	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A [,] B ) ) -> C e. ( RR \ { 0 } ) )
62	59, 61	fvmpt2d 6293	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) = C )
63	62	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( abs ` ( F ` x ) ) = ( abs ` C ) )
64	63	mpteq2dva 4744	. . . . . . . . 9 |- ( ph -> ( x e. ( A [,] B ) |-> ( abs ` ( F ` x ) ) ) = ( x e. ( A [,] B ) |-> ( abs ` C ) ) )
65	50, 58, 64	3eqtrd 2660	. . . . . . . 8 |- ( ph -> ( abs o. F ) = ( x e. ( A [,] B ) |-> ( abs ` C ) ) )
66	46, 61	sseldi 3601	. . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] B ) ) -> C e. CC )
67	66	abscld 14175	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( abs ` C ) e. RR )
68	65, 67	fvmpt2d 6293	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( abs o. F ) ` x ) = ( abs ` C ) )
69	68	ad4ant14 1293	. . . . . 6 |- ( ( ( ( ph /\ c e. ( A [,] B ) ) /\ A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) ) /\ x e. ( A [,] B ) ) -> ( ( abs o. F ) ` x ) = ( abs ` C ) )
70	41, 69	breqtrd 4679	. . . . 5 |- ( ( ( ( ph /\ c e. ( A [,] B ) ) /\ A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) ) /\ x e. ( A [,] B ) ) -> ( ( abs o. F ) ` c ) <_ ( abs ` C ) )
71	70	ex 450	. . . 4 |- ( ( ( ph /\ c e. ( A [,] B ) ) /\ A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) ) -> ( x e. ( A [,] B ) -> ( ( abs o. F ) ` c ) <_ ( abs ` C ) ) )
72	37, 71	ralrimi 2957	. . 3 |- ( ( ( ph /\ c e. ( A [,] B ) ) /\ A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) ) -> A. x e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( abs ` C ) )
73	31	nfeq2 2780	. . . . 5 |- F/ x y = ( ( abs o. F ) ` c )
74		breq1 4656	. . . . 5 |- ( y = ( ( abs o. F ) ` c ) -> ( y <_ ( abs ` C ) <-> ( ( abs o. F ) ` c ) <_ ( abs ` C ) ) )
75	73, 74	ralbid 2983	. . . 4 |- ( y = ( ( abs o. F ) ` c ) -> ( A. x e. ( A [,] B ) y <_ ( abs ` C ) <-> A. x e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( abs ` C ) ) )
76	75	rspcev 3309	. . 3 |- ( ( ( ( abs o. F ) ` c ) e. RR+ /\ A. x e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( abs ` C ) ) -> E. y e. RR+ A. x e. ( A [,] B ) y <_ ( abs ` C ) )
77	22, 72, 76	syl2anc 693	. 2 |- ( ( ( ph /\ c e. ( A [,] B ) ) /\ A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) ) -> E. y e. RR+ A. x e. ( A [,] B ) y <_ ( abs ` C ) )
78		cncficcgt0.a	. . . 4 |- ( ph -> A e. RR )
79		cncficcgt0.b	. . . 4 |- ( ph -> B e. RR )
80		cncficcgt0.aleb	. . . 4 |- ( ph -> A <_ B )
81		ssid 3624	. . . . . . . 8 |- CC C_ CC
82	81	a1i 11	. . . . . . 7 |- ( ph -> CC C_ CC )
83		cncfss 22702	. . . . . . 7 |- ( ( ( RR \ { 0 } ) C_ CC /\ CC C_ CC ) -> ( ( A [,] B ) -cn-> ( RR \ { 0 } ) ) C_ ( ( A [,] B ) -cn-> CC ) )
84	47, 82, 83	syl2anc 693	. . . . . 6 |- ( ph -> ( ( A [,] B ) -cn-> ( RR \ { 0 } ) ) C_ ( ( A [,] B ) -cn-> CC ) )
85	84, 1	sseldd 3604	. . . . 5 |- ( ph -> F e. ( ( A [,] B ) -cn-> CC ) )
86		abscncf 22704	. . . . . 6 |- abs e. ( CC -cn-> RR )
87	86	a1i 11	. . . . 5 |- ( ph -> abs e. ( CC -cn-> RR ) )
88	85, 87	cncfco 22710	. . . 4 |- ( ph -> ( abs o. F ) e. ( ( A [,] B ) -cn-> RR ) )
89	78, 79, 80, 88	evthicc 23228	. . 3 |- ( ph -> ( E. a e. ( A [,] B ) A. b e. ( A [,] B ) ( ( abs o. F ) ` b ) <_ ( ( abs o. F ) ` a ) /\ E. c e. ( A [,] B ) A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) ) )
90	89	simprd 479	. 2 |- ( ph -> E. c e. ( A [,] B ) A. d e. ( A [,] B ) ( ( abs o. F ) ` c ) <_ ( ( abs o. F ) ` d ) )
91	77, 90	r19.29a 3078	1 |- ( ph -> E. y e. RR+ A. x e. ( A [,] B ) y <_ ( abs ` C ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   <_ cle 10075   RR+crp 11832   [,]cicc 12178   abscabs 13974   -cn->ccncf 22679

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-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-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-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-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  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-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681

This theorem is referenced by:  fourierdlem68  40391