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Theorem cncficcgt0 37059
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.
StepHypRef Expression
1 cncficcgt0.fcn . . . . . . . 8  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> ( RR  \  {
0 } ) ) )
2 cncff 21689 . . . . . . . 8  |-  ( F  e.  ( ( A [,] B ) -cn-> ( RR  \  { 0 } ) )  ->  F : ( A [,] B ) --> ( RR 
\  { 0 } ) )
3 ffun 5716 . . . . . . . 8  |-  ( F : ( A [,] B ) --> ( RR 
\  { 0 } )  ->  Fun  F )
41, 2, 33syl 18 . . . . . . 7  |-  ( ph  ->  Fun  F )
54adantr 463 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  Fun  F )
6 simpr 459 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  c  e.  ( A [,] B ) )
71, 2syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> ( RR  \  { 0 } ) )
8 fdm 5718 . . . . . . . . . 10  |-  ( F : ( A [,] B ) --> ( RR 
\  { 0 } )  ->  dom  F  =  ( A [,] B
) )
97, 8syl 17 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  ( A [,] B ) )
109eqcomd 2410 . . . . . . . 8  |-  ( ph  ->  ( A [,] B
)  =  dom  F
)
1110adantr 463 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( A [,] B )  =  dom  F )
126, 11eleqtrd 2492 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  c  e.  dom  F )
13 fvco 5925 . . . . . 6  |-  ( ( Fun  F  /\  c  e.  dom  F )  -> 
( ( abs  o.  F ) `  c
)  =  ( abs `  ( F `  c
) ) )
145, 12, 13syl2anc 659 . . . . 5  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( ( abs  o.  F ) `  c )  =  ( abs `  ( F `
 c ) ) )
157ffvelrnda 6009 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  e.  ( RR  \  { 0 } ) )
1615eldifad 3426 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  e.  RR )
1716recnd 9652 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  e.  CC )
18 eldifsni 4098 . . . . . . 7  |-  ( ( F `  c )  e.  ( RR  \  { 0 } )  ->  ( F `  c )  =/=  0
)
1915, 18syl 17 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  =/=  0
)
2017, 19absrpcld 13428 . . . . 5  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( abs `  ( F `  c
) )  e.  RR+ )
2114, 20eqeltrd 2490 . . . 4  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( ( abs  o.  F ) `  c )  e.  RR+ )
2221adantr 463 . . 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 1728 . . . . 5  |-  F/ x
( ph  /\  c  e.  ( A [,] B
) )
24 nfcv 2564 . . . . . 6  |-  F/_ x
( A [,] B
)
25 nfcv 2564 . . . . . . . . 9  |-  F/_ x abs
26 cncficcgt0.f . . . . . . . . . 10  |-  F  =  ( x  e.  ( A [,] B ) 
|->  C )
27 nfmpt1 4484 . . . . . . . . . 10  |-  F/_ x
( x  e.  ( A [,] B ) 
|->  C )
2826, 27nfcxfr 2562 . . . . . . . . 9  |-  F/_ x F
2925, 28nfco 4989 . . . . . . . 8  |-  F/_ x
( abs  o.  F
)
30 nfcv 2564 . . . . . . . 8  |-  F/_ x
c
3129, 30nffv 5856 . . . . . . 7  |-  F/_ x
( ( abs  o.  F ) `  c
)
32 nfcv 2564 . . . . . . 7  |-  F/_ x  <_
33 nfcv 2564 . . . . . . . 8  |-  F/_ x
d
3429, 33nffv 5856 . . . . . . 7  |-  F/_ x
( ( abs  o.  F ) `  d
)
3531, 32, 34nfbr 4439 . . . . . 6  |-  F/ x
( ( abs  o.  F ) `  c
)  <_  ( ( abs  o.  F ) `  d )
3624, 35nfral 2790 . . . . 5  |-  F/ x A. d  e.  ( A [,] B ) ( ( abs  o.  F
) `  c )  <_  ( ( abs  o.  F ) `  d
)
3723, 36nfan 1956 . . . 4  |-  F/ x
( ( ph  /\  c  e.  ( A [,] B ) )  /\  A. d  e.  ( A [,] B ) ( ( abs  o.  F
) `  c )  <_  ( ( abs  o.  F ) `  d
) )
38 fveq2 5849 . . . . . . . . 9  |-  ( d  =  x  ->  (
( abs  o.  F
) `  d )  =  ( ( abs 
o.  F ) `  x ) )
3938breq2d 4407 . . . . . . . 8  |-  ( d  =  x  ->  (
( ( abs  o.  F ) `  c
)  <_  ( ( abs  o.  F ) `  d )  <->  ( ( abs  o.  F ) `  c )  <_  (
( abs  o.  F
) `  x )
) )
4039rspccva 3159 . . . . . . 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
) )
4140adantll 712 . . . . . 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 13319 . . . . . . . . . . 11  |-  abs : CC
--> RR
4342a1i 11 . . . . . . . . . 10  |-  ( ph  ->  abs : CC --> RR )
44 difss 3570 . . . . . . . . . . . . 13  |-  ( RR 
\  { 0 } )  C_  RR
45 ax-resscn 9579 . . . . . . . . . . . . 13  |-  RR  C_  CC
4644, 45sstri 3451 . . . . . . . . . . . 12  |-  ( RR 
\  { 0 } )  C_  CC
4746a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  \  {
0 } )  C_  CC )
487, 47fssd 5723 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
49 fcompt 6046 . . . . . . . . . 10  |-  ( ( abs : CC --> RR  /\  F : ( A [,] B ) --> CC )  ->  ( abs  o.  F )  =  ( z  e.  ( A [,] B )  |->  ( abs `  ( F `
 z ) ) ) )
5043, 48, 49syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( abs  o.  F
)  =  ( z  e.  ( A [,] B )  |->  ( abs `  ( F `  z
) ) ) )
51 nfcv 2564 . . . . . . . . . . . . 13  |-  F/_ x
z
5228, 51nffv 5856 . . . . . . . . . . . 12  |-  F/_ x
( F `  z
)
5325, 52nffv 5856 . . . . . . . . . . 11  |-  F/_ x
( abs `  ( F `  z )
)
54 nfcv 2564 . . . . . . . . . . 11  |-  F/_ z
( abs `  ( F `  x )
)
55 fveq2 5849 . . . . . . . . . . . 12  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
5655fveq2d 5853 . . . . . . . . . . 11  |-  ( z  =  x  ->  ( abs `  ( F `  z ) )  =  ( abs `  ( F `  x )
) )
5753, 54, 56cbvmpt 4486 . . . . . . . . . 10  |-  ( z  e.  ( A [,] B )  |->  ( abs `  ( F `  z
) ) )  =  ( x  e.  ( A [,] B ) 
|->  ( abs `  ( F `  x )
) )
5857a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( abs `  ( F `  z )
) )  =  ( x  e.  ( A [,] B )  |->  ( abs `  ( F `
 x ) ) ) )
5926a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  F  =  ( x  e.  ( A [,] B )  |->  C ) )
6059, 7feq1dd 36817 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  C ) : ( A [,] B ) --> ( RR  \  {
0 } ) )
6160mptex2 36820 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  C  e.  ( RR  \  { 0 } ) )
6259, 61fvmpt2d 5943 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  =  C )
6362fveq2d 5853 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( F `  x
) )  =  ( abs `  C ) )
6463mpteq2dva 4481 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( abs `  ( F `  x )
) )  =  ( x  e.  ( A [,] B )  |->  ( abs `  C ) ) )
6550, 58, 643eqtrd 2447 . . . . . . . 8  |-  ( ph  ->  ( abs  o.  F
)  =  ( x  e.  ( A [,] B )  |->  ( abs `  C ) ) )
6646, 61sseldi 3440 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  C  e.  CC )
6766abscld 13416 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  C )  e.  RR )
6865, 67fvmpt2d 5943 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( abs  o.  F ) `  x )  =  ( abs `  C ) )
6968adant423 36802 . . . . . 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
) )
7041, 69breqtrd 4419 . . . . 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
) )
7170ex 432 . . . 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 ) ) )
7237, 71ralrimi 2804 . . 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 ) )
7331nfeq2 2581 . . . . 5  |-  F/ x  y  =  ( ( abs  o.  F ) `  c )
74 breq1 4398 . . . . 5  |-  ( y  =  ( ( abs 
o.  F ) `  c )  ->  (
y  <_  ( abs `  C )  <->  ( ( abs  o.  F ) `  c )  <_  ( abs `  C ) ) )
7573, 74ralbid 2838 . . . 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 ) ) )
7675rspcev 3160 . . 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 ) )
7722, 72, 76syl2anc 659 . 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 3461 . . . . . . . 8  |-  CC  C_  CC
8281a1i 11 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
83 cncfss 21695 . . . . . . 7  |-  ( ( ( RR  \  {
0 } )  C_  CC  /\  CC  C_  CC )  ->  ( ( A [,] B ) -cn-> ( RR  \  { 0 } ) )  C_  ( ( A [,] B ) -cn-> CC ) )
8447, 82, 83syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ( A [,] B ) -cn-> ( RR 
\  { 0 } ) )  C_  (
( A [,] B
) -cn-> CC ) )
8584, 1sseldd 3443 . . . . 5  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
86 abscncf 21697 . . . . . 6  |-  abs  e.  ( CC -cn-> RR )
8786a1i 11 . . . . 5  |-  ( ph  ->  abs  e.  ( CC
-cn-> RR ) )
8885, 87cncfco 21703 . . . 4  |-  ( ph  ->  ( abs  o.  F
)  e.  ( ( A [,] B )
-cn-> RR ) )
8978, 79, 80, 88evthicc 22163 . . 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 ) ) )
9089simprd 461 . 2  |-  ( ph  ->  E. c  e.  ( A [,] B ) A. d  e.  ( A [,] B ) ( ( abs  o.  F ) `  c
)  <_  ( ( abs  o.  F ) `  d ) )
9177, 90r19.29a 2949 1  |-  ( ph  ->  E. y  e.  RR+  A. x  e.  ( A [,] B ) y  <_  ( abs `  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755    \ cdif 3411    C_ wss 3414   {csn 3972   class class class wbr 4395    |-> cmpt 4453   dom cdm 4823    o. ccom 4827   Fun wfun 5563   -->wf 5565   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522    <_ cle 9659   RR+crp 11265   [,]cicc 11585   abscabs 13216   -cn->ccncf 21672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-icc 11589  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cn 20021  df-cnp 20022  df-cmp 20180  df-tx 20355  df-hmeo 20548  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674
This theorem is referenced by:  fourierdlem68  37325
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