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Theorem cncficcgt0 37760
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 21918 . . . . . . . 8  |-  ( F  e.  ( ( A [,] B ) -cn-> ( RR  \  { 0 } ) )  ->  F : ( A [,] B ) --> ( RR 
\  { 0 } ) )
3 ffun 5729 . . . . . . . 8  |-  ( F : ( A [,] B ) --> ( RR 
\  { 0 } )  ->  Fun  F )
41, 2, 33syl 18 . . . . . . 7  |-  ( ph  ->  Fun  F )
54adantr 467 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  Fun  F )
6 simpr 463 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  c  e.  ( A [,] B ) )
71, 2syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> ( RR  \  { 0 } ) )
8 fdm 5731 . . . . . . . . . 10  |-  ( F : ( A [,] B ) --> ( RR 
\  { 0 } )  ->  dom  F  =  ( A [,] B
) )
97, 8syl 17 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  ( A [,] B ) )
109eqcomd 2456 . . . . . . . 8  |-  ( ph  ->  ( A [,] B
)  =  dom  F
)
1110adantr 467 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( A [,] B )  =  dom  F )
126, 11eleqtrd 2530 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  c  e.  dom  F )
13 fvco 5939 . . . . . 6  |-  ( ( Fun  F  /\  c  e.  dom  F )  -> 
( ( abs  o.  F ) `  c
)  =  ( abs `  ( F `  c
) ) )
145, 12, 13syl2anc 666 . . . . 5  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( ( abs  o.  F ) `  c )  =  ( abs `  ( F `
 c ) ) )
157ffvelrnda 6020 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  e.  ( RR  \  { 0 } ) )
1615eldifad 3415 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  e.  RR )
1716recnd 9666 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  e.  CC )
18 eldifsni 4097 . . . . . . 7  |-  ( ( F `  c )  e.  ( RR  \  { 0 } )  ->  ( F `  c )  =/=  0
)
1915, 18syl 17 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  =/=  0
)
2017, 19absrpcld 13503 . . . . 5  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( abs `  ( F `  c
) )  e.  RR+ )
2114, 20eqeltrd 2528 . . . 4  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( ( abs  o.  F ) `  c )  e.  RR+ )
2221adantr 467 . . 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 1760 . . . . 5  |-  F/ x
( ph  /\  c  e.  ( A [,] B
) )
24 nfcv 2591 . . . . . 6  |-  F/_ x
( A [,] B
)
25 nfcv 2591 . . . . . . . . 9  |-  F/_ x abs
26 cncficcgt0.f . . . . . . . . . 10  |-  F  =  ( x  e.  ( A [,] B ) 
|->  C )
27 nfmpt1 4491 . . . . . . . . . 10  |-  F/_ x
( x  e.  ( A [,] B ) 
|->  C )
2826, 27nfcxfr 2589 . . . . . . . . 9  |-  F/_ x F
2925, 28nfco 4999 . . . . . . . 8  |-  F/_ x
( abs  o.  F
)
30 nfcv 2591 . . . . . . . 8  |-  F/_ x
c
3129, 30nffv 5870 . . . . . . 7  |-  F/_ x
( ( abs  o.  F ) `  c
)
32 nfcv 2591 . . . . . . 7  |-  F/_ x  <_
33 nfcv 2591 . . . . . . . 8  |-  F/_ x
d
3429, 33nffv 5870 . . . . . . 7  |-  F/_ x
( ( abs  o.  F ) `  d
)
3531, 32, 34nfbr 4446 . . . . . 6  |-  F/ x
( ( abs  o.  F ) `  c
)  <_  ( ( abs  o.  F ) `  d )
3624, 35nfral 2773 . . . . 5  |-  F/ x A. d  e.  ( A [,] B ) ( ( abs  o.  F
) `  c )  <_  ( ( abs  o.  F ) `  d
)
3723, 36nfan 2010 . . . 4  |-  F/ x
( ( ph  /\  c  e.  ( A [,] B ) )  /\  A. d  e.  ( A [,] B ) ( ( abs  o.  F
) `  c )  <_  ( ( abs  o.  F ) `  d
) )
38 fveq2 5863 . . . . . . . . 9  |-  ( d  =  x  ->  (
( abs  o.  F
) `  d )  =  ( ( abs 
o.  F ) `  x ) )
3938breq2d 4413 . . . . . . . 8  |-  ( d  =  x  ->  (
( ( abs  o.  F ) `  c
)  <_  ( ( abs  o.  F ) `  d )  <->  ( ( abs  o.  F ) `  c )  <_  (
( abs  o.  F
) `  x )
) )
4039rspccva 3148 . . . . . . 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 719 . . . . . 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 13393 . . . . . . . . . . 11  |-  abs : CC
--> RR
4342a1i 11 . . . . . . . . . 10  |-  ( ph  ->  abs : CC --> RR )
44 difss 3559 . . . . . . . . . . . . 13  |-  ( RR 
\  { 0 } )  C_  RR
45 ax-resscn 9593 . . . . . . . . . . . . 13  |-  RR  C_  CC
4644, 45sstri 3440 . . . . . . . . . . . 12  |-  ( RR 
\  { 0 } )  C_  CC
4746a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  \  {
0 } )  C_  CC )
487, 47fssd 5736 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
49 fcompt 6057 . . . . . . . . . 10  |-  ( ( abs : CC --> RR  /\  F : ( A [,] B ) --> CC )  ->  ( abs  o.  F )  =  ( z  e.  ( A [,] B )  |->  ( abs `  ( F `
 z ) ) ) )
5043, 48, 49syl2anc 666 . . . . . . . . 9  |-  ( ph  ->  ( abs  o.  F
)  =  ( z  e.  ( A [,] B )  |->  ( abs `  ( F `  z
) ) ) )
51 nfcv 2591 . . . . . . . . . . . . 13  |-  F/_ x
z
5228, 51nffv 5870 . . . . . . . . . . . 12  |-  F/_ x
( F `  z
)
5325, 52nffv 5870 . . . . . . . . . . 11  |-  F/_ x
( abs `  ( F `  z )
)
54 nfcv 2591 . . . . . . . . . . 11  |-  F/_ z
( abs `  ( F `  x )
)
55 fveq2 5863 . . . . . . . . . . . 12  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
5655fveq2d 5867 . . . . . . . . . . 11  |-  ( z  =  x  ->  ( abs `  ( F `  z ) )  =  ( abs `  ( F `  x )
) )
5753, 54, 56cbvmpt 4493 . . . . . . . . . 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 37424 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  C ) : ( A [,] B ) --> ( RR  \  {
0 } ) )
6160mptex2 37427 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  C  e.  ( RR  \  { 0 } ) )
6259, 61fvmpt2d 5957 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  =  C )
6362fveq2d 5867 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( F `  x
) )  =  ( abs `  C ) )
6463mpteq2dva 4488 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( abs `  ( F `  x )
) )  =  ( x  e.  ( A [,] B )  |->  ( abs `  C ) ) )
6550, 58, 643eqtrd 2488 . . . . . . . 8  |-  ( ph  ->  ( abs  o.  F
)  =  ( x  e.  ( A [,] B )  |->  ( abs `  C ) ) )
6646, 61sseldi 3429 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  C  e.  CC )
6766abscld 13491 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  C )  e.  RR )
6865, 67fvmpt2d 5957 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( abs  o.  F ) `  x )  =  ( abs `  C ) )
6968adant423 37361 . . . . . 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 4426 . . . . 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 436 . . . 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 2787 . . 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 2606 . . . . 5  |-  F/ x  y  =  ( ( abs  o.  F ) `  c )
74 breq1 4404 . . . . 5  |-  ( y  =  ( ( abs 
o.  F ) `  c )  ->  (
y  <_  ( abs `  C )  <->  ( ( abs  o.  F ) `  c )  <_  ( abs `  C ) ) )
7573, 74ralbid 2821 . . . 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 3149 . . 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 666 . 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 3450 . . . . . . . 8  |-  CC  C_  CC
8281a1i 11 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
83 cncfss 21924 . . . . . . 7  |-  ( ( ( RR  \  {
0 } )  C_  CC  /\  CC  C_  CC )  ->  ( ( A [,] B ) -cn-> ( RR  \  { 0 } ) )  C_  ( ( A [,] B ) -cn-> CC ) )
8447, 82, 83syl2anc 666 . . . . . 6  |-  ( ph  ->  ( ( A [,] B ) -cn-> ( RR 
\  { 0 } ) )  C_  (
( A [,] B
) -cn-> CC ) )
8584, 1sseldd 3432 . . . . 5  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
86 abscncf 21926 . . . . . 6  |-  abs  e.  ( CC -cn-> RR )
8786a1i 11 . . . . 5  |-  ( ph  ->  abs  e.  ( CC
-cn-> RR ) )
8885, 87cncfco 21932 . . . 4  |-  ( ph  ->  ( abs  o.  F
)  e.  ( ( A [,] B )
-cn-> RR ) )
8978, 79, 80, 88evthicc 22403 . . 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 465 . 2  |-  ( ph  ->  E. c  e.  ( A [,] B ) A. d  e.  ( A [,] B ) ( ( abs  o.  F ) `  c
)  <_  ( ( abs  o.  F ) `  d ) )
9177, 90r19.29a 2931 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 371    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737    \ cdif 3400    C_ wss 3403   {csn 3967   class class class wbr 4401    |-> cmpt 4460   dom cdm 4833    o. ccom 4837   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536    <_ cle 9673   RR+crp 11299   [,]cicc 11635   abscabs 13290   -cn->ccncf 21901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-icc 11639  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cn 20236  df-cnp 20237  df-cmp 20395  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903
This theorem is referenced by:  fourierdlem68  38032
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