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Theorem cncficcgt0 31894
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 21523 . . . . . . . 8  |-  ( F  e.  ( ( A [,] B ) -cn-> ( RR  \  { 0 } ) )  ->  F : ( A [,] B ) --> ( RR 
\  { 0 } ) )
3 ffun 5739 . . . . . . . 8  |-  ( F : ( A [,] B ) --> ( RR 
\  { 0 } )  ->  Fun  F )
41, 2, 33syl 20 . . . . . . 7  |-  ( ph  ->  Fun  F )
54adantr 465 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  Fun  F )
6 simpr 461 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  c  e.  ( A [,] B ) )
71, 2syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> ( RR  \  { 0 } ) )
8 fdm 5741 . . . . . . . . . 10  |-  ( F : ( A [,] B ) --> ( RR 
\  { 0 } )  ->  dom  F  =  ( A [,] B
) )
97, 8syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  ( A [,] B ) )
109eqcomd 2465 . . . . . . . 8  |-  ( ph  ->  ( A [,] B
)  =  dom  F
)
1110adantr 465 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( A [,] B )  =  dom  F )
126, 11eleqtrd 2547 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  c  e.  dom  F )
13 fvco 5949 . . . . . 6  |-  ( ( Fun  F  /\  c  e.  dom  F )  -> 
( ( abs  o.  F ) `  c
)  =  ( abs `  ( F `  c
) ) )
145, 12, 13syl2anc 661 . . . . 5  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( ( abs  o.  F ) `  c )  =  ( abs `  ( F `
 c ) ) )
157ffvelrnda 6032 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  e.  ( RR  \  { 0 } ) )
1615eldifad 3483 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  e.  RR )
1716recnd 9639 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  e.  CC )
18 eldifsni 4158 . . . . . . 7  |-  ( ( F `  c )  e.  ( RR  \  { 0 } )  ->  ( F `  c )  =/=  0
)
1915, 18syl 16 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( F `  c )  =/=  0
)
2017, 19absrpcld 13291 . . . . 5  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( abs `  ( F `  c
) )  e.  RR+ )
2114, 20eqeltrd 2545 . . . 4  |-  ( (
ph  /\  c  e.  ( A [,] B ) )  ->  ( ( abs  o.  F ) `  c )  e.  RR+ )
2221adantr 465 . . 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 1708 . . . . 5  |-  F/ x
( ph  /\  c  e.  ( A [,] B
) )
24 nfcv 2619 . . . . . 6  |-  F/_ x
( A [,] B
)
25 nfcv 2619 . . . . . . . . 9  |-  F/_ x abs
26 cncficcgt0.f . . . . . . . . . 10  |-  F  =  ( x  e.  ( A [,] B ) 
|->  C )
27 nfmpt1 4546 . . . . . . . . . 10  |-  F/_ x
( x  e.  ( A [,] B ) 
|->  C )
2826, 27nfcxfr 2617 . . . . . . . . 9  |-  F/_ x F
2925, 28nfco 5178 . . . . . . . 8  |-  F/_ x
( abs  o.  F
)
30 nfcv 2619 . . . . . . . 8  |-  F/_ x
c
3129, 30nffv 5879 . . . . . . 7  |-  F/_ x
( ( abs  o.  F ) `  c
)
32 nfcv 2619 . . . . . . 7  |-  F/_ x  <_
33 nfcv 2619 . . . . . . . 8  |-  F/_ x
d
3429, 33nffv 5879 . . . . . . 7  |-  F/_ x
( ( abs  o.  F ) `  d
)
3531, 32, 34nfbr 4500 . . . . . 6  |-  F/ x
( ( abs  o.  F ) `  c
)  <_  ( ( abs  o.  F ) `  d )
3624, 35nfral 2843 . . . . 5  |-  F/ x A. d  e.  ( A [,] B ) ( ( abs  o.  F
) `  c )  <_  ( ( abs  o.  F ) `  d
)
3723, 36nfan 1929 . . . 4  |-  F/ x
( ( ph  /\  c  e.  ( A [,] B ) )  /\  A. d  e.  ( A [,] B ) ( ( abs  o.  F
) `  c )  <_  ( ( abs  o.  F ) `  d
) )
38 fveq2 5872 . . . . . . . . 9  |-  ( d  =  x  ->  (
( abs  o.  F
) `  d )  =  ( ( abs 
o.  F ) `  x ) )
3938breq2d 4468 . . . . . . . 8  |-  ( d  =  x  ->  (
( ( abs  o.  F ) `  c
)  <_  ( ( abs  o.  F ) `  d )  <->  ( ( abs  o.  F ) `  c )  <_  (
( abs  o.  F
) `  x )
) )
4039rspccva 3209 . . . . . . 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 713 . . . . . 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 13182 . . . . . . . . . . 11  |-  abs : CC
--> RR
4342a1i 11 . . . . . . . . . 10  |-  ( ph  ->  abs : CC --> RR )
44 difss 3627 . . . . . . . . . . . . 13  |-  ( RR 
\  { 0 } )  C_  RR
45 ax-resscn 9566 . . . . . . . . . . . . 13  |-  RR  C_  CC
4644, 45sstri 3508 . . . . . . . . . . . 12  |-  ( RR 
\  { 0 } )  C_  CC
4746a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  \  {
0 } )  C_  CC )
487, 47fssd 5746 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
49 fcompt 6068 . . . . . . . . . 10  |-  ( ( abs : CC --> RR  /\  F : ( A [,] B ) --> CC )  ->  ( abs  o.  F )  =  ( z  e.  ( A [,] B )  |->  ( abs `  ( F `
 z ) ) ) )
5043, 48, 49syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( abs  o.  F
)  =  ( z  e.  ( A [,] B )  |->  ( abs `  ( F `  z
) ) ) )
51 nfcv 2619 . . . . . . . . . . . . 13  |-  F/_ x
z
5228, 51nffv 5879 . . . . . . . . . . . 12  |-  F/_ x
( F `  z
)
5325, 52nffv 5879 . . . . . . . . . . 11  |-  F/_ x
( abs `  ( F `  z )
)
54 nfcv 2619 . . . . . . . . . . 11  |-  F/_ z
( abs `  ( F `  x )
)
55 fveq2 5872 . . . . . . . . . . . 12  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
5655fveq2d 5876 . . . . . . . . . . 11  |-  ( z  =  x  ->  ( abs `  ( F `  z ) )  =  ( abs `  ( F `  x )
) )
5753, 54, 56cbvmpt 4547 . . . . . . . . . 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 31645 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  C ) : ( A [,] B ) --> ( RR  \  {
0 } ) )
6160mptex2 31648 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  C  e.  ( RR  \  { 0 } ) )
6259, 61fvmpt2d 5966 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  =  C )
6362fveq2d 5876 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( F `  x
) )  =  ( abs `  C ) )
6463mpteq2dva 4543 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( abs `  ( F `  x )
) )  =  ( x  e.  ( A [,] B )  |->  ( abs `  C ) ) )
6550, 58, 643eqtrd 2502 . . . . . . . 8  |-  ( ph  ->  ( abs  o.  F
)  =  ( x  e.  ( A [,] B )  |->  ( abs `  C ) ) )
6646, 61sseldi 3497 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  C  e.  CC )
6766abscld 13279 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  C )  e.  RR )
6865, 67fvmpt2d 5966 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( abs  o.  F ) `  x )  =  ( abs `  C ) )
6968adant423 31628 . . . . . 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 4480 . . . . 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 434 . . . 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 2857 . . 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 2636 . . . . 5  |-  F/ x  y  =  ( ( abs  o.  F ) `  c )
74 breq1 4459 . . . . 5  |-  ( y  =  ( ( abs 
o.  F ) `  c )  ->  (
y  <_  ( abs `  C )  <->  ( ( abs  o.  F ) `  c )  <_  ( abs `  C ) ) )
7573, 74ralbid 2891 . . . 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 3210 . . 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 661 . 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 3518 . . . . . . . 8  |-  CC  C_  CC
8281a1i 11 . . . . . . 7  |-  ( ph  ->  CC  C_  CC )
83 cncfss 21529 . . . . . . 7  |-  ( ( ( RR  \  {
0 } )  C_  CC  /\  CC  C_  CC )  ->  ( ( A [,] B ) -cn-> ( RR  \  { 0 } ) )  C_  ( ( A [,] B ) -cn-> CC ) )
8447, 82, 83syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( A [,] B ) -cn-> ( RR 
\  { 0 } ) )  C_  (
( A [,] B
) -cn-> CC ) )
8584, 1sseldd 3500 . . . . 5  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
86 abscncf 21531 . . . . . 6  |-  abs  e.  ( CC -cn-> RR )
8786a1i 11 . . . . 5  |-  ( ph  ->  abs  e.  ( CC
-cn-> RR ) )
8885, 87cncfco 21537 . . . 4  |-  ( ph  ->  ( abs  o.  F
)  e.  ( ( A [,] B )
-cn-> RR ) )
8978, 79, 80, 88evthicc 21997 . . 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 463 . 2  |-  ( ph  ->  E. c  e.  ( A [,] B ) A. d  e.  ( A [,] B ) ( ( abs  o.  F ) `  c
)  <_  ( ( abs  o.  F ) `  d ) )
9177, 90r19.29a 2999 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 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808    \ cdif 3468    C_ wss 3471   {csn 4032   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008    o. ccom 5012   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509    <_ cle 9646   RR+crp 11245   [,]cicc 11557   abscabs 13079   -cn->ccncf 21506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-icc 11561  df-fz 11698  df-fzo 11822  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cn 19855  df-cnp 19856  df-cmp 20014  df-tx 20189  df-hmeo 20382  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508
This theorem is referenced by:  fourierdlem68  32160
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