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Theorem dvgt0 22587
Description: A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.)
Hypotheses
Ref Expression
dvgt0.a  |-  ( ph  ->  A  e.  RR )
dvgt0.b  |-  ( ph  ->  B  e.  RR )
dvgt0.f  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
dvgt0.d  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> RR+ )
Assertion
Ref Expression
dvgt0  |-  ( ph  ->  F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
) )

Proof of Theorem dvgt0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvgt0.a . 2  |-  ( ph  ->  A  e.  RR )
2 dvgt0.b . 2  |-  ( ph  ->  B  e.  RR )
3 dvgt0.f . 2  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
4 dvgt0.d . 2  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> RR+ )
5 ltso 9614 . 2  |-  <  Or  RR
61, 2, 3, 4dvgt0lem1 22585 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  RR+ )
76rpgt0d 11223 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  0  <  ( ( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) ) )
8 cncff 21579 . . . . . . . . 9  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
93, 8syl 17 . . . . . . . 8  |-  ( ph  ->  F : ( A [,] B ) --> RR )
109ad2antrr 724 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F :
( A [,] B
) --> RR )
11 simplrr 761 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( A [,] B ) )
1210, 11ffvelrnd 5964 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  e.  RR )
13 simplrl 760 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( A [,] B ) )
1410, 13ffvelrnd 5964 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  e.  RR )
1512, 14resubcld 9946 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  e.  RR )
16 iccssre 11575 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
171, 2, 16syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( A [,] B
)  C_  RR )
1817ad2antrr 724 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  RR )
1918, 11sseldd 3440 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR )
2018, 13sseldd 3440 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR )
2119, 20resubcld 9946 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR )
22 simpr 459 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <  y )
2320, 19posdifd 10097 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x  <  y  <->  0  <  (
y  -  x ) ) )
2422, 23mpbid 210 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  0  <  ( y  -  x ) )
25 gt0div 10367 . . . . 5  |-  ( ( ( ( F `  y )  -  ( F `  x )
)  e.  RR  /\  ( y  -  x
)  e.  RR  /\  0  <  ( y  -  x ) )  -> 
( 0  <  (
( F `  y
)  -  ( F `
 x ) )  <->  0  <  ( ( ( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) ) ) )
2615, 21, 24, 25syl3anc 1228 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( 0  <  ( ( F `
 y )  -  ( F `  x ) )  <->  0  <  (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) ) ) )
277, 26mpbird 232 . . 3  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  0  <  ( ( F `  y
)  -  ( F `
 x ) ) )
2814, 12posdifd 10097 . . 3  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  x )  <  ( F `  y
)  <->  0  <  (
( F `  y
)  -  ( F `
 x ) ) ) )
2927, 28mpbird 232 . 2  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  <  ( F `  y )
)
301, 2, 3, 4, 5, 29dvgt0lem2 22586 1  |-  ( ph  ->  F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1840    C_ wss 3411   class class class wbr 4392   ran crn 4941   -->wf 5519   ` cfv 5523    Isom wiso 5524  (class class class)co 6232   RRcr 9439   0cc0 9440    < clt 9576    - cmin 9759    / cdiv 10165   RR+crp 11181   (,)cioo 11498   [,]cicc 11501   -cn->ccncf 21562    _D cdv 22449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-fi 7823  df-sup 7853  df-oi 7887  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-q 11144  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-ioo 11502  df-ico 11504  df-icc 11505  df-fz 11642  df-fzo 11766  df-seq 12060  df-exp 12119  df-hash 12358  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-hom 14823  df-cco 14824  df-rest 14927  df-topn 14928  df-0g 14946  df-gsum 14947  df-topgen 14948  df-pt 14949  df-prds 14952  df-xrs 15006  df-qtop 15011  df-imas 15012  df-xps 15014  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-submnd 16181  df-mulg 16274  df-cntz 16569  df-cmn 17014  df-psmet 18621  df-xmet 18622  df-met 18623  df-bl 18624  df-mopn 18625  df-fbas 18626  df-fg 18627  df-cnfld 18631  df-top 19581  df-bases 19583  df-topon 19584  df-topsp 19585  df-cld 19702  df-ntr 19703  df-cls 19704  df-nei 19782  df-lp 19820  df-perf 19821  df-cn 19911  df-cnp 19912  df-haus 19999  df-cmp 20070  df-tx 20245  df-hmeo 20438  df-fil 20529  df-fm 20621  df-flim 20622  df-flf 20623  df-xms 21005  df-ms 21006  df-tms 21007  df-cncf 21564  df-limc 22452  df-dv 22453
This theorem is referenced by:  dvne0  22594
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