Theorem dvlt0 22138

Description: A function on a closed interval with negative derivative is decreasing. (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 ) )
dvlt0.d	|- ( ph -> ( RR _D F ) : ( A (,) B ) --> ( -oo (,) 0 ) )

Assertion

Ref	Expression
dvlt0	|- ( ph -> F Isom < , `' < ( ( A [,] B ) , ran F ) )

Proof of Theorem dvlt0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	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		dvlt0.d	. 2 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> ( -oo (,) 0 ) )
5		gtso 9662	. 2 |- `' < Or RR
6	1, 2, 3, 4	dvgt0lem1 22135	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. ( -oo (,) 0 ) )
7		eliooord 11580	. . . . . . . . 9 |- ( ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. ( -oo (,) 0 ) -> ( -oo < ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) /\ ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 ) )
8	6, 7	syl 16	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( -oo < ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) /\ ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 ) )
9	8	simprd 463	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 )
10		cncff 21129	. . . . . . . . . . . 12 |- ( F e. ( ( A [,] B ) -cn-> RR ) -> F : ( A [,] B ) --> RR )
11	3, 10	syl 16	. . . . . . . . . . 11 |- ( ph -> F : ( A [,] B ) --> RR )
12	11	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> F : ( A [,] B ) --> RR )
13		simplrr 760	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( A [,] B ) )
14	12, 13	ffvelrnd 6020	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) e. RR )
15		simplrl 759	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( A [,] B ) )
16	12, 15	ffvelrnd 6020	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. RR )
17	14, 16	resubcld 9983	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) e. RR )
18		0red 9593	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> 0 e. RR )
19		iccssre 11602	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
20	1, 2, 19	syl2anc 661	. . . . . . . . . . 11 |- ( ph -> ( A [,] B ) C_ RR )
21	20	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ RR )
22	21, 13	sseldd 3505	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR )
23	21, 15	sseldd 3505	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR )
24	22, 23	resubcld 9983	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR )
25		simpr 461	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x < y )
26	23, 22	posdifd 10135	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( x < y <-> 0 < ( y - x ) ) )
27	25, 26	mpbid 210	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> 0 < ( y - x ) )
28		ltdivmul 10413	. . . . . . . 8 |- ( ( ( ( F ` y ) - ( F ` x ) ) e. RR /\ 0 e. RR /\ ( ( y - x ) e. RR /\ 0 < ( y - x ) ) ) -> ( ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 <-> ( ( F ` y ) - ( F ` x ) ) < ( ( y - x ) x. 0 ) ) )
29	17, 18, 24, 27, 28	syl112anc 1232	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 <-> ( ( F ` y ) - ( F ` x ) ) < ( ( y - x ) x. 0 ) ) )
30	9, 29	mpbid 210	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) < ( ( y - x ) x. 0 ) )
31	24	recnd 9618	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. CC )
32	31	mul01d 9774	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( y - x ) x. 0 ) = 0 )
33	30, 32	breqtrd 4471	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) < 0 )
34	14, 16, 18	ltsubaddd 10144	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) < 0 <-> ( F ` y ) < ( 0 + ( F ` x ) ) ) )
35	33, 34	mpbid 210	. . . 4 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) < ( 0 + ( F ` x ) ) )
36	16	recnd 9618	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. CC )
37	36	addid2d 9776	. . . 4 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( 0 + ( F ` x ) ) = ( F ` x ) )
38	35, 37	breqtrd 4471	. . 3 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) < ( F ` x ) )
39		fvex 5874	. . . 4 |- ( F ` x ) e. _V
40		fvex 5874	. . . 4 |- ( F ` y ) e. _V
41	39, 40	brcnv 5183	. . 3 |- ( ( F ` x ) `' < ( F ` y ) <-> ( F ` y ) < ( F ` x ) )
42	38, 41	sylibr 212	. 2 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) `' < ( F ` y ) )
43	1, 2, 3, 4, 5, 42	dvgt0lem2 22136	1 |- ( ph -> F Isom < , `' < ( ( A [,] B ) , ran F ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1767   C_ wss 3476   class class class wbr 4447   `'ccnv 4998   ran crn 5000   -->wf 5582   ` cfv 5586   Isom wiso 5587  (class class class)co 6282   RRcr 9487   0cc0 9488   + caddc 9491   x. cmul 9493   -oocmnf 9622   < clt 9624   - cmin 9801   / cdiv 10202   (,)cioo 11525   [,]cicc 11528   -cn->ccncf 21112   _D cdv 21999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-0g 14690  df-gsum 14691  df-topgen 14692  df-pt 14693  df-prds 14696  df-xrs 14750  df-qtop 14755  df-imas 14756  df-xps 14758  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-submnd 15775  df-mulg 15858  df-cntz 16147  df-cmn 16593  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-fbas 18184  df-fg 18185  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-ntr 19284  df-cls 19285  df-nei 19362  df-lp 19400  df-perf 19401  df-cn 19491  df-cnp 19492  df-haus 19579  df-cmp 19650  df-tx 19795  df-hmeo 19988  df-fil 20079  df-fm 20171  df-flim 20172  df-flf 20173  df-xms 20555  df-ms 20556  df-tms 20557  df-cncf 21114  df-limc 22002  df-dv 22003

This theorem is referenced by:  dvne0  22144