Theorem dvlt0 22698

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 9697	. 2 |- `' < Or RR
6	1, 2, 3, 4	dvgt0lem1 22695	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. ( -oo (,) 0 ) )
7		eliooord 11638	. . . . . . . . 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 17	. . . . . . . 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 461	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) < 0 )
10		cncff 21689	. . . . . . . . . . . 12 |- ( F e. ( ( A [,] B ) -cn-> RR ) -> F : ( A [,] B ) --> RR )
11	3, 10	syl 17	. . . . . . . . . . 11 |- ( ph -> F : ( A [,] B ) --> RR )
12	11	ad2antrr 724	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> F : ( A [,] B ) --> RR )
13		simplrr 763	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. ( A [,] B ) )
14	12, 13	ffvelrnd 6010	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) e. RR )
15		simplrl 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. ( A [,] B ) )
16	12, 15	ffvelrnd 6010	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. RR )
17	14, 16	resubcld 10028	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) e. RR )
18		0red 9627	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> 0 e. RR )
19		iccssre 11660	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
20	1, 2, 19	syl2anc 659	. . . . . . . . . . 11 |- ( ph -> ( A [,] B ) C_ RR )
21	20	ad2antrr 724	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( A [,] B ) C_ RR )
22	21, 13	sseldd 3443	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR )
23	21, 15	sseldd 3443	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR )
24	22, 23	resubcld 10028	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. RR )
25		simpr 459	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x < y )
26	23, 22	posdifd 10179	. . . . . . . . 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 10458	. . . . . . . 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 1234	. . . . . . 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 9652	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. CC )
32	31	mul01d 9813	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( y - x ) x. 0 ) = 0 )
33	30, 32	breqtrd 4419	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) < 0 )
34	14, 16, 18	ltsubaddd 10188	. . . . 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 9652	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. CC )
37	36	addid2d 9815	. . . 4 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( 0 + ( F ` x ) ) = ( F ` x ) )
38	35, 37	breqtrd 4419	. . 3 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) < ( F ` x ) )
39		fvex 5859	. . . 4 |- ( F ` x ) e. _V
40		fvex 5859	. . . 4 |- ( F ` y ) e. _V
41	39, 40	brcnv 5006	. . 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 22696	1 |- ( ph -> F Isom < , `' < ( ( A [,] B ) , ran F ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   e. wcel 1842   C_ wss 3414   class class class wbr 4395   `'ccnv 4822   ran crn 4824   -->wf 5565   ` cfv 5569   Isom wiso 5570  (class class class)co 6278   RRcr 9521   0cc0 9522   + caddc 9525   x. cmul 9527   -oocmnf 9656   < clt 9658   - cmin 9841   / cdiv 10247   (,)cioo 11582   [,]cicc 11585   -cn->ccncf 21672   _D cdv 22559

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-addf 9601  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-pm 7460  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-ico 11588  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-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-haus 20109  df-cmp 20180  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674  df-limc 22562  df-dv 22563

This theorem is referenced by:  dvne0  22704