Theorem dvlt0 21613

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 9570	. 2 |- `' < Or RR
6	1, 2, 3, 4	dvgt0lem1 21610	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( ( F ` y ) - ( F ` x ) ) / ( y - x ) ) e. ( -oo (,) 0 ) )
7		eliooord 11469	. . . . . . . . 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 20604	. . . . . . . . . . . 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 5956	. . . . . . . . 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 5956	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. RR )
17	14, 16	resubcld 9890	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) e. RR )
18		0red 9501	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> 0 e. RR )
19		iccssre 11491	. . . . . . . . . . . 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 3468	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> y e. RR )
23	21, 15	sseldd 3468	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> x e. RR )
24	22, 23	resubcld 9890	. . . . . . . 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 10040	. . . . . . . . 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 10318	. . . . . . . 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 1223	. . . . . . 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 9526	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( y - x ) e. CC )
32	31	mul01d 9682	. . . . . 6 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( y - x ) x. 0 ) = 0 )
33	30, 32	breqtrd 4427	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( ( F ` y ) - ( F ` x ) ) < 0 )
34	14, 16, 18	ltsubaddd 10049	. . . . 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 9526	. . . . 5 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` x ) e. CC )
37	36	addid2d 9684	. . . 4 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( 0 + ( F ` x ) ) = ( F ` x ) )
38	35, 37	breqtrd 4427	. . 3 |- ( ( ( ph /\ ( x e. ( A [,] B ) /\ y e. ( A [,] B ) ) ) /\ x < y ) -> ( F ` y ) < ( F ` x ) )
39		fvex 5812	. . . 4 |- ( F ` x ) e. _V
40		fvex 5812	. . . 4 |- ( F ` y ) e. _V
41	39, 40	brcnv 5133	. . 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 21611	1 |- ( ph -> F Isom < , `' < ( ( A [,] B ) , ran F ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1758   C_ wss 3439   class class class wbr 4403   `'ccnv 4950   ran crn 4952   -->wf 5525   ` cfv 5529   Isom wiso 5530  (class class class)co 6203   RRcr 9395   0cc0 9396   + caddc 9399   x. cmul 9401   -oocmnf 9530   < clt 9532   - cmin 9709   / cdiv 10107   (,)cioo 11414   [,]cicc 11417   -cn->ccncf 20587   _D cdv 21474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474  ax-addf 9475  ax-mulf 9476

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-fi 7775  df-sup 7805  df-oi 7838  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-q 11068  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-ioo 11418  df-ico 11420  df-icc 11421  df-fz 11558  df-fzo 11669  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-starv 14375  df-sca 14376  df-vsca 14377  df-ip 14378  df-tset 14379  df-ple 14380  df-ds 14382  df-unif 14383  df-hom 14384  df-cco 14385  df-rest 14483  df-topn 14484  df-0g 14502  df-gsum 14503  df-topgen 14504  df-pt 14505  df-prds 14508  df-xrs 14562  df-qtop 14567  df-imas 14568  df-xps 14570  df-mre 14646  df-mrc 14647  df-acs 14649  df-mnd 15537  df-submnd 15587  df-mulg 15670  df-cntz 15957  df-cmn 16403  df-psmet 17937  df-xmet 17938  df-met 17939  df-bl 17940  df-mopn 17941  df-fbas 17942  df-fg 17943  df-cnfld 17947  df-top 18638  df-bases 18640  df-topon 18641  df-topsp 18642  df-cld 18758  df-ntr 18759  df-cls 18760  df-nei 18837  df-lp 18875  df-perf 18876  df-cn 18966  df-cnp 18967  df-haus 19054  df-cmp 19125  df-tx 19270  df-hmeo 19463  df-fil 19554  df-fm 19646  df-flim 19647  df-flf 19648  df-xms 20030  df-ms 20031  df-tms 20032  df-cncf 20589  df-limc 21477  df-dv 21478

This theorem is referenced by:  dvne0  21619