Theorem monoord2 12832

Description: Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)

Hypotheses

Ref	Expression
monoord2.1	|- ( ph -> N e. ( ZZ>= ` M ) )
monoord2.2	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR )
monoord2.3	|- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )

Assertion

Ref	Expression
monoord2	|- ( ph -> ( F ` N ) <_ ( F ` M ) )

Distinct variable groups:   k, F   k, M   k, N   ph, k

Proof of Theorem monoord2

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		monoord2.1	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
2		monoord2.2	. . . . . . 7 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR )
3	2	renegcld 10457	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> -u ( F ` k ) e. RR )
4		eqid 2622	. . . . . 6 |- ( k e. ( M ... N ) |-> -u ( F ` k ) ) = ( k e. ( M ... N ) |-> -u ( F ` k ) )
5	3, 4	fmptd 6385	. . . . 5 |- ( ph -> ( k e. ( M ... N ) |-> -u ( F ` k ) ) : ( M ... N ) --> RR )
6	5	ffvelrnda 6359	. . . 4 |- ( ( ph /\ n e. ( M ... N ) ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` n ) e. RR )
7		monoord2.3	. . . . . . . . 9 |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )
8	7	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. k e. ( M ... ( N - 1 ) ) ( F ` ( k + 1 ) ) <_ ( F ` k ) )
9		oveq1 6657	. . . . . . . . . . 11 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
10	9	fveq2d 6195	. . . . . . . . . 10 |- ( k = n -> ( F ` ( k + 1 ) ) = ( F ` ( n + 1 ) ) )
11		fveq2 6191	. . . . . . . . . 10 |- ( k = n -> ( F ` k ) = ( F ` n ) )
12	10, 11	breq12d 4666	. . . . . . . . 9 |- ( k = n -> ( ( F ` ( k + 1 ) ) <_ ( F ` k ) <-> ( F ` ( n + 1 ) ) <_ ( F ` n ) ) )
13	12	cbvralv 3171	. . . . . . . 8 |- ( A. k e. ( M ... ( N - 1 ) ) ( F ` ( k + 1 ) ) <_ ( F ` k ) <-> A. n e. ( M ... ( N - 1 ) ) ( F ` ( n + 1 ) ) <_ ( F ` n ) )
14	8, 13	sylib 208	. . . . . . 7 |- ( ph -> A. n e. ( M ... ( N - 1 ) ) ( F ` ( n + 1 ) ) <_ ( F ` n ) )
15	14	r19.21bi 2932	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` ( n + 1 ) ) <_ ( F ` n ) )
16		fzp1elp1 12394	. . . . . . . . . 10 |- ( n e. ( M ... ( N - 1 ) ) -> ( n + 1 ) e. ( M ... ( ( N - 1 ) + 1 ) ) )
17	16	adantl 482	. . . . . . . . 9 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( n + 1 ) e. ( M ... ( ( N - 1 ) + 1 ) ) )
18		eluzelz 11697	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
19	1, 18	syl 17	. . . . . . . . . . . . 13 |- ( ph -> N e. ZZ )
20	19	zcnd 11483	. . . . . . . . . . . 12 |- ( ph -> N e. CC )
21		ax-1cn 9994	. . . . . . . . . . . 12 |- 1 e. CC
22		npcan 10290	. . . . . . . . . . . 12 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
23	20, 21, 22	sylancl 694	. . . . . . . . . . 11 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
24	23	oveq2d 6666	. . . . . . . . . 10 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
25	24	adantr 481	. . . . . . . . 9 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
26	17, 25	eleqtrd 2703	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( n + 1 ) e. ( M ... N ) )
27	2	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. k e. ( M ... N ) ( F ` k ) e. RR )
28	27	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> A. k e. ( M ... N ) ( F ` k ) e. RR )
29		fveq2 6191	. . . . . . . . . 10 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
30	29	eleq1d 2686	. . . . . . . . 9 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. RR <-> ( F ` ( n + 1 ) ) e. RR ) )
31	30	rspcv 3305	. . . . . . . 8 |- ( ( n + 1 ) e. ( M ... N ) -> ( A. k e. ( M ... N ) ( F ` k ) e. RR -> ( F ` ( n + 1 ) ) e. RR ) )
32	26, 28, 31	sylc 65	. . . . . . 7 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` ( n + 1 ) ) e. RR )
33		fzssp1 12384	. . . . . . . . . 10 |- ( M ... ( N - 1 ) ) C_ ( M ... ( ( N - 1 ) + 1 ) )
34	33, 24	syl5sseq 3653	. . . . . . . . 9 |- ( ph -> ( M ... ( N - 1 ) ) C_ ( M ... N ) )
35	34	sselda 3603	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ( M ... N ) )
36	11	eleq1d 2686	. . . . . . . . 9 |- ( k = n -> ( ( F ` k ) e. RR <-> ( F ` n ) e. RR ) )
37	36	rspcv 3305	. . . . . . . 8 |- ( n e. ( M ... N ) -> ( A. k e. ( M ... N ) ( F ` k ) e. RR -> ( F ` n ) e. RR ) )
38	35, 28, 37	sylc 65	. . . . . . 7 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` n ) e. RR )
39	32, 38	lenegd 10606	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( F ` ( n + 1 ) ) <_ ( F ` n ) <-> -u ( F ` n ) <_ -u ( F ` ( n + 1 ) ) ) )
40	15, 39	mpbid 222	. . . . 5 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> -u ( F ` n ) <_ -u ( F ` ( n + 1 ) ) )
41	11	negeqd 10275	. . . . . . 7 |- ( k = n -> -u ( F ` k ) = -u ( F ` n ) )
42		negex 10279	. . . . . . 7 |- -u ( F ` n ) e. _V
43	41, 4, 42	fvmpt 6282	. . . . . 6 |- ( n e. ( M ... N ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` n ) = -u ( F ` n ) )
44	35, 43	syl 17	. . . . 5 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` n ) = -u ( F ` n ) )
45	29	negeqd 10275	. . . . . . 7 |- ( k = ( n + 1 ) -> -u ( F ` k ) = -u ( F ` ( n + 1 ) ) )
46		negex 10279	. . . . . . 7 |- -u ( F ` ( n + 1 ) ) e. _V
47	45, 4, 46	fvmpt 6282	. . . . . 6 |- ( ( n + 1 ) e. ( M ... N ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` ( n + 1 ) ) = -u ( F ` ( n + 1 ) ) )
48	26, 47	syl 17	. . . . 5 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` ( n + 1 ) ) = -u ( F ` ( n + 1 ) ) )
49	40, 44, 48	3brtr4d 4685	. . . 4 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` n ) <_ ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` ( n + 1 ) ) )
50	1, 6, 49	monoord 12831	. . 3 |- ( ph -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` M ) <_ ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` N ) )
51		eluzfz1 12348	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
52	1, 51	syl 17	. . . 4 |- ( ph -> M e. ( M ... N ) )
53		fveq2 6191	. . . . . 6 |- ( k = M -> ( F ` k ) = ( F ` M ) )
54	53	negeqd 10275	. . . . 5 |- ( k = M -> -u ( F ` k ) = -u ( F ` M ) )
55		negex 10279	. . . . 5 |- -u ( F ` M ) e. _V
56	54, 4, 55	fvmpt 6282	. . . 4 |- ( M e. ( M ... N ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` M ) = -u ( F ` M ) )
57	52, 56	syl 17	. . 3 |- ( ph -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` M ) = -u ( F ` M ) )
58		eluzfz2 12349	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
59	1, 58	syl 17	. . . 4 |- ( ph -> N e. ( M ... N ) )
60		fveq2 6191	. . . . . 6 |- ( k = N -> ( F ` k ) = ( F ` N ) )
61	60	negeqd 10275	. . . . 5 |- ( k = N -> -u ( F ` k ) = -u ( F ` N ) )
62		negex 10279	. . . . 5 |- -u ( F ` N ) e. _V
63	61, 4, 62	fvmpt 6282	. . . 4 |- ( N e. ( M ... N ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` N ) = -u ( F ` N ) )
64	59, 63	syl 17	. . 3 |- ( ph -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` N ) = -u ( F ` N ) )
65	50, 57, 64	3brtr3d 4684	. 2 |- ( ph -> -u ( F ` M ) <_ -u ( F ` N ) )
66	60	eleq1d 2686	. . . . 5 |- ( k = N -> ( ( F ` k ) e. RR <-> ( F ` N ) e. RR ) )
67	66	rspcv 3305	. . . 4 |- ( N e. ( M ... N ) -> ( A. k e. ( M ... N ) ( F ` k ) e. RR -> ( F ` N ) e. RR ) )
68	59, 27, 67	sylc 65	. . 3 |- ( ph -> ( F ` N ) e. RR )
69	53	eleq1d 2686	. . . . 5 |- ( k = M -> ( ( F ` k ) e. RR <-> ( F ` M ) e. RR ) )
70	69	rspcv 3305	. . . 4 |- ( M e. ( M ... N ) -> ( A. k e. ( M ... N ) ( F ` k ) e. RR -> ( F ` M ) e. RR ) )
71	52, 27, 70	sylc 65	. . 3 |- ( ph -> ( F ` M ) e. RR )
72	68, 71	lenegd 10606	. 2 |- ( ph -> ( ( F ` N ) <_ ( F ` M ) <-> -u ( F ` M ) <_ -u ( F ` N ) ) )
73	65, 72	mpbird 247	1 |- ( ph -> ( F ` N ) <_ ( F ` M ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   -ucneg 10267   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  iseraltlem1  14412  climcndslem1  14581  climcndslem2  14582  dvfsumlem3  23791  emcllem7  24728  climinf  39838