Theorem monoord2 10433

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 8299	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> -u ( F ` k ) e. RR )
4		eqid 2170	. . . . . 6 |- ( k e. ( M ... N ) |-> -u ( F ` k ) ) = ( k e. ( M ... N ) |-> -u ( F ` k ) )
5	3, 4	fmptd 5650	. . . . 5 |- ( ph -> ( k e. ( M ... N ) |-> -u ( F ` k ) ) : ( M ... N ) --> RR )
6	5	ffvelrnda 5631	. . . 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 2543	. . . . . . . 8 |- ( ph -> A. k e. ( M ... ( N - 1 ) ) ( F ` ( k + 1 ) ) <_ ( F ` k ) )
9		oveq1 5860	. . . . . . . . . . 11 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
10	9	fveq2d 5500	. . . . . . . . . 10 |- ( k = n -> ( F ` ( k + 1 ) ) = ( F ` ( n + 1 ) ) )
11		fveq2 5496	. . . . . . . . . 10 |- ( k = n -> ( F ` k ) = ( F ` n ) )
12	10, 11	breq12d 4002	. . . . . . . . 9 |- ( k = n -> ( ( F ` ( k + 1 ) ) <_ ( F ` k ) <-> ( F ` ( n + 1 ) ) <_ ( F ` n ) ) )
13	12	cbvralv 2696	. . . . . . . 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 121	. . . . . . 7 |- ( ph -> A. n e. ( M ... ( N - 1 ) ) ( F ` ( n + 1 ) ) <_ ( F ` n ) )
15	14	r19.21bi 2558	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` ( n + 1 ) ) <_ ( F ` n ) )
16		fveq2 5496	. . . . . . . . 9 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
17	16	eleq1d 2239	. . . . . . . 8 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. RR <-> ( F ` ( n + 1 ) ) e. RR ) )
18	2	ralrimiva 2543	. . . . . . . . 9 |- ( ph -> A. k e. ( M ... N ) ( F ` k ) e. RR )
19	18	adantr 274	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> A. k e. ( M ... N ) ( F ` k ) e. RR )
20		fzp1elp1 10031	. . . . . . . . . 10 |- ( n e. ( M ... ( N - 1 ) ) -> ( n + 1 ) e. ( M ... ( ( N - 1 ) + 1 ) ) )
21	20	adantl 275	. . . . . . . . 9 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( n + 1 ) e. ( M ... ( ( N - 1 ) + 1 ) ) )
22		eluzelz 9496	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
23	1, 22	syl 14	. . . . . . . . . . . . 13 |- ( ph -> N e. ZZ )
24	23	zcnd 9335	. . . . . . . . . . . 12 |- ( ph -> N e. CC )
25		ax-1cn 7867	. . . . . . . . . . . 12 |- 1 e. CC
26		npcan 8128	. . . . . . . . . . . 12 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
27	24, 25, 26	sylancl 411	. . . . . . . . . . 11 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
28	27	oveq2d 5869	. . . . . . . . . 10 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
29	28	adantr 274	. . . . . . . . 9 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
30	21, 29	eleqtrd 2249	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( n + 1 ) e. ( M ... N ) )
31	17, 19, 30	rspcdva 2839	. . . . . . 7 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` ( n + 1 ) ) e. RR )
32	11	eleq1d 2239	. . . . . . . 8 |- ( k = n -> ( ( F ` k ) e. RR <-> ( F ` n ) e. RR ) )
33		fzssp1 10023	. . . . . . . . . 10 |- ( M ... ( N - 1 ) ) C_ ( M ... ( ( N - 1 ) + 1 ) )
34	33, 28	sseqtrid 3197	. . . . . . . . 9 |- ( ph -> ( M ... ( N - 1 ) ) C_ ( M ... N ) )
35	34	sselda 3147	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ( M ... N ) )
36	32, 19, 35	rspcdva 2839	. . . . . . 7 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` n ) e. RR )
37	31, 36	lenegd 8443	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( F ` ( n + 1 ) ) <_ ( F ` n ) <-> -u ( F ` n ) <_ -u ( F ` ( n + 1 ) ) ) )
38	15, 37	mpbid 146	. . . . 5 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> -u ( F ` n ) <_ -u ( F ` ( n + 1 ) ) )
39	36	renegcld 8299	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> -u ( F ` n ) e. RR )
40	11	negeqd 8114	. . . . . . 7 |- ( k = n -> -u ( F ` k ) = -u ( F ` n ) )
41	40, 4	fvmptg 5572	. . . . . 6 |- ( ( n e. ( M ... N ) /\ -u ( F ` n ) e. RR ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` n ) = -u ( F ` n ) )
42	35, 39, 41	syl2anc 409	. . . . 5 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` n ) = -u ( F ` n ) )
43	31	renegcld 8299	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> -u ( F ` ( n + 1 ) ) e. RR )
44	16	negeqd 8114	. . . . . . 7 |- ( k = ( n + 1 ) -> -u ( F ` k ) = -u ( F ` ( n + 1 ) ) )
45	44, 4	fvmptg 5572	. . . . . 6 |- ( ( ( n + 1 ) e. ( M ... N ) /\ -u ( F ` ( n + 1 ) ) e. RR ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` ( n + 1 ) ) = -u ( F ` ( n + 1 ) ) )
46	30, 43, 45	syl2anc 409	. . . . 5 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` ( n + 1 ) ) = -u ( F ` ( n + 1 ) ) )
47	38, 42, 46	3brtr4d 4021	. . . 4 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` n ) <_ ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` ( n + 1 ) ) )
48	1, 6, 47	monoord 10432	. . 3 |- ( ph -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` M ) <_ ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` N ) )
49		eluzfz1 9987	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
50	1, 49	syl 14	. . . 4 |- ( ph -> M e. ( M ... N ) )
51		fveq2 5496	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
52	51	eleq1d 2239	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. RR <-> ( F ` M ) e. RR ) )
53	52, 18, 50	rspcdva 2839	. . . . 5 |- ( ph -> ( F ` M ) e. RR )
54	53	renegcld 8299	. . . 4 |- ( ph -> -u ( F ` M ) e. RR )
55	51	negeqd 8114	. . . . 5 |- ( k = M -> -u ( F ` k ) = -u ( F ` M ) )
56	55, 4	fvmptg 5572	. . . 4 |- ( ( M e. ( M ... N ) /\ -u ( F ` M ) e. RR ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` M ) = -u ( F ` M ) )
57	50, 54, 56	syl2anc 409	. . 3 |- ( ph -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` M ) = -u ( F ` M ) )
58		eluzfz2 9988	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
59	1, 58	syl 14	. . . 4 |- ( ph -> N e. ( M ... N ) )
60		fveq2 5496	. . . . . . 7 |- ( k = N -> ( F ` k ) = ( F ` N ) )
61	60	eleq1d 2239	. . . . . 6 |- ( k = N -> ( ( F ` k ) e. RR <-> ( F ` N ) e. RR ) )
62	61, 18, 59	rspcdva 2839	. . . . 5 |- ( ph -> ( F ` N ) e. RR )
63	62	renegcld 8299	. . . 4 |- ( ph -> -u ( F ` N ) e. RR )
64	60	negeqd 8114	. . . . 5 |- ( k = N -> -u ( F ` k ) = -u ( F ` N ) )
65	64, 4	fvmptg 5572	. . . 4 |- ( ( N e. ( M ... N ) /\ -u ( F ` N ) e. RR ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` N ) = -u ( F ` N ) )
66	59, 63, 65	syl2anc 409	. . 3 |- ( ph -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` N ) = -u ( F ` N ) )
67	48, 57, 66	3brtr3d 4020	. 2 |- ( ph -> -u ( F ` M ) <_ -u ( F ` N ) )
68	62, 53	lenegd 8443	. 2 |- ( ph -> ( ( F ` N ) <_ ( F ` M ) <-> -u ( F ` M ) <_ -u ( F ` N ) ) )
69	67, 68	mpbird 166	1 |- ( ph -> ( F ` N ) <_ ( F ` M ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   class class class wbr 3989   |-> cmpt 4050   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   1c1 7775   + caddc 7777   <_ cle 7955   - cmin 8090   -ucneg 8091   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966

This theorem is referenced by: (None)