Theorem monoord2 10811

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 8618	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> -u ( F ` k ) e. RR )
4		eqid 2231	. . . . . 6 |- ( k e. ( M ... N ) |-> -u ( F ` k ) ) = ( k e. ( M ... N ) |-> -u ( F ` k ) )
5	3, 4	fmptd 5809	. . . . 5 |- ( ph -> ( k e. ( M ... N ) |-> -u ( F ` k ) ) : ( M ... N ) --> RR )
6	5	ffvelcdmda 5790	. . . 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 2606	. . . . . . . 8 |- ( ph -> A. k e. ( M ... ( N - 1 ) ) ( F ` ( k + 1 ) ) <_ ( F ` k ) )
9		oveq1 6035	. . . . . . . . . . 11 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
10	9	fveq2d 5652	. . . . . . . . . 10 |- ( k = n -> ( F ` ( k + 1 ) ) = ( F ` ( n + 1 ) ) )
11		fveq2 5648	. . . . . . . . . 10 |- ( k = n -> ( F ` k ) = ( F ` n ) )
12	10, 11	breq12d 4106	. . . . . . . . 9 |- ( k = n -> ( ( F ` ( k + 1 ) ) <_ ( F ` k ) <-> ( F ` ( n + 1 ) ) <_ ( F ` n ) ) )
13	12	cbvralv 2768	. . . . . . . 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 122	. . . . . . 7 |- ( ph -> A. n e. ( M ... ( N - 1 ) ) ( F ` ( n + 1 ) ) <_ ( F ` n ) )
15	14	r19.21bi 2621	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` ( n + 1 ) ) <_ ( F ` n ) )
16		fveq2 5648	. . . . . . . . 9 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
17	16	eleq1d 2300	. . . . . . . 8 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. RR <-> ( F ` ( n + 1 ) ) e. RR ) )
18	2	ralrimiva 2606	. . . . . . . . 9 |- ( ph -> A. k e. ( M ... N ) ( F ` k ) e. RR )
19	18	adantr 276	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> A. k e. ( M ... N ) ( F ` k ) e. RR )
20		fzp1elp1 10372	. . . . . . . . . 10 |- ( n e. ( M ... ( N - 1 ) ) -> ( n + 1 ) e. ( M ... ( ( N - 1 ) + 1 ) ) )
21	20	adantl 277	. . . . . . . . 9 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( n + 1 ) e. ( M ... ( ( N - 1 ) + 1 ) ) )
22		eluzelz 9826	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
23	1, 22	syl 14	. . . . . . . . . . . . 13 |- ( ph -> N e. ZZ )
24	23	zcnd 9664	. . . . . . . . . . . 12 |- ( ph -> N e. CC )
25		ax-1cn 8185	. . . . . . . . . . . 12 |- 1 e. CC
26		npcan 8447	. . . . . . . . . . . 12 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
27	24, 25, 26	sylancl 413	. . . . . . . . . . 11 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
28	27	oveq2d 6044	. . . . . . . . . 10 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
29	28	adantr 276	. . . . . . . . 9 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
30	21, 29	eleqtrd 2310	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( n + 1 ) e. ( M ... N ) )
31	17, 19, 30	rspcdva 2916	. . . . . . 7 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` ( n + 1 ) ) e. RR )
32	11	eleq1d 2300	. . . . . . . 8 |- ( k = n -> ( ( F ` k ) e. RR <-> ( F ` n ) e. RR ) )
33		fzssp1 10364	. . . . . . . . . 10 |- ( M ... ( N - 1 ) ) C_ ( M ... ( ( N - 1 ) + 1 ) )
34	33, 28	sseqtrid 3278	. . . . . . . . 9 |- ( ph -> ( M ... ( N - 1 ) ) C_ ( M ... N ) )
35	34	sselda 3228	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ( M ... N ) )
36	32, 19, 35	rspcdva 2916	. . . . . . 7 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` n ) e. RR )
37	31, 36	lenegd 8763	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( F ` ( n + 1 ) ) <_ ( F ` n ) <-> -u ( F ` n ) <_ -u ( F ` ( n + 1 ) ) ) )
38	15, 37	mpbid 147	. . . . 5 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> -u ( F ` n ) <_ -u ( F ` ( n + 1 ) ) )
39	36	renegcld 8618	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> -u ( F ` n ) e. RR )
40	11	negeqd 8433	. . . . . . 7 |- ( k = n -> -u ( F ` k ) = -u ( F ` n ) )
41	40, 4	fvmptg 5731	. . . . . 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 411	. . . . 5 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` n ) = -u ( F ` n ) )
43	31	renegcld 8618	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> -u ( F ` ( n + 1 ) ) e. RR )
44	16	negeqd 8433	. . . . . . 7 |- ( k = ( n + 1 ) -> -u ( F ` k ) = -u ( F ` ( n + 1 ) ) )
45	44, 4	fvmptg 5731	. . . . . 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 411	. . . . 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 4125	. . . 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 10810	. . 3 |- ( ph -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` M ) <_ ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` N ) )
49		eluzfz1 10328	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
50	1, 49	syl 14	. . . 4 |- ( ph -> M e. ( M ... N ) )
51		fveq2 5648	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
52	51	eleq1d 2300	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. RR <-> ( F ` M ) e. RR ) )
53	52, 18, 50	rspcdva 2916	. . . . 5 |- ( ph -> ( F ` M ) e. RR )
54	53	renegcld 8618	. . . 4 |- ( ph -> -u ( F ` M ) e. RR )
55	51	negeqd 8433	. . . . 5 |- ( k = M -> -u ( F ` k ) = -u ( F ` M ) )
56	55, 4	fvmptg 5731	. . . 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 411	. . 3 |- ( ph -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` M ) = -u ( F ` M ) )
58		eluzfz2 10329	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
59	1, 58	syl 14	. . . 4 |- ( ph -> N e. ( M ... N ) )
60		fveq2 5648	. . . . . . 7 |- ( k = N -> ( F ` k ) = ( F ` N ) )
61	60	eleq1d 2300	. . . . . 6 |- ( k = N -> ( ( F ` k ) e. RR <-> ( F ` N ) e. RR ) )
62	61, 18, 59	rspcdva 2916	. . . . 5 |- ( ph -> ( F ` N ) e. RR )
63	62	renegcld 8618	. . . 4 |- ( ph -> -u ( F ` N ) e. RR )
64	60	negeqd 8433	. . . . 5 |- ( k = N -> -u ( F ` k ) = -u ( F ` N ) )
65	64, 4	fvmptg 5731	. . . 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 411	. . 3 |- ( ph -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` N ) = -u ( F ` N ) )
67	48, 57, 66	3brtr3d 4124	. 2 |- ( ph -> -u ( F ` M ) <_ -u ( F ` N ) )
68	62, 53	lenegd 8763	. 2 |- ( ph -> ( ( F ` N ) <_ ( F ` M ) <-> -u ( F ` M ) <_ -u ( F ` N ) ) )
69	67, 68	mpbird 167	1 |- ( ph -> ( F ` N ) <_ ( F ` M ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   class class class wbr 4093   |-> cmpt 4155   ` cfv 5333  (class class class)co 6028   CCcc 8090   RRcr 8091   1c1 8093   + caddc 8095   <_ cle 8274   - cmin 8409   -ucneg 8410   ZZcz 9540   ZZ>=cuz 9816   ...cfz 10305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306

This theorem is referenced by: (None)