Theorem monoord2 10847

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 8652	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> -u ( F ` k ) e. RR )
4		eqid 2232	. . . . . 6 |- ( k e. ( M ... N ) |-> -u ( F ` k ) ) = ( k e. ( M ... N ) |-> -u ( F ` k ) )
5	3, 4	fmptd 5830	. . . . 5 |- ( ph -> ( k e. ( M ... N ) |-> -u ( F ` k ) ) : ( M ... N ) --> RR )
6	5	ffvelcdmda 5811	. . . 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 2615	. . . . . . . 8 |- ( ph -> A. k e. ( M ... ( N - 1 ) ) ( F ` ( k + 1 ) ) <_ ( F ` k ) )
9		oveq1 6056	. . . . . . . . . . 11 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
10	9	fveq2d 5673	. . . . . . . . . 10 |- ( k = n -> ( F ` ( k + 1 ) ) = ( F ` ( n + 1 ) ) )
11		fveq2 5669	. . . . . . . . . 10 |- ( k = n -> ( F ` k ) = ( F ` n ) )
12	10, 11	breq12d 4121	. . . . . . . . 9 |- ( k = n -> ( ( F ` ( k + 1 ) ) <_ ( F ` k ) <-> ( F ` ( n + 1 ) ) <_ ( F ` n ) ) )
13	12	cbvralv 2777	. . . . . . . 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 2630	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` ( n + 1 ) ) <_ ( F ` n ) )
16		fveq2 5669	. . . . . . . . 9 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
17	16	eleq1d 2301	. . . . . . . 8 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. RR <-> ( F ` ( n + 1 ) ) e. RR ) )
18	2	ralrimiva 2615	. . . . . . . . 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 10408	. . . . . . . . . 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 9862	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
23	1, 22	syl 14	. . . . . . . . . . . . 13 |- ( ph -> N e. ZZ )
24	23	zcnd 9700	. . . . . . . . . . . 12 |- ( ph -> N e. CC )
25		ax-1cn 8219	. . . . . . . . . . . 12 |- 1 e. CC
26		npcan 8481	. . . . . . . . . . . 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 6065	. . . . . . . . . 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 2311	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( n + 1 ) e. ( M ... N ) )
31	17, 19, 30	rspcdva 2925	. . . . . . 7 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` ( n + 1 ) ) e. RR )
32	11	eleq1d 2301	. . . . . . . 8 |- ( k = n -> ( ( F ` k ) e. RR <-> ( F ` n ) e. RR ) )
33		fzssp1 10400	. . . . . . . . . 10 |- ( M ... ( N - 1 ) ) C_ ( M ... ( ( N - 1 ) + 1 ) )
34	33, 28	sseqtrid 3287	. . . . . . . . 9 |- ( ph -> ( M ... ( N - 1 ) ) C_ ( M ... N ) )
35	34	sselda 3237	. . . . . . . 8 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ( M ... N ) )
36	32, 19, 35	rspcdva 2925	. . . . . . 7 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` n ) e. RR )
37	31, 36	lenegd 8797	. . . . . 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 8652	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> -u ( F ` n ) e. RR )
40	11	negeqd 8467	. . . . . . 7 |- ( k = n -> -u ( F ` k ) = -u ( F ` n ) )
41	40, 4	fvmptg 5752	. . . . . 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 8652	. . . . . 6 |- ( ( ph /\ n e. ( M ... ( N - 1 ) ) ) -> -u ( F ` ( n + 1 ) ) e. RR )
44	16	negeqd 8467	. . . . . . 7 |- ( k = ( n + 1 ) -> -u ( F ` k ) = -u ( F ` ( n + 1 ) ) )
45	44, 4	fvmptg 5752	. . . . . 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 4140	. . . 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 10846	. . 3 |- ( ph -> ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` M ) <_ ( ( k e. ( M ... N ) |-> -u ( F ` k ) ) ` N ) )
49		eluzfz1 10364	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
50	1, 49	syl 14	. . . 4 |- ( ph -> M e. ( M ... N ) )
51		fveq2 5669	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
52	51	eleq1d 2301	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. RR <-> ( F ` M ) e. RR ) )
53	52, 18, 50	rspcdva 2925	. . . . 5 |- ( ph -> ( F ` M ) e. RR )
54	53	renegcld 8652	. . . 4 |- ( ph -> -u ( F ` M ) e. RR )
55	51	negeqd 8467	. . . . 5 |- ( k = M -> -u ( F ` k ) = -u ( F ` M ) )
56	55, 4	fvmptg 5752	. . . 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 10365	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
59	1, 58	syl 14	. . . 4 |- ( ph -> N e. ( M ... N ) )
60		fveq2 5669	. . . . . . 7 |- ( k = N -> ( F ` k ) = ( F ` N ) )
61	60	eleq1d 2301	. . . . . 6 |- ( k = N -> ( ( F ` k ) e. RR <-> ( F ` N ) e. RR ) )
62	61, 18, 59	rspcdva 2925	. . . . 5 |- ( ph -> ( F ` N ) e. RR )
63	62	renegcld 8652	. . . 4 |- ( ph -> -u ( F ` N ) e. RR )
64	60	negeqd 8467	. . . . 5 |- ( k = N -> -u ( F ` k ) = -u ( F ` N ) )
65	64, 4	fvmptg 5752	. . . 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 4139	. 2 |- ( ph -> -u ( F ` M ) <_ -u ( F ` N ) )
68	62, 53	lenegd 8797	. 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 2203   A.wral 2520   class class class wbr 4108   |-> cmpt 4170   ` cfv 5351  (class class class)co 6049   CCcc 8124   RRcr 8125   1c1 8127   + caddc 8129   <_ cle 8308   - cmin 8443   -ucneg 8444   ZZcz 9576   ZZ>=cuz 9852   ...cfz 10341

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-ltadd 8242

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-n0 9496  df-z 9577  df-uz 9853  df-fz 10342

This theorem is referenced by: (None)