Theorem monoord2 12694

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

Hypotheses

Ref	Expression
monoord2.1	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
monoord2.2	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
monoord2.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))

Assertion

Ref	Expression
monoord2	⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))

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

Proof of Theorem monoord2

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		monoord2.1	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		monoord2.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
3	2	renegcld 10336	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → -(𝐹‘𝑘) ∈ ℝ)
4		eqid 2610	. . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘)) = (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))
5	3, 4	fmptd 6292	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘)):(𝑀...𝑁)⟶ℝ)
6	5	ffvelrnda 6267	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑛) ∈ ℝ)
7		monoord2.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
8	7	ralrimiva 2949	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
9		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
10	9	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
11		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
12	10, 11	breq12d 4596	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛)))
13	12	cbvralv 3147	. . . . . . . 8 ⊢ (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
14	8, 13	sylib 207	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
15	14	r19.21bi 2916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
16		fzp1elp1 12264	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
17	16	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
18		eluzelz 11573	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
19	1, 18	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
20	19	zcnd 11359	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
21		ax-1cn 9873	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
22		npcan 10169	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
23	20, 21, 22	sylancl 693	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
24	23	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
25	24	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
26	17, 25	eleqtrd 2690	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...𝑁))
27	2	ralrimiva 2949	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
29		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
30	29	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
31	30	rspcv 3278	. . . . . . . 8 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (𝐹‘(𝑛 + 1)) ∈ ℝ))
32	26, 28, 31	sylc 63	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
33		fzssp1 12255	. . . . . . . . . 10 ⊢ (𝑀...(𝑁 − 1)) ⊆ (𝑀...((𝑁 − 1) + 1))
34	33, 24	syl5sseq 3616	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...(𝑁 − 1)) ⊆ (𝑀...𝑁))
35	34	sselda 3568	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝑛 ∈ (𝑀...𝑁))
36	11	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑛) ∈ ℝ))
37	36	rspcv 3278	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (𝐹‘𝑛) ∈ ℝ))
38	35, 28, 37	sylc 63	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑛) ∈ ℝ)
39	32, 38	lenegd 10485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛) ↔ -(𝐹‘𝑛) ≤ -(𝐹‘(𝑛 + 1))))
40	15, 39	mpbid 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹‘𝑛) ≤ -(𝐹‘(𝑛 + 1)))
41	11	negeqd 10154	. . . . . . 7 ⊢ (𝑘 = 𝑛 → -(𝐹‘𝑘) = -(𝐹‘𝑛))
42		negex 10158	. . . . . . 7 ⊢ -(𝐹‘𝑛) ∈ V
43	41, 4, 42	fvmpt 6191	. . . . . 6 ⊢ (𝑛 ∈ (𝑀...𝑁) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑛) = -(𝐹‘𝑛))
44	35, 43	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑛) = -(𝐹‘𝑛))
45	29	negeqd 10154	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → -(𝐹‘𝑘) = -(𝐹‘(𝑛 + 1)))
46		negex 10158	. . . . . . 7 ⊢ -(𝐹‘(𝑛 + 1)) ∈ V
47	45, 4, 46	fvmpt 6191	. . . . . 6 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘(𝑛 + 1)) = -(𝐹‘(𝑛 + 1)))
48	26, 47	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘(𝑛 + 1)) = -(𝐹‘(𝑛 + 1)))
49	40, 44, 48	3brtr4d 4615	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑛) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘(𝑛 + 1)))
50	1, 6, 49	monoord 12693	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑀) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑁))
51		eluzfz1 12219	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
52	1, 51	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
53		fveq2 6103	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
54	53	negeqd 10154	. . . . 5 ⊢ (𝑘 = 𝑀 → -(𝐹‘𝑘) = -(𝐹‘𝑀))
55		negex 10158	. . . . 5 ⊢ -(𝐹‘𝑀) ∈ V
56	54, 4, 55	fvmpt 6191	. . . 4 ⊢ (𝑀 ∈ (𝑀...𝑁) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑀) = -(𝐹‘𝑀))
57	52, 56	syl 17	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑀) = -(𝐹‘𝑀))
58		eluzfz2 12220	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
59	1, 58	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
60		fveq2 6103	. . . . . 6 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
61	60	negeqd 10154	. . . . 5 ⊢ (𝑘 = 𝑁 → -(𝐹‘𝑘) = -(𝐹‘𝑁))
62		negex 10158	. . . . 5 ⊢ -(𝐹‘𝑁) ∈ V
63	61, 4, 62	fvmpt 6191	. . . 4 ⊢ (𝑁 ∈ (𝑀...𝑁) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑁) = -(𝐹‘𝑁))
64	59, 63	syl 17	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑁) = -(𝐹‘𝑁))
65	50, 57, 64	3brtr3d 4614	. 2 ⊢ (𝜑 → -(𝐹‘𝑀) ≤ -(𝐹‘𝑁))
66	60	eleq1d 2672	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑁) ∈ ℝ))
67	66	rspcv 3278	. . . 4 ⊢ (𝑁 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (𝐹‘𝑁) ∈ ℝ))
68	59, 27, 67	sylc 63	. . 3 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ)
69	53	eleq1d 2672	. . . . 5 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑀) ∈ ℝ))
70	69	rspcv 3278	. . . 4 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (𝐹‘𝑀) ∈ ℝ))
71	52, 27, 70	sylc 63	. . 3 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ)
72	68, 71	lenegd 10485	. 2 ⊢ (𝜑 → ((𝐹‘𝑁) ≤ (𝐹‘𝑀) ↔ -(𝐹‘𝑀) ≤ -(𝐹‘𝑁)))
73	65, 72	mpbird 246	1 ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  1c1 9816   + caddc 9818   ≤ cle 9954   − cmin 10145  -cneg 10146  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198

This theorem is referenced by:  iseraltlem1  14260  climcndslem1  14420  climcndslem2  14421  dvfsumlem3  23595  emcllem7  24528  climinf  38673