Theorem monoord2 10420

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 8286	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → -(𝐹‘𝑘) ∈ ℝ)
4		eqid 2170	. . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘)) = (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))
5	3, 4	fmptd 5647	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘)):(𝑀...𝑁)⟶ℝ)
6	5	ffvelrnda 5628	. . . 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 2543	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
9		oveq1 5857	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
10	9	fveq2d 5498	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
11		fveq2 5494	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
12	10, 11	breq12d 4000	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛)))
13	12	cbvralv 2696	. . . . . . . 8 ⊢ (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
14	8, 13	sylib 121	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
15	14	r19.21bi 2558	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
16		fveq2 5494	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
17	16	eleq1d 2239	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
18	2	ralrimiva 2543	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
19	18	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
20		fzp1elp1 10018	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
21	20	adantl 275	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
22		eluzelz 9483	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
23	1, 22	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24	23	zcnd 9322	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
25		ax-1cn 7854	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
26		npcan 8115	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
27	24, 25, 26	sylancl 411	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
28	27	oveq2d 5866	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
29	28	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
30	21, 29	eleqtrd 2249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...𝑁))
31	17, 19, 30	rspcdva 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
32	11	eleq1d 2239	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑛) ∈ ℝ))
33		fzssp1 10010	. . . . . . . . . 10 ⊢ (𝑀...(𝑁 − 1)) ⊆ (𝑀...((𝑁 − 1) + 1))
34	33, 28	sseqtrid 3197	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...(𝑁 − 1)) ⊆ (𝑀...𝑁))
35	34	sselda 3147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝑛 ∈ (𝑀...𝑁))
36	32, 19, 35	rspcdva 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑛) ∈ ℝ)
37	31, 36	lenegd 8430	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛) ↔ -(𝐹‘𝑛) ≤ -(𝐹‘(𝑛 + 1))))
38	15, 37	mpbid 146	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹‘𝑛) ≤ -(𝐹‘(𝑛 + 1)))
39	36	renegcld 8286	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹‘𝑛) ∈ ℝ)
40	11	negeqd 8101	. . . . . . 7 ⊢ (𝑘 = 𝑛 → -(𝐹‘𝑘) = -(𝐹‘𝑛))
41	40, 4	fvmptg 5570	. . . . . 6 ⊢ ((𝑛 ∈ (𝑀...𝑁) ∧ -(𝐹‘𝑛) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑛) = -(𝐹‘𝑛))
42	35, 39, 41	syl2anc 409	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑛) = -(𝐹‘𝑛))
43	31	renegcld 8286	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹‘(𝑛 + 1)) ∈ ℝ)
44	16	negeqd 8101	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → -(𝐹‘𝑘) = -(𝐹‘(𝑛 + 1)))
45	44, 4	fvmptg 5570	. . . . . 6 ⊢ (((𝑛 + 1) ∈ (𝑀...𝑁) ∧ -(𝐹‘(𝑛 + 1)) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘(𝑛 + 1)) = -(𝐹‘(𝑛 + 1)))
46	30, 43, 45	syl2anc 409	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘(𝑛 + 1)) = -(𝐹‘(𝑛 + 1)))
47	38, 42, 46	3brtr4d 4019	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑛) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘(𝑛 + 1)))
48	1, 6, 47	monoord 10419	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑀) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑁))
49		eluzfz1 9974	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
50	1, 49	syl 14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
51		fveq2 5494	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
52	51	eleq1d 2239	. . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑀) ∈ ℝ))
53	52, 18, 50	rspcdva 2839	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ)
54	53	renegcld 8286	. . . 4 ⊢ (𝜑 → -(𝐹‘𝑀) ∈ ℝ)
55	51	negeqd 8101	. . . . 5 ⊢ (𝑘 = 𝑀 → -(𝐹‘𝑘) = -(𝐹‘𝑀))
56	55, 4	fvmptg 5570	. . . 4 ⊢ ((𝑀 ∈ (𝑀...𝑁) ∧ -(𝐹‘𝑀) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑀) = -(𝐹‘𝑀))
57	50, 54, 56	syl2anc 409	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑀) = -(𝐹‘𝑀))
58		eluzfz2 9975	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
59	1, 58	syl 14	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
60		fveq2 5494	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
61	60	eleq1d 2239	. . . . . 6 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑁) ∈ ℝ))
62	61, 18, 59	rspcdva 2839	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ)
63	62	renegcld 8286	. . . 4 ⊢ (𝜑 → -(𝐹‘𝑁) ∈ ℝ)
64	60	negeqd 8101	. . . . 5 ⊢ (𝑘 = 𝑁 → -(𝐹‘𝑘) = -(𝐹‘𝑁))
65	64, 4	fvmptg 5570	. . . 4 ⊢ ((𝑁 ∈ (𝑀...𝑁) ∧ -(𝐹‘𝑁) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑁) = -(𝐹‘𝑁))
66	59, 63, 65	syl2anc 409	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑁) = -(𝐹‘𝑁))
67	48, 57, 66	3brtr3d 4018	. 2 ⊢ (𝜑 → -(𝐹‘𝑀) ≤ -(𝐹‘𝑁))
68	62, 53	lenegd 8430	. 2 ⊢ (𝜑 → ((𝐹‘𝑁) ≤ (𝐹‘𝑀) ↔ -(𝐹‘𝑀) ≤ -(𝐹‘𝑁)))
69	67, 68	mpbird 166	1 ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ∀wral 2448   class class class wbr 3987   ↦ cmpt 4048  ‘cfv 5196  (class class class)co 5850  ℂcc 7759  ℝcr 7760  1c1 7762   + caddc 7764   ≤ cle 7942   − cmin 8077  -cneg 8078  ℤcz 9199  ℤ_≥cuz 9474  ...cfz 9952

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 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-ltadd 7877

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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-fz 9953

This theorem is referenced by: (None)