Theorem monoord2 10848

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 8653	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → -(𝐹‘𝑘) ∈ ℝ)
4		eqid 2232	. . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘)) = (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))
5	3, 4	fmptd 5831	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘)):(𝑀...𝑁)⟶ℝ)
6	5	ffvelcdmda 5812	. . . 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 2615	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
9		oveq1 6057	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
10	9	fveq2d 5674	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
11		fveq2 5670	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
12	10, 11	breq12d 4122	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛)))
13	12	cbvralv 2778	. . . . . . . 8 ⊢ (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
14	8, 13	sylib 122	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
15	14	r19.21bi 2630	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
16		fveq2 5670	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
17	16	eleq1d 2301	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
18	2	ralrimiva 2615	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
19	18	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
20		fzp1elp1 10409	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
21	20	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
22		eluzelz 9863	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
23	1, 22	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24	23	zcnd 9701	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
25		ax-1cn 8220	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
26		npcan 8482	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
27	24, 25, 26	sylancl 413	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
28	27	oveq2d 6066	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
29	28	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
30	21, 29	eleqtrd 2311	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...𝑁))
31	17, 19, 30	rspcdva 2926	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
32	11	eleq1d 2301	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑛) ∈ ℝ))
33		fzssp1 10401	. . . . . . . . . 10 ⊢ (𝑀...(𝑁 − 1)) ⊆ (𝑀...((𝑁 − 1) + 1))
34	33, 28	sseqtrid 3288	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...(𝑁 − 1)) ⊆ (𝑀...𝑁))
35	34	sselda 3238	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝑛 ∈ (𝑀...𝑁))
36	32, 19, 35	rspcdva 2926	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑛) ∈ ℝ)
37	31, 36	lenegd 8798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛) ↔ -(𝐹‘𝑛) ≤ -(𝐹‘(𝑛 + 1))))
38	15, 37	mpbid 147	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹‘𝑛) ≤ -(𝐹‘(𝑛 + 1)))
39	36	renegcld 8653	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹‘𝑛) ∈ ℝ)
40	11	negeqd 8468	. . . . . . 7 ⊢ (𝑘 = 𝑛 → -(𝐹‘𝑘) = -(𝐹‘𝑛))
41	40, 4	fvmptg 5753	. . . . . 6 ⊢ ((𝑛 ∈ (𝑀...𝑁) ∧ -(𝐹‘𝑛) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑛) = -(𝐹‘𝑛))
42	35, 39, 41	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑛) = -(𝐹‘𝑛))
43	31	renegcld 8653	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹‘(𝑛 + 1)) ∈ ℝ)
44	16	negeqd 8468	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → -(𝐹‘𝑘) = -(𝐹‘(𝑛 + 1)))
45	44, 4	fvmptg 5753	. . . . . 6 ⊢ (((𝑛 + 1) ∈ (𝑀...𝑁) ∧ -(𝐹‘(𝑛 + 1)) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘(𝑛 + 1)) = -(𝐹‘(𝑛 + 1)))
46	30, 43, 45	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘(𝑛 + 1)) = -(𝐹‘(𝑛 + 1)))
47	38, 42, 46	3brtr4d 4141	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑛) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘(𝑛 + 1)))
48	1, 6, 47	monoord 10847	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑀) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑁))
49		eluzfz1 10365	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
50	1, 49	syl 14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
51		fveq2 5670	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
52	51	eleq1d 2301	. . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑀) ∈ ℝ))
53	52, 18, 50	rspcdva 2926	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ)
54	53	renegcld 8653	. . . 4 ⊢ (𝜑 → -(𝐹‘𝑀) ∈ ℝ)
55	51	negeqd 8468	. . . . 5 ⊢ (𝑘 = 𝑀 → -(𝐹‘𝑘) = -(𝐹‘𝑀))
56	55, 4	fvmptg 5753	. . . 4 ⊢ ((𝑀 ∈ (𝑀...𝑁) ∧ -(𝐹‘𝑀) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑀) = -(𝐹‘𝑀))
57	50, 54, 56	syl2anc 411	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑀) = -(𝐹‘𝑀))
58		eluzfz2 10366	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
59	1, 58	syl 14	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
60		fveq2 5670	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
61	60	eleq1d 2301	. . . . . 6 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑁) ∈ ℝ))
62	61, 18, 59	rspcdva 2926	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ)
63	62	renegcld 8653	. . . 4 ⊢ (𝜑 → -(𝐹‘𝑁) ∈ ℝ)
64	60	negeqd 8468	. . . . 5 ⊢ (𝑘 = 𝑁 → -(𝐹‘𝑘) = -(𝐹‘𝑁))
65	64, 4	fvmptg 5753	. . . 4 ⊢ ((𝑁 ∈ (𝑀...𝑁) ∧ -(𝐹‘𝑁) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑁) = -(𝐹‘𝑁))
66	59, 63, 65	syl2anc 411	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹‘𝑘))‘𝑁) = -(𝐹‘𝑁))
67	48, 57, 66	3brtr3d 4140	. 2 ⊢ (𝜑 → -(𝐹‘𝑀) ≤ -(𝐹‘𝑁))
68	62, 53	lenegd 8798	. 2 ⊢ (𝜑 → ((𝐹‘𝑁) ≤ (𝐹‘𝑀) ↔ -(𝐹‘𝑀) ≤ -(𝐹‘𝑁)))
69	67, 68	mpbird 167	1 ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ∀wral 2520   class class class wbr 4109   ↦ cmpt 4171  ‘cfv 5352  (class class class)co 6050  ℂcc 8125  ℝcr 8126  1c1 8128   + caddc 8130   ≤ cle 8309   − cmin 8444  -cneg 8445  ℤcz 9577  ℤ_≥cuz 9853  ...cfz 10342

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 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

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 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343

This theorem is referenced by: (None)