monoord2xrv - Mathbox for Glauco Siliprandi

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Theorem monoord2xrv 44194

Description: Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)

**Hypotheses**
Ref	Expression
monoord2xrv.n	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
monoord2xrv.x	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ^*)
monoord2xrv.l	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))

**Assertion**
Ref	Expression
monoord2xrv	⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))

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

Proof of Theorem monoord2xrv Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		monoord2xrv.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		monoord2xrv.x	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ^*)
3	2	xnegcld 13279	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → -_𝑒(𝐹‘𝑘) ∈ ℝ^*)
4	3	fmpttd 7115	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘)):(𝑀...𝑁)⟶ℝ^*)
5	4	ffvelcdmda 7087	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘𝑛) ∈ ℝ^*)
6		monoord2xrv.l	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
7	6	ralrimiva 3147	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
8		fvoveq1 7432	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
9		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
10	8, 9	breq12d 5162	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛)))
11	10	cbvralvw 3235	. . . . . . . 8 ⊢ (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
12	7, 11	sylib 217	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
13	12	r19.21bi 3249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
14		fzp1elp1 13554	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
15	14	adantl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
16		eluzelz 12832	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
17	1, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	17	zcnd 12667	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
19		ax-1cn 11168	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
20		npcan 11469	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
21	18, 19, 20	sylancl 587	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
22	21	oveq2d 7425	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
23	22	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
24	15, 23	eleqtrd 2836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...𝑁))
25	2	ralrimiva 3147	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^*)
26	25	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^*)
27		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
28	27	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ^* ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ^*))
29	28	rspcv 3609	. . . . . . . 8 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^* → (𝐹‘(𝑛 + 1)) ∈ ℝ^*))
30	24, 26, 29	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ∈ ℝ^*)
31		fzssp1 13544	. . . . . . . . . 10 ⊢ (𝑀...(𝑁 − 1)) ⊆ (𝑀...((𝑁 − 1) + 1))
32	31, 22	sseqtrid 4035	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...(𝑁 − 1)) ⊆ (𝑀...𝑁))
33	32	sselda 3983	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝑛 ∈ (𝑀...𝑁))
34	9	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ^* ↔ (𝐹‘𝑛) ∈ ℝ^*))
35	34	rspcv 3609	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^* → (𝐹‘𝑛) ∈ ℝ^*))
36	33, 26, 35	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑛) ∈ ℝ^*)
37		xleneg 13197	. . . . . . 7 ⊢ (((𝐹‘(𝑛 + 1)) ∈ ℝ^* ∧ (𝐹‘𝑛) ∈ ℝ^*) → ((𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛) ↔ -_𝑒(𝐹‘𝑛) ≤ -_𝑒(𝐹‘(𝑛 + 1))))
38	30, 36, 37	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛) ↔ -_𝑒(𝐹‘𝑛) ≤ -_𝑒(𝐹‘(𝑛 + 1))))
39	13, 38	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → -_𝑒(𝐹‘𝑛) ≤ -_𝑒(𝐹‘(𝑛 + 1)))
40	9	xnegeqd 44147	. . . . . . 7 ⊢ (𝑘 = 𝑛 → -_𝑒(𝐹‘𝑘) = -_𝑒(𝐹‘𝑛))
41		eqid 2733	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘)) = (𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))
42		xnegex 13187	. . . . . . 7 ⊢ -_𝑒(𝐹‘𝑛) ∈ V
43	40, 41, 42	fvmpt 6999	. . . . . 6 ⊢ (𝑛 ∈ (𝑀...𝑁) → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘𝑛) = -_𝑒(𝐹‘𝑛))
44	33, 43	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘𝑛) = -_𝑒(𝐹‘𝑛))
45	27	xnegeqd 44147	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → -_𝑒(𝐹‘𝑘) = -_𝑒(𝐹‘(𝑛 + 1)))
46		xnegex 13187	. . . . . . 7 ⊢ -_𝑒(𝐹‘(𝑛 + 1)) ∈ V
47	45, 41, 46	fvmpt 6999	. . . . . 6 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘(𝑛 + 1)) = -_𝑒(𝐹‘(𝑛 + 1)))
48	24, 47	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘(𝑛 + 1)) = -_𝑒(𝐹‘(𝑛 + 1)))
49	39, 44, 48	3brtr4d 5181	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘𝑛) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘(𝑛 + 1)))
50	1, 5, 49	monoordxrv 44192	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘𝑀) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘𝑁))
51		eluzfz1 13508	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
52	1, 51	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
53		fveq2 6892	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
54	53	xnegeqd 44147	. . . . 5 ⊢ (𝑘 = 𝑀 → -_𝑒(𝐹‘𝑘) = -_𝑒(𝐹‘𝑀))
55		xnegex 13187	. . . . 5 ⊢ -_𝑒(𝐹‘𝑀) ∈ V
56	54, 41, 55	fvmpt 6999	. . . 4 ⊢ (𝑀 ∈ (𝑀...𝑁) → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘𝑀) = -_𝑒(𝐹‘𝑀))
57	52, 56	syl 17	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘𝑀) = -_𝑒(𝐹‘𝑀))
58		eluzfz2 13509	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
59	1, 58	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
60		fveq2 6892	. . . . . 6 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
61	60	xnegeqd 44147	. . . . 5 ⊢ (𝑘 = 𝑁 → -_𝑒(𝐹‘𝑘) = -_𝑒(𝐹‘𝑁))
62		xnegex 13187	. . . . 5 ⊢ -_𝑒(𝐹‘𝑁) ∈ V
63	61, 41, 62	fvmpt 6999	. . . 4 ⊢ (𝑁 ∈ (𝑀...𝑁) → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘𝑁) = -_𝑒(𝐹‘𝑁))
64	59, 63	syl 17	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -_𝑒(𝐹‘𝑘))‘𝑁) = -_𝑒(𝐹‘𝑁))
65	50, 57, 64	3brtr3d 5180	. 2 ⊢ (𝜑 → -_𝑒(𝐹‘𝑀) ≤ -_𝑒(𝐹‘𝑁))
66	60	eleq1d 2819	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ^* ↔ (𝐹‘𝑁) ∈ ℝ^*))
67	66	rspcv 3609	. . . 4 ⊢ (𝑁 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^* → (𝐹‘𝑁) ∈ ℝ^*))
68	59, 25, 67	sylc 65	. . 3 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ^*)
69	53	eleq1d 2819	. . . . 5 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ^* ↔ (𝐹‘𝑀) ∈ ℝ^*))
70	69	rspcv 3609	. . . 4 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^* → (𝐹‘𝑀) ∈ ℝ^*))
71	52, 25, 70	sylc 65	. . 3 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ^*)
72		xleneg 13197	. . 3 ⊢ (((𝐹‘𝑁) ∈ ℝ^* ∧ (𝐹‘𝑀) ∈ ℝ^*) → ((𝐹‘𝑁) ≤ (𝐹‘𝑀) ↔ -_𝑒(𝐹‘𝑀) ≤ -_𝑒(𝐹‘𝑁)))
73	68, 71, 72	syl2anc 585	. 2 ⊢ (𝜑 → ((𝐹‘𝑁) ≤ (𝐹‘𝑀) ↔ -_𝑒(𝐹‘𝑀) ≤ -_𝑒(𝐹‘𝑁)))
74	65, 73	mpbird 257	1 ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 1c1 11111 + caddc 11113 ℝ^*cxr 11247 ≤ cle 11249 − cmin 11444 ℤcz 12558 ℤ_≥cuz 12822 -_𝑒cxne 13089 ...cfz 13484

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-xneg 13092 df-fz 13485

This theorem is referenced by: monoord2xr 44195

W3C validator