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Theorem natglobalincr 47322

Description: Local monotonicity on half-open integer range implies global monotonicity. Inference form. (Contributed by Ender Ting, 23-Nov-2024.)

**Hypotheses**
Ref	Expression
natglobalincr.1	⊢ ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘) < (𝐵‘(𝑘 + 1))
natglobalincr.2	⊢ 𝑇 ∈ ℤ

**Assertion**
Ref	Expression
natglobalincr	⊢ ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ ((𝑘 + 1)...𝑇)(𝐵‘𝑘) < (𝐵‘𝑡)

Distinct variable groups: 𝐵,𝑘 𝑇,𝑘,𝑡 Allowed substitution hint: 𝐵(𝑡)

Proof of Theorem natglobalincr Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfzoelz 13604	. . . . 5 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
2	1	peano2zd 12627	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑘 + 1) ∈ ℤ)
3		natglobalincr.2	. . . 4 ⊢ 𝑇 ∈ ℤ
4		elfz1 13457	. . . 4 ⊢ (((𝑘 + 1) ∈ ℤ ∧ 𝑇 ∈ ℤ) → (𝑡 ∈ ((𝑘 + 1)...𝑇) ↔ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))
5	2, 3, 4	sylancl 592	. . 3 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑡 ∈ ((𝑘 + 1)...𝑇) ↔ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))
6		fveq2 6827	. . . . 5 ⊢ (𝑎 = (𝑘 + 1) → (𝐵‘𝑎) = (𝐵‘(𝑘 + 1)))
7	6	breq2d 5084	. . . 4 ⊢ (𝑎 = (𝑘 + 1) → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘(𝑘 + 1))))
8		fveq2 6827	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐵‘𝑎) = (𝐵‘𝑏))
9	8	breq2d 5084	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘𝑏)))
10		fveq2 6827	. . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝐵‘𝑎) = (𝐵‘(𝑏 + 1)))
11	10	breq2d 5084	. . . 4 ⊢ (𝑎 = (𝑏 + 1) → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘(𝑏 + 1))))
12		fveq2 6827	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝐵‘𝑎) = (𝐵‘𝑡))
13	12	breq2d 5084	. . . 4 ⊢ (𝑎 = 𝑡 → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘𝑡)))
14		natglobalincr.1	. . . . 5 ⊢ ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘) < (𝐵‘(𝑘 + 1))
15	14	rspec 3230	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝐵‘𝑘) < (𝐵‘(𝑘 + 1)))
16		df-br 5073	. . . . . . . 8 ⊢ ((𝐵‘𝑘) < (𝐵‘𝑏) ↔ ⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ < )
17		ltrelxr 11197	. . . . . . . . 9 ⊢ < ⊆ (ℝ^* × ℝ^*)
18	17	sseli 3911	. . . . . . . 8 ⊢ (⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ < → ⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ (ℝ^* × ℝ^*))
19	16, 18	sylbi 218	. . . . . . 7 ⊢ ((𝐵‘𝑘) < (𝐵‘𝑏) → ⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ (ℝ^* × ℝ^*))
20		opelxp1 5660	. . . . . . 7 ⊢ (⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ (ℝ^* × ℝ^) → (𝐵‘𝑘) ∈ ℝ^)
21	19, 20	syl 17	. . . . . 6 ⊢ ((𝐵‘𝑘) < (𝐵‘𝑏) → (𝐵‘𝑘) ∈ ℝ^*)
22	21	3ad2ant3 1141	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑘) ∈ ℝ^*)
23		opelxp2 5661	. . . . . . 7 ⊢ (⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ (ℝ^* × ℝ^) → (𝐵‘𝑏) ∈ ℝ^)
24	19, 23	syl 17	. . . . . 6 ⊢ ((𝐵‘𝑘) < (𝐵‘𝑏) → (𝐵‘𝑏) ∈ ℝ^*)
25	24	3ad2ant3 1141	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑏) ∈ ℝ^*)
26		0red 11138	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ∈ ℝ)
27		simp1 1142	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
28		zre 12519	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
29		peano2re 11310	. . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
30	27, 1, 28, 29	4syl 19	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝑘 + 1) ∈ ℝ)
31		simp21 1213	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 ∈ ℤ)
32	31	zred 12624	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 ∈ ℝ)
33		elfzole1 13613	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ 𝑘)
34	28	ltp1d 12077	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ → 𝑘 < (𝑘 + 1))
35	1, 34	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 < (𝑘 + 1))
36		0red 11138	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℝ → 0 ∈ ℝ)
37		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ)
38	36, 37, 29	3jca 1134	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℝ → (0 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ))
39		leltletr 11228	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → ((0 ≤ 𝑘 ∧ 𝑘 < (𝑘 + 1)) → 0 ≤ (𝑘 + 1)))
40	1, 28, 38, 39	4syl 19	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → ((0 ≤ 𝑘 ∧ 𝑘 < (𝑘 + 1)) → 0 ≤ (𝑘 + 1)))
41	33, 35, 40	mp2and 705	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ (𝑘 + 1))
42	41	3ad2ant1 1139	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ≤ (𝑘 + 1))
43		simp22 1214	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝑘 + 1) ≤ 𝑏)
44	26, 30, 32, 42, 43	letrd 11294	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ≤ 𝑏)
45		simp23 1215	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 < 𝑇)
46		0zd 12527	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ → 0 ∈ ℤ)
47	3	a1i 11	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ → 𝑇 ∈ ℤ)
48		elfzo 13606	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑇 ∈ ℤ) → (𝑏 ∈ (0..^𝑇) ↔ (0 ≤ 𝑏 ∧ 𝑏 < 𝑇)))
49	46, 47, 48	mpd3an23 1471	. . . . . . . . 9 ⊢ (𝑏 ∈ ℤ → (𝑏 ∈ (0..^𝑇) ↔ (0 ≤ 𝑏 ∧ 𝑏 < 𝑇)))
50		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑏 → (𝐵‘𝑘) = (𝐵‘𝑏))
51		fvoveq1 7379	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑏 → (𝐵‘(𝑘 + 1)) = (𝐵‘(𝑏 + 1)))
52	50, 51	breq12d 5085	. . . . . . . . . 10 ⊢ (𝑘 = 𝑏 → ((𝐵‘𝑘) < (𝐵‘(𝑘 + 1)) ↔ (𝐵‘𝑏) < (𝐵‘(𝑏 + 1))))
53	52, 14	vtoclri 3528	. . . . . . . . 9 ⊢ (𝑏 ∈ (0..^𝑇) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1)))
54	49, 53	biimtrrdi 255	. . . . . . . 8 ⊢ (𝑏 ∈ ℤ → ((0 ≤ 𝑏 ∧ 𝑏 < 𝑇) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1))))
55	31, 54	syl 17	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → ((0 ≤ 𝑏 ∧ 𝑏 < 𝑇) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1))))
56	44, 45, 55	mp2and 705	. . . . . 6 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1)))
57		df-br 5073	. . . . . . 7 ⊢ ((𝐵‘𝑏) < (𝐵‘(𝑏 + 1)) ↔ ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ < )
58	17	sseli 3911	. . . . . . 7 ⊢ (⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ < → ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ (ℝ^* × ℝ^*))
59	57, 58	sylbi 218	. . . . . 6 ⊢ ((𝐵‘𝑏) < (𝐵‘(𝑏 + 1)) → ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ (ℝ^* × ℝ^*))
60		opelxp2 5661	. . . . . 6 ⊢ (⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ (ℝ^* × ℝ^) → (𝐵‘(𝑏 + 1)) ∈ ℝ^)
61	56, 59, 60	3syl 18	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘(𝑏 + 1)) ∈ ℝ^*)
62		simp3 1144	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑘) < (𝐵‘𝑏))
63	22, 25, 61, 62, 56	xrlttrd 13101	. . . 4 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑘) < (𝐵‘(𝑏 + 1)))
64		elfzoel2 13603	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
65		elfzop1le2 13618	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑘 + 1) ≤ 𝑇)
66	7, 9, 11, 13, 15, 63, 2, 64, 65	fzindd 12622	. . 3 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)) → (𝐵‘𝑘) < (𝐵‘𝑡))
67	5, 66	sylbida 598	. 2 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ ((𝑘 + 1)...𝑇)) → (𝐵‘𝑘) < (𝐵‘𝑡))
68	67	rgen2 3179	1 ⊢ ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ ((𝑘 + 1)...𝑇)(𝐵‘𝑘) < (𝐵‘𝑡)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ⟨cop 4561 class class class wbr 5072 × cxp 5616 ‘cfv 6485 (class class class)co 7356 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 ℤcz 12515 ...cfz 13452 ..^cfzo 13599

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600

This theorem is referenced by: (None)

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