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Theorem natglobalincr 45891

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 13637	. . . . 5 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
2	1	peano2zd 12674	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑘 + 1) ∈ ℤ)
3		natglobalincr.2	. . . 4 ⊢ 𝑇 ∈ ℤ
4		elfz1 13494	. . . 4 ⊢ (((𝑘 + 1) ∈ ℤ ∧ 𝑇 ∈ ℤ) → (𝑡 ∈ ((𝑘 + 1)...𝑇) ↔ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))
5	2, 3, 4	sylancl 585	. . 3 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑡 ∈ ((𝑘 + 1)...𝑇) ↔ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))
6		fveq2 6892	. . . . 5 ⊢ (𝑎 = (𝑘 + 1) → (𝐵‘𝑎) = (𝐵‘(𝑘 + 1)))
7	6	breq2d 5161	. . . 4 ⊢ (𝑎 = (𝑘 + 1) → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘(𝑘 + 1))))
8		fveq2 6892	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐵‘𝑎) = (𝐵‘𝑏))
9	8	breq2d 5161	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘𝑏)))
10		fveq2 6892	. . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝐵‘𝑎) = (𝐵‘(𝑏 + 1)))
11	10	breq2d 5161	. . . 4 ⊢ (𝑎 = (𝑏 + 1) → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘(𝑏 + 1))))
12		fveq2 6892	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝐵‘𝑎) = (𝐵‘𝑡))
13	12	breq2d 5161	. . . 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 3246	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝐵‘𝑘) < (𝐵‘(𝑘 + 1)))
16		df-br 5150	. . . . . . . 8 ⊢ ((𝐵‘𝑘) < (𝐵‘𝑏) ↔ ⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ < )
17		ltrelxr 11280	. . . . . . . . 9 ⊢ < ⊆ (ℝ^* × ℝ^*)
18	17	sseli 3979	. . . . . . . 8 ⊢ (⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ < → ⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ (ℝ^* × ℝ^*))
19	16, 18	sylbi 216	. . . . . . 7 ⊢ ((𝐵‘𝑘) < (𝐵‘𝑏) → ⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ (ℝ^* × ℝ^*))
20		opelxp1 5719	. . . . . . 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 1134	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑘) ∈ ℝ^*)
23		opelxp2 5720	. . . . . . 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 1134	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑏) ∈ ℝ^*)
26		0red 11222	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ∈ ℝ)
27		simp1 1135	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
28		zre 12567	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
29	27, 1, 28	3syl 18	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑘 ∈ ℝ)
30		peano2re 11392	. . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
31	29, 30	syl 17	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝑘 + 1) ∈ ℝ)
32		simp21 1205	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 ∈ ℤ)
33	32	zred 12671	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 ∈ ℝ)
34		elfzole1 13645	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ 𝑘)
35	28	ltp1d 12149	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ → 𝑘 < (𝑘 + 1))
36	1, 35	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 < (𝑘 + 1))
37		0red 11222	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℝ → 0 ∈ ℝ)
38		id 22	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ)
39	37, 38, 30	3jca 1127	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℝ → (0 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ))
40	1, 28, 39	3syl 18	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑇) → (0 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ))
41		leltletr 11310	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → ((0 ≤ 𝑘 ∧ 𝑘 < (𝑘 + 1)) → 0 ≤ (𝑘 + 1)))
42	40, 41	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → ((0 ≤ 𝑘 ∧ 𝑘 < (𝑘 + 1)) → 0 ≤ (𝑘 + 1)))
43	34, 36, 42	mp2and 696	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ (𝑘 + 1))
44	43	3ad2ant1 1132	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ≤ (𝑘 + 1))
45		simp22 1206	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝑘 + 1) ≤ 𝑏)
46	26, 31, 33, 44, 45	letrd 11376	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ≤ 𝑏)
47		simp23 1207	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 < 𝑇)
48		0zd 12575	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ → 0 ∈ ℤ)
49	3	a1i 11	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ → 𝑇 ∈ ℤ)
50		elfzo 13639	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑇 ∈ ℤ) → (𝑏 ∈ (0..^𝑇) ↔ (0 ≤ 𝑏 ∧ 𝑏 < 𝑇)))
51	48, 49, 50	mpd3an23 1462	. . . . . . . . 9 ⊢ (𝑏 ∈ ℤ → (𝑏 ∈ (0..^𝑇) ↔ (0 ≤ 𝑏 ∧ 𝑏 < 𝑇)))
52		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑏 → (𝐵‘𝑘) = (𝐵‘𝑏))
53		fvoveq1 7435	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑏 → (𝐵‘(𝑘 + 1)) = (𝐵‘(𝑏 + 1)))
54	52, 53	breq12d 5162	. . . . . . . . . 10 ⊢ (𝑘 = 𝑏 → ((𝐵‘𝑘) < (𝐵‘(𝑘 + 1)) ↔ (𝐵‘𝑏) < (𝐵‘(𝑏 + 1))))
55	54, 14	vtoclri 3577	. . . . . . . . 9 ⊢ (𝑏 ∈ (0..^𝑇) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1)))
56	51, 55	syl6bir 253	. . . . . . . 8 ⊢ (𝑏 ∈ ℤ → ((0 ≤ 𝑏 ∧ 𝑏 < 𝑇) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1))))
57	32, 56	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))))
58	46, 47, 57	mp2and 696	. . . . . 6 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1)))
59		df-br 5150	. . . . . . 7 ⊢ ((𝐵‘𝑏) < (𝐵‘(𝑏 + 1)) ↔ ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ < )
60	17	sseli 3979	. . . . . . 7 ⊢ (⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ < → ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ (ℝ^* × ℝ^*))
61	59, 60	sylbi 216	. . . . . 6 ⊢ ((𝐵‘𝑏) < (𝐵‘(𝑏 + 1)) → ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ (ℝ^* × ℝ^*))
62		opelxp2 5720	. . . . . 6 ⊢ (⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ (ℝ^* × ℝ^) → (𝐵‘(𝑏 + 1)) ∈ ℝ^)
63	58, 61, 62	3syl 18	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘(𝑏 + 1)) ∈ ℝ^*)
64		simp3 1137	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑘) < (𝐵‘𝑏))
65	22, 25, 63, 64, 58	xrlttrd 13143	. . . 4 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑘) < (𝐵‘(𝑏 + 1)))
66		elfzoel2 13636	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
67		elfzop1le2 13650	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑘 + 1) ≤ 𝑇)
68	7, 9, 11, 13, 15, 65, 2, 66, 67	fzindd 12669	. . 3 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)) → (𝐵‘𝑘) < (𝐵‘𝑡))
69	5, 68	sylbida 591	. 2 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ ((𝑘 + 1)...𝑇)) → (𝐵‘𝑘) < (𝐵‘𝑡))
70	69	rgen2 3196	1 ⊢ ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ ((𝑘 + 1)...𝑇)(𝐵‘𝑘) < (𝐵‘𝑡)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ⟨cop 4635 class class class wbr 5149 × cxp 5675 ‘cfv 6544 (class class class)co 7412 ℝcr 11112 0cc0 11113 1c1 11114 + caddc 11116 ℝ^*cxr 11252 < clt 11253 ≤ cle 11254 ℤcz 12563 ...cfz 13489 ..^cfzo 13632

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633

This theorem is referenced by: (None)

W3C validator