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Theorem natglobalincr 46868

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 13596	. . . . 5 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
2	1	peano2zd 12617	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑘 + 1) ∈ ℤ)
3		natglobalincr.2	. . . 4 ⊢ 𝑇 ∈ ℤ
4		elfz1 13449	. . . 4 ⊢ (((𝑘 + 1) ∈ ℤ ∧ 𝑇 ∈ ℤ) → (𝑡 ∈ ((𝑘 + 1)...𝑇) ↔ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))
5	2, 3, 4	sylancl 586	. . 3 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑡 ∈ ((𝑘 + 1)...𝑇) ↔ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))
6		fveq2 6840	. . . . 5 ⊢ (𝑎 = (𝑘 + 1) → (𝐵‘𝑎) = (𝐵‘(𝑘 + 1)))
7	6	breq2d 5114	. . . 4 ⊢ (𝑎 = (𝑘 + 1) → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘(𝑘 + 1))))
8		fveq2 6840	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐵‘𝑎) = (𝐵‘𝑏))
9	8	breq2d 5114	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘𝑏)))
10		fveq2 6840	. . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝐵‘𝑎) = (𝐵‘(𝑏 + 1)))
11	10	breq2d 5114	. . . 4 ⊢ (𝑎 = (𝑏 + 1) → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘(𝑏 + 1))))
12		fveq2 6840	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝐵‘𝑎) = (𝐵‘𝑡))
13	12	breq2d 5114	. . . 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 3226	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝐵‘𝑘) < (𝐵‘(𝑘 + 1)))
16		df-br 5103	. . . . . . . 8 ⊢ ((𝐵‘𝑘) < (𝐵‘𝑏) ↔ ⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ < )
17		ltrelxr 11211	. . . . . . . . 9 ⊢ < ⊆ (ℝ^* × ℝ^*)
18	17	sseli 3939	. . . . . . . 8 ⊢ (⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ < → ⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ (ℝ^* × ℝ^*))
19	16, 18	sylbi 217	. . . . . . 7 ⊢ ((𝐵‘𝑘) < (𝐵‘𝑏) → ⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ (ℝ^* × ℝ^*))
20		opelxp1 5673	. . . . . . 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 1135	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑘) ∈ ℝ^*)
23		opelxp2 5674	. . . . . . 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 1135	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑏) ∈ ℝ^*)
26		0red 11153	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ∈ ℝ)
27		simp1 1136	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
28		zre 12509	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
29		peano2re 11323	. . . . . . . . 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 1207	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 ∈ ℤ)
32	31	zred 12614	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 ∈ ℝ)
33		elfzole1 13604	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ 𝑘)
34	28	ltp1d 12089	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ → 𝑘 < (𝑘 + 1))
35	1, 34	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 < (𝑘 + 1))
36		0red 11153	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℝ → 0 ∈ ℝ)
37		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ)
38	36, 37, 29	3jca 1128	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℝ → (0 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ))
39		leltletr 11241	. . . . . . . . . . 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 699	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ (𝑘 + 1))
42	41	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ≤ (𝑘 + 1))
43		simp22 1208	. . . . . . . 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 11307	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ≤ 𝑏)
45		simp23 1209	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 < 𝑇)
46		0zd 12517	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ → 0 ∈ ℤ)
47	3	a1i 11	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ → 𝑇 ∈ ℤ)
48		elfzo 13598	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑇 ∈ ℤ) → (𝑏 ∈ (0..^𝑇) ↔ (0 ≤ 𝑏 ∧ 𝑏 < 𝑇)))
49	46, 47, 48	mpd3an23 1465	. . . . . . . . 9 ⊢ (𝑏 ∈ ℤ → (𝑏 ∈ (0..^𝑇) ↔ (0 ≤ 𝑏 ∧ 𝑏 < 𝑇)))
50		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑏 → (𝐵‘𝑘) = (𝐵‘𝑏))
51		fvoveq1 7392	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑏 → (𝐵‘(𝑘 + 1)) = (𝐵‘(𝑏 + 1)))
52	50, 51	breq12d 5115	. . . . . . . . . 10 ⊢ (𝑘 = 𝑏 → ((𝐵‘𝑘) < (𝐵‘(𝑘 + 1)) ↔ (𝐵‘𝑏) < (𝐵‘(𝑏 + 1))))
53	52, 14	vtoclri 3553	. . . . . . . . 9 ⊢ (𝑏 ∈ (0..^𝑇) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1)))
54	49, 53	biimtrrdi 254	. . . . . . . 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 699	. . . . . 6 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1)))
57		df-br 5103	. . . . . . 7 ⊢ ((𝐵‘𝑏) < (𝐵‘(𝑏 + 1)) ↔ ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ < )
58	17	sseli 3939	. . . . . . 7 ⊢ (⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ < → ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ (ℝ^* × ℝ^*))
59	57, 58	sylbi 217	. . . . . 6 ⊢ ((𝐵‘𝑏) < (𝐵‘(𝑏 + 1)) → ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ (ℝ^* × ℝ^*))
60		opelxp2 5674	. . . . . 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 1138	. . . . 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 13095	. . . 4 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑘) < (𝐵‘(𝑏 + 1)))
64		elfzoel2 13595	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
65		elfzop1le2 13609	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑘 + 1) ≤ 𝑇)
66	7, 9, 11, 13, 15, 63, 2, 64, 65	fzindd 12612	. . 3 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)) → (𝐵‘𝑘) < (𝐵‘𝑡))
67	5, 66	sylbida 592	. 2 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ ((𝑘 + 1)...𝑇)) → (𝐵‘𝑘) < (𝐵‘𝑡))
68	67	rgen2 3175	1 ⊢ ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ ((𝑘 + 1)...𝑇)(𝐵‘𝑘) < (𝐵‘𝑡)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⟨cop 4591 class class class wbr 5102 × cxp 5629 ‘cfv 6499 (class class class)co 7369 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 ℤcz 12505 ...cfz 13444 ..^cfzo 13591

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592

This theorem is referenced by: (None)

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