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Theorem natglobalincr 46496

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 13686	. . . . 5 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
2	1	peano2zd 12721	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑘 + 1) ∈ ℤ)
3		natglobalincr.2	. . . 4 ⊢ 𝑇 ∈ ℤ
4		elfz1 13543	. . . 4 ⊢ (((𝑘 + 1) ∈ ℤ ∧ 𝑇 ∈ ℤ) → (𝑡 ∈ ((𝑘 + 1)...𝑇) ↔ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))
5	2, 3, 4	sylancl 584	. . 3 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑡 ∈ ((𝑘 + 1)...𝑇) ↔ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))
6		fveq2 6901	. . . . 5 ⊢ (𝑎 = (𝑘 + 1) → (𝐵‘𝑎) = (𝐵‘(𝑘 + 1)))
7	6	breq2d 5165	. . . 4 ⊢ (𝑎 = (𝑘 + 1) → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘(𝑘 + 1))))
8		fveq2 6901	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐵‘𝑎) = (𝐵‘𝑏))
9	8	breq2d 5165	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘𝑏)))
10		fveq2 6901	. . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝐵‘𝑎) = (𝐵‘(𝑏 + 1)))
11	10	breq2d 5165	. . . 4 ⊢ (𝑎 = (𝑏 + 1) → ((𝐵‘𝑘) < (𝐵‘𝑎) ↔ (𝐵‘𝑘) < (𝐵‘(𝑏 + 1))))
12		fveq2 6901	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝐵‘𝑎) = (𝐵‘𝑡))
13	12	breq2d 5165	. . . 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 3238	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝐵‘𝑘) < (𝐵‘(𝑘 + 1)))
16		df-br 5154	. . . . . . . 8 ⊢ ((𝐵‘𝑘) < (𝐵‘𝑏) ↔ ⟨(𝐵‘𝑘), (𝐵‘𝑏)⟩ ∈ < )
17		ltrelxr 11325	. . . . . . . . 9 ⊢ < ⊆ (ℝ^* × ℝ^*)
18	17	sseli 3975	. . . . . . . 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 5724	. . . . . . 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 1132	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑘) ∈ ℝ^*)
23		opelxp2 5725	. . . . . . 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 1132	. . . . 5 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑏) ∈ ℝ^*)
26		0red 11267	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ∈ ℝ)
27		simp1 1133	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
28		zre 12614	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
29		peano2re 11437	. . . . . . . . 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 1203	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 ∈ ℤ)
32	31	zred 12718	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 ∈ ℝ)
33		elfzole1 13694	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ 𝑘)
34	28	ltp1d 12196	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ → 𝑘 < (𝑘 + 1))
35	1, 34	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 < (𝑘 + 1))
36		0red 11267	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℝ → 0 ∈ ℝ)
37		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ)
38	36, 37, 29	3jca 1125	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℝ → (0 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ))
39		leltletr 11355	. . . . . . . . . . 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 697	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ (𝑘 + 1))
42	41	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ≤ (𝑘 + 1))
43		simp22 1204	. . . . . . . 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 11421	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 0 ≤ 𝑏)
45		simp23 1205	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → 𝑏 < 𝑇)
46		0zd 12622	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ → 0 ∈ ℤ)
47	3	a1i 11	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ → 𝑇 ∈ ℤ)
48		elfzo 13688	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑇 ∈ ℤ) → (𝑏 ∈ (0..^𝑇) ↔ (0 ≤ 𝑏 ∧ 𝑏 < 𝑇)))
49	46, 47, 48	mpd3an23 1460	. . . . . . . . 9 ⊢ (𝑏 ∈ ℤ → (𝑏 ∈ (0..^𝑇) ↔ (0 ≤ 𝑏 ∧ 𝑏 < 𝑇)))
50		fveq2 6901	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑏 → (𝐵‘𝑘) = (𝐵‘𝑏))
51		fvoveq1 7447	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑏 → (𝐵‘(𝑘 + 1)) = (𝐵‘(𝑏 + 1)))
52	50, 51	breq12d 5166	. . . . . . . . . 10 ⊢ (𝑘 = 𝑏 → ((𝐵‘𝑘) < (𝐵‘(𝑘 + 1)) ↔ (𝐵‘𝑏) < (𝐵‘(𝑏 + 1))))
53	52, 14	vtoclri 3572	. . . . . . . . 9 ⊢ (𝑏 ∈ (0..^𝑇) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1)))
54	49, 53	biimtrrdi 253	. . . . . . . 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 697	. . . . . 6 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑏) < (𝐵‘(𝑏 + 1)))
57		df-br 5154	. . . . . . 7 ⊢ ((𝐵‘𝑏) < (𝐵‘(𝑏 + 1)) ↔ ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ < )
58	17	sseli 3975	. . . . . . 7 ⊢ (⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ < → ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ (ℝ^* × ℝ^*))
59	57, 58	sylbi 216	. . . . . 6 ⊢ ((𝐵‘𝑏) < (𝐵‘(𝑏 + 1)) → ⟨(𝐵‘𝑏), (𝐵‘(𝑏 + 1))⟩ ∈ (ℝ^* × ℝ^*))
60		opelxp2 5725	. . . . . 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 1135	. . . . 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 13192	. . . 4 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘) < (𝐵‘𝑏)) → (𝐵‘𝑘) < (𝐵‘(𝑏 + 1)))
64		elfzoel2 13685	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
65		elfzop1le2 13699	. . . 4 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑘 + 1) ≤ 𝑇)
66	7, 9, 11, 13, 15, 63, 2, 64, 65	fzindd 12716	. . 3 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)) → (𝐵‘𝑘) < (𝐵‘𝑡))
67	5, 66	sylbida 590	. 2 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ ((𝑘 + 1)...𝑇)) → (𝐵‘𝑘) < (𝐵‘𝑡))
68	67	rgen2 3188	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 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ⟨cop 4639 class class class wbr 5153 × cxp 5680 ‘cfv 6554 (class class class)co 7424 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 ℝ^*cxr 11297 < clt 11298 ≤ cle 11299 ℤcz 12610 ...cfz 13538 ..^cfzo 13681

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682

This theorem is referenced by: (None)

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