oexpled - Mathbox for Thierry Arnoux

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Theorem oexpled 32793

Description: Odd power monomials are monotonic. (Contributed by Thierry Arnoux, 9-Nov-2025.)

**Hypotheses**
Ref	Expression
oexpled.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
oexpled.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
oexpled.3	⊢ (𝜑 → 𝑁 ∈ ℕ)
oexpled.4	⊢ (𝜑 → ¬ 2 ∥ 𝑁)
oexpled.5	⊢ (𝜑 → 𝐴 ≤ 𝐵)

**Assertion**
Ref	Expression
oexpled	⊢ (𝜑 → (𝐴↑𝑁) ≤ (𝐵↑𝑁))

**Proof of Theorem oexpled**
Step	Hyp	Ref	Expression
1		0red 11118	. 2 ⊢ (𝜑 → 0 ∈ ℝ)
2		oexpled.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		0red 11118	. . 3 ⊢ ((𝜑 ∧ 0 ≤ 𝐵) → 0 ∈ ℝ)
4		oexpled.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0 ≤ 𝐵) → 𝐴 ∈ ℝ)
6	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
7	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → 𝐵 ∈ ℝ)
8		oexpled.3	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	8	nnnn0d 12445	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → 𝑁 ∈ ℕ₀)
11		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → 0 ≤ 𝐴)
12		oexpled.5	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
13	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → 𝐴 ≤ 𝐵)
14	6, 7, 10, 11, 13	leexp1ad 14083	. . . 4 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
15	14	adantlr 715	. . 3 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 0 ≤ 𝐴) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
16	4	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝐴 ∈ ℝ)
17	9	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝑁 ∈ ℕ₀)
18	16, 17	reexpcld 14070	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ∈ ℝ)
19		0red 11118	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ∈ ℝ)
20	2	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝐵 ∈ ℝ)
21	20, 17	reexpcld 14070	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐵↑𝑁) ∈ ℝ)
22	8	nncnd 12144	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
23		1cnd 11110	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
24	22, 23	npcand 11479	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
25	24	oveq2d 7365	. . . . . . 7 ⊢ (𝜑 → (𝐴↑((𝑁 − 1) + 1)) = (𝐴↑𝑁))
26	4	recnd 11143	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
27		nnm1nn0 12425	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
28	8, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
29	26, 28	expp1d 14054	. . . . . . 7 ⊢ (𝜑 → (𝐴↑((𝑁 − 1) + 1)) = ((𝐴↑(𝑁 − 1)) · 𝐴))
30	25, 29	eqtr3d 2766	. . . . . 6 ⊢ (𝜑 → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
31	30	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
32	4, 28	reexpcld 14070	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑁 − 1)) ∈ ℝ)
33	32	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑(𝑁 − 1)) ∈ ℝ)
34	8	nnzd 12498	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
35		oexpled.4	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝑁)
36		oddm1even 16254	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1)))
37	36	biimpa 476	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → 2 ∥ (𝑁 − 1))
38	34, 35, 37	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 2 ∥ (𝑁 − 1))
39	4, 28, 38	expevenpos 32792	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝐴↑(𝑁 − 1)))
40	39	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ≤ (𝐴↑(𝑁 − 1)))
41		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝐴 ≤ 0)
42	16, 19, 33, 40, 41	lemul2ad 12065	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 𝐴) ≤ ((𝐴↑(𝑁 − 1)) · 0))
43	33	recnd 11143	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑(𝑁 − 1)) ∈ ℂ)
44	43	mul01d 11315	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 0) = 0)
45	42, 44	breqtrd 5118	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 𝐴) ≤ 0)
46	31, 45	eqbrtrd 5114	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ≤ 0)
47		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ≤ 𝐵)
48	20, 17, 47	expge0d 14071	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ≤ (𝐵↑𝑁))
49	18, 19, 21, 46, 48	letrd 11273	. . 3 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
50	3, 5, 15, 49	lecasei 11222	. 2 ⊢ ((𝜑 ∧ 0 ≤ 𝐵) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
51	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℝ)
52	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 𝑁 ∈ ℕ₀)
53	51, 52	reexpcld 14070	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐴↑𝑁) ∈ ℝ)
54	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 𝐵 ∈ ℝ)
55	54, 52	reexpcld 14070	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐵↑𝑁) ∈ ℝ)
56	2	renegcld 11547	. . . . . 6 ⊢ (𝜑 → -𝐵 ∈ ℝ)
57	56	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
58	4	renegcld 11547	. . . . . 6 ⊢ (𝜑 → -𝐴 ∈ ℝ)
59	58	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐴 ∈ ℝ)
60	2	le0neg1d 11691	. . . . . 6 ⊢ (𝜑 → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
61	60	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
62	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 𝐴 ≤ 𝐵)
63		leneg 11623	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴))
64	63	biimpa 476	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → -𝐵 ≤ -𝐴)
65	51, 54, 62, 64	syl21anc 837	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐵 ≤ -𝐴)
66	57, 59, 52, 61, 65	leexp1ad 14083	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (-𝐵↑𝑁) ≤ (-𝐴↑𝑁))
67	2	recnd 11143	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
68		oexpneg 16256	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (-𝐵↑𝑁) = -(𝐵↑𝑁))
69	67, 8, 35, 68	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (-𝐵↑𝑁) = -(𝐵↑𝑁))
70	69	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (-𝐵↑𝑁) = -(𝐵↑𝑁))
71		oexpneg 16256	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (-𝐴↑𝑁) = -(𝐴↑𝑁))
72	26, 8, 35, 71	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (-𝐴↑𝑁) = -(𝐴↑𝑁))
73	72	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (-𝐴↑𝑁) = -(𝐴↑𝑁))
74	66, 70, 73	3brtr3d 5123	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -(𝐵↑𝑁) ≤ -(𝐴↑𝑁))
75		leneg 11623	. . . 4 ⊢ (((𝐴↑𝑁) ∈ ℝ ∧ (𝐵↑𝑁) ∈ ℝ) → ((𝐴↑𝑁) ≤ (𝐵↑𝑁) ↔ -(𝐵↑𝑁) ≤ -(𝐴↑𝑁)))
76	75	biimpar 477	. . 3 ⊢ ((((𝐴↑𝑁) ∈ ℝ ∧ (𝐵↑𝑁) ∈ ℝ) ∧ -(𝐵↑𝑁) ≤ -(𝐴↑𝑁)) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
77	53, 55, 74, 76	syl21anc 837	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
78	1, 2, 50, 77	lecasei 11222	1 ⊢ (𝜑 → (𝐴↑𝑁) ≤ (𝐵↑𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 class class class wbr 5092 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 ≤ cle 11150 − cmin 11347 -cneg 11348 ℕcn 12128 2c2 12183 ℕ₀cn0 12384 ℤcz 12471 ↑cexp 13968 ∥ cdvds 16163

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 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

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-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-n0 12385 df-z 12472 df-uz 12736 df-rp 12894 df-seq 13909 df-exp 13969 df-dvds 16164

This theorem is referenced by: cos9thpiminplylem1 33755

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