oexpled - Mathbox for Thierry Arnoux

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Theorem oexpled 33480

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 11321	. 2 ⊢ (𝜑 → 0 ∈ ℝ)
2		oexpled.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		0red 11321	. . 3 ⊢ ((𝜑 ∧ 0 ≤ 𝐵) → 0 ∈ ℝ)
4		oexpled.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 0 ≤ 𝐵) → 𝐴 ∈ ℝ)
6	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
7	2	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → 𝐵 ∈ ℝ)
8		oexpled.3	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	8	nnnn0d 12677	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → 𝑁 ∈ ℕ₀)
11		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → 0 ≤ 𝐴)
12		oexpled.5	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
13	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → 𝐴 ≤ 𝐵)
14	6, 7, 10, 11, 13	leexp1ad 14329	. . . 4 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
15	14	adantlr 728	. . 3 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 0 ≤ 𝐴) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
16	4	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝐴 ∈ ℝ)
17	9	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝑁 ∈ ℕ₀)
18	16, 17	reexpcld 14316	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ∈ ℝ)
19		0red 11321	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ∈ ℝ)
20	2	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝐵 ∈ ℝ)
21	20, 17	reexpcld 14316	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐵↑𝑁) ∈ ℝ)
22	8	nncnd 12361	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
23		1cnd 11312	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
24	22, 23	npcand 11683	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
25	24	oveq2d 7443	. . . . . . 7 ⊢ (𝜑 → (𝐴↑((𝑁 − 1) + 1)) = (𝐴↑𝑁))
26	4	recnd 11347	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
27		nnm1nn0 12657	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
28	8, 27	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
29	26, 28	expp1d 14300	. . . . . . 7 ⊢ (𝜑 → (𝐴↑((𝑁 − 1) + 1)) = ((𝐴↑(𝑁 − 1)) · 𝐴))
30	25, 29	eqtr3d 2800	. . . . . 6 ⊢ (𝜑 → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
31	30	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
32	4, 28	reexpcld 14316	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑁 − 1)) ∈ ℝ)
33	32	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑(𝑁 − 1)) ∈ ℝ)
34	8	nnzd 12729	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
35		oexpled.4	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝑁)
36		oddm1even 16525	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1)))
37	36	biimpa 482	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → 2 ∥ (𝑁 − 1))
38	34, 35, 37	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → 2 ∥ (𝑁 − 1))
39	4, 28, 38	expevenpos 33479	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝐴↑(𝑁 − 1)))
40	39	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ≤ (𝐴↑(𝑁 − 1)))
41		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝐴 ≤ 0)
42	16, 19, 33, 40, 41	lemul2ad 12267	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 𝐴) ≤ ((𝐴↑(𝑁 − 1)) · 0))
43	33	recnd 11347	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑(𝑁 − 1)) ∈ ℂ)
44	43	mul01d 11519	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 0) = 0)
45	42, 44	breqtrd 5133	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 𝐴) ≤ 0)
46	31, 45	eqbrtrd 5129	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ≤ 0)
47		simplr 781	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ≤ 𝐵)
48	20, 17, 47	expge0d 14317	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ≤ (𝐵↑𝑁))
49	18, 19, 21, 46, 48	letrd 11477	. . 3 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
50	3, 5, 15, 49	lecasei 11426	. 2 ⊢ ((𝜑 ∧ 0 ≤ 𝐵) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
51	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℝ)
52	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 𝑁 ∈ ℕ₀)
53	51, 52	reexpcld 14316	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐴↑𝑁) ∈ ℝ)
54	2	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 𝐵 ∈ ℝ)
55	54, 52	reexpcld 14316	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐵↑𝑁) ∈ ℝ)
56	2	renegcld 11753	. . . . . 6 ⊢ (𝜑 → -𝐵 ∈ ℝ)
57	56	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
58	4	renegcld 11753	. . . . . 6 ⊢ (𝜑 → -𝐴 ∈ ℝ)
59	58	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐴 ∈ ℝ)
60	2	le0neg1d 11897	. . . . . 6 ⊢ (𝜑 → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
61	60	biimpa 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
62	12	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 𝐴 ≤ 𝐵)
63		leneg 11829	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴))
64	63	biimpa 482	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → -𝐵 ≤ -𝐴)
65	51, 54, 62, 64	syl21anc 851	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐵 ≤ -𝐴)
66	57, 59, 52, 61, 65	leexp1ad 14329	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → ( -𝐵↑𝑁) ≤ ( -𝐴↑𝑁))
67	2	recnd 11347	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
68		oexpneg 16527	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → ( -𝐵↑𝑁) = -(𝐵↑𝑁))
69	67, 8, 35, 68	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ( -𝐵↑𝑁) = -(𝐵↑𝑁))
70	69	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → ( -𝐵↑𝑁) = -(𝐵↑𝑁))
71		oexpneg 16527	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → ( -𝐴↑𝑁) = -(𝐴↑𝑁))
72	26, 8, 35, 71	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ( -𝐴↑𝑁) = -(𝐴↑𝑁))
73	72	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → ( -𝐴↑𝑁) = -(𝐴↑𝑁))
74	66, 70, 73	3brtr3d 5138	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -(𝐵↑𝑁) ≤ -(𝐴↑𝑁))
75		leneg 11829	. . . 4 ⊢ (((𝐴↑𝑁) ∈ ℝ ∧ (𝐵↑𝑁) ∈ ℝ) → ((𝐴↑𝑁) ≤ (𝐵↑𝑁) ↔ -(𝐵↑𝑁) ≤ -(𝐴↑𝑁)))
76	75	biimpar 483	. . 3 ⊢ ((((𝐴↑𝑁) ∈ ℝ ∧ (𝐵↑𝑁) ∈ ℝ) ∧ -(𝐵↑𝑁) ≤ -(𝐴↑𝑁)) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
77	53, 55, 74, 76	syl21anc 851	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
78	1, 2, 50, 77	lecasei 11426	1 ⊢ (𝜑 → (𝐴↑𝑁) ≤ (𝐵↑𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2146 class class class wbr 5105 (class class class)co 7427 ℂcc 11208 ℝcr 11209 0cc0 11210 1c1 11211 + caddc 11213 · cmul 11215 ≤ cle 11354 − cmin 11551 -cneg 11552 ℕcn 12345 2c2 12407 ℕ₀cn0 12616 ℤcz 12703 ↑cexp 14214 ∥ cdvds 16434

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-er 8728 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-n0 12617 df-z 12704 df-uz 12976 df-rp 13131 df-seq 14155 df-exp 14215 df-dvds 16435

This theorem is used by: cos9thpiminplylem1 34467

W3C validator