oexpled - Mathbox for Thierry Arnoux

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Theorem oexpled 32939

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 11147	. 2 ⊢ (𝜑 → 0 ∈ ℝ)
2		oexpled.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		0red 11147	. . 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 12474	. . . . . 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 14111	. . . 4 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
15	14	adantlr 716	. . 3 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 0 ≤ 𝐴) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
16	4	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝐴 ∈ ℝ)
17	9	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝑁 ∈ ℕ₀)
18	16, 17	reexpcld 14098	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ∈ ℝ)
19		0red 11147	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ∈ ℝ)
20	2	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝐵 ∈ ℝ)
21	20, 17	reexpcld 14098	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐵↑𝑁) ∈ ℝ)
22	8	nncnd 12173	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
23		1cnd 11139	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
24	22, 23	npcand 11508	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
25	24	oveq2d 7384	. . . . . . 7 ⊢ (𝜑 → (𝐴↑((𝑁 − 1) + 1)) = (𝐴↑𝑁))
26	4	recnd 11172	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
27		nnm1nn0 12454	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
28	8, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
29	26, 28	expp1d 14082	. . . . . . 7 ⊢ (𝜑 → (𝐴↑((𝑁 − 1) + 1)) = ((𝐴↑(𝑁 − 1)) · 𝐴))
30	25, 29	eqtr3d 2774	. . . . . 6 ⊢ (𝜑 → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
31	30	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
32	4, 28	reexpcld 14098	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑁 − 1)) ∈ ℝ)
33	32	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑(𝑁 − 1)) ∈ ℝ)
34	8	nnzd 12526	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
35		oexpled.4	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝑁)
36		oddm1even 16282	. . . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (𝜑 → 2 ∥ (𝑁 − 1))
39	4, 28, 38	expevenpos 32938	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝐴↑(𝑁 − 1)))
40	39	ad2antrr 727	. . . . . . 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 12094	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 𝐴) ≤ ((𝐴↑(𝑁 − 1)) · 0))
43	33	recnd 11172	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑(𝑁 − 1)) ∈ ℂ)
44	43	mul01d 11344	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 0) = 0)
45	42, 44	breqtrd 5126	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 𝐴) ≤ 0)
46	31, 45	eqbrtrd 5122	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ≤ 0)
47		simplr 769	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ≤ 𝐵)
48	20, 17, 47	expge0d 14099	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ≤ (𝐵↑𝑁))
49	18, 19, 21, 46, 48	letrd 11302	. . 3 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
50	3, 5, 15, 49	lecasei 11251	. 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 14098	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐴↑𝑁) ∈ ℝ)
54	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 𝐵 ∈ ℝ)
55	54, 52	reexpcld 14098	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐵↑𝑁) ∈ ℝ)
56	2	renegcld 11576	. . . . . 6 ⊢ (𝜑 → -𝐵 ∈ ℝ)
57	56	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
58	4	renegcld 11576	. . . . . 6 ⊢ (𝜑 → -𝐴 ∈ ℝ)
59	58	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐴 ∈ ℝ)
60	2	le0neg1d 11720	. . . . . 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 11652	. . . . . . 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 838	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐵 ≤ -𝐴)
66	57, 59, 52, 61, 65	leexp1ad 14111	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (-𝐵↑𝑁) ≤ (-𝐴↑𝑁))
67	2	recnd 11172	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
68		oexpneg 16284	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (-𝐵↑𝑁) = -(𝐵↑𝑁))
69	67, 8, 35, 68	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (-𝐵↑𝑁) = -(𝐵↑𝑁))
70	69	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (-𝐵↑𝑁) = -(𝐵↑𝑁))
71		oexpneg 16284	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (-𝐴↑𝑁) = -(𝐴↑𝑁))
72	26, 8, 35, 71	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (-𝐴↑𝑁) = -(𝐴↑𝑁))
73	72	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (-𝐴↑𝑁) = -(𝐴↑𝑁))
74	66, 70, 73	3brtr3d 5131	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -(𝐵↑𝑁) ≤ -(𝐴↑𝑁))
75		leneg 11652	. . . 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 838	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
78	1, 2, 50, 77	lecasei 11251	1 ⊢ (𝜑 → (𝐴↑𝑁) ≤ (𝐵↑𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 class class class wbr 5100 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ≤ cle 11179 − cmin 11376 -cneg 11377 ℕcn 12157 2c2 12212 ℕ₀cn0 12413 ℤcz 12500 ↑cexp 13996 ∥ cdvds 16191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-seq 13937 df-exp 13997 df-dvds 16192

This theorem is referenced by: cos9thpiminplylem1 33960

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