oexpled - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > oexpled		Structured version Visualization version GIF version

Theorem oexpled 32822

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 11153	. 2 ⊢ (𝜑 → 0 ∈ ℝ)
2		oexpled.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		0red 11153	. . 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 12479	. . . . . 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 14117	. . . 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 14104	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ∈ ℝ)
19		0red 11153	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ∈ ℝ)
20	2	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 𝐵 ∈ ℝ)
21	20, 17	reexpcld 14104	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐵↑𝑁) ∈ ℝ)
22	8	nncnd 12178	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
23		1cnd 11145	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
24	22, 23	npcand 11513	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
25	24	oveq2d 7385	. . . . . . 7 ⊢ (𝜑 → (𝐴↑((𝑁 − 1) + 1)) = (𝐴↑𝑁))
26	4	recnd 11178	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
27		nnm1nn0 12459	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
28	8, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
29	26, 28	expp1d 14088	. . . . . . 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 14104	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑁 − 1)) ∈ ℝ)
33	32	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑(𝑁 − 1)) ∈ ℝ)
34	8	nnzd 12532	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
35		oexpled.4	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝑁)
36		oddm1even 16289	. . . . . . . . . . 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 32821	. . . . . . . 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 12099	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 𝐴) ≤ ((𝐴↑(𝑁 − 1)) · 0))
43	33	recnd 11178	. . . . . . 7 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑(𝑁 − 1)) ∈ ℂ)
44	43	mul01d 11349	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 0) = 0)
45	42, 44	breqtrd 5128	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → ((𝐴↑(𝑁 − 1)) · 𝐴) ≤ 0)
46	31, 45	eqbrtrd 5124	. . . 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 14105	. . . 4 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → 0 ≤ (𝐵↑𝑁))
49	18, 19, 21, 46, 48	letrd 11307	. . 3 ⊢ (((𝜑 ∧ 0 ≤ 𝐵) ∧ 𝐴 ≤ 0) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
50	3, 5, 15, 49	lecasei 11256	. 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 14104	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐴↑𝑁) ∈ ℝ)
54	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → 𝐵 ∈ ℝ)
55	54, 52	reexpcld 14104	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝐵↑𝑁) ∈ ℝ)
56	2	renegcld 11581	. . . . . 6 ⊢ (𝜑 → -𝐵 ∈ ℝ)
57	56	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
58	4	renegcld 11581	. . . . . 6 ⊢ (𝜑 → -𝐴 ∈ ℝ)
59	58	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -𝐴 ∈ ℝ)
60	2	le0neg1d 11725	. . . . . 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 11657	. . . . . . 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 14117	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (-𝐵↑𝑁) ≤ (-𝐴↑𝑁))
67	2	recnd 11178	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
68		oexpneg 16291	. . . . . 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 16291	. . . . . 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 5133	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → -(𝐵↑𝑁) ≤ -(𝐴↑𝑁))
75		leneg 11657	. . . 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 11256	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 5102 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 ≤ cle 11185 − cmin 11381 -cneg 11382 ℕcn 12162 2c2 12217 ℕ₀cn0 12418 ℤcz 12505 ↑cexp 14002 ∥ cdvds 16198

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-rmo 3351 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-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-div 11812 df-nn 12163 df-2 12225 df-n0 12419 df-z 12506 df-uz 12770 df-rp 12928 df-seq 13943 df-exp 14003 df-dvds 16199

This theorem is referenced by: cos9thpiminplylem1 33765

W3C validator