oexpreposd - Mathbox for Steven Nguyen

	Mathbox for Steven Nguyen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > oexpreposd		Structured version Visualization version GIF version

Theorem oexpreposd 42754

Description: Lemma for dffltz 43067. For a more standard version, see expgt0b 32890. TODO-SN?: This can be used to show exp11d 42758 holds for all integers when the exponent is odd. (Contributed by SN, 4-Mar-2023.)

**Hypotheses**
Ref	Expression
oexpreposd.n	⊢ (𝜑 → 𝑁 ∈ ℝ)
oexpreposd.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
oexpreposd.1	⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℕ)

**Assertion**
Ref	Expression
oexpreposd	⊢ (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁↑𝑀)))

**Proof of Theorem oexpreposd**
Step	Hyp	Ref	Expression
1		oexpreposd.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑁 ∈ ℝ)
3		oexpreposd.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4	3	nnzd 12550	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑀 ∈ ℤ)
6		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → 0 < 𝑁)
7		expgt0 14057	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑁) → 0 < (𝑁↑𝑀))
8	2, 5, 6, 7	syl3anc 1374	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑁) → 0 < (𝑁↑𝑀))
9	8	ex 412	. 2 ⊢ (𝜑 → (0 < 𝑁 → 0 < (𝑁↑𝑀)))
10		0red 11147	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
11	10, 1	lttrid 11284	. . . 4 ⊢ (𝜑 → (0 < 𝑁 ↔ ¬ (0 = 𝑁 ∨ 𝑁 < 0)))
12	11	notbid 318	. . 3 ⊢ (𝜑 → (¬ 0 < 𝑁 ↔ ¬ ¬ (0 = 𝑁 ∨ 𝑁 < 0)))
13		notnotr 130	. . . 4 ⊢ (¬ ¬ (0 = 𝑁 ∨ 𝑁 < 0) → (0 = 𝑁 ∨ 𝑁 < 0))
14		0re 11146	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
15	14	ltnri 11255	. . . . . . . . 9 ⊢ ¬ 0 < 0
16	3	0expd 14101	. . . . . . . . . 10 ⊢ (𝜑 → (0↑𝑀) = 0)
17	16	breq2d 5097	. . . . . . . . 9 ⊢ (𝜑 → (0 < (0↑𝑀) ↔ 0 < 0))
18	15, 17	mtbiri 327	. . . . . . . 8 ⊢ (𝜑 → ¬ 0 < (0↑𝑀))
19	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 = 𝑁) → ¬ 0 < (0↑𝑀))
20		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 = 𝑁) → 0 = 𝑁)
21	20	eqcomd 2742	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 = 𝑁) → 𝑁 = 0)
22	21	oveq1d 7382	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 = 𝑁) → (𝑁↑𝑀) = (0↑𝑀))
23	22	breq2d 5097	. . . . . . 7 ⊢ ((𝜑 ∧ 0 = 𝑁) → (0 < (𝑁↑𝑀) ↔ 0 < (0↑𝑀)))
24	19, 23	mtbird 325	. . . . . 6 ⊢ ((𝜑 ∧ 0 = 𝑁) → ¬ 0 < (𝑁↑𝑀))
25	24	ex 412	. . . . 5 ⊢ (𝜑 → (0 = 𝑁 → ¬ 0 < (𝑁↑𝑀)))
26	1	renegcld 11577	. . . . . . . . . 10 ⊢ (𝜑 → -𝑁 ∈ ℝ)
27	26	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < -𝑁) → -𝑁 ∈ ℝ)
28	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < -𝑁) → 𝑀 ∈ ℤ)
29		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < -𝑁) → 0 < -𝑁)
30		expgt0 14057	. . . . . . . . 9 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 0 < -𝑁) → 0 < (-𝑁↑𝑀))
31	27, 28, 29, 30	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < -𝑁) → 0 < (-𝑁↑𝑀))
32	31	ex 412	. . . . . . 7 ⊢ (𝜑 → (0 < -𝑁 → 0 < (-𝑁↑𝑀)))
33	1	recnd 11173	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
34		oexpreposd.1	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℕ)
35		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℤ)
36		zq 12904	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 / 2) ∈ ℤ → (𝑀 / 2) ∈ ℚ)
37	36	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℚ)
38		qden1elz 16727	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 / 2) ∈ ℚ → ((denom‘(𝑀 / 2)) = 1 ↔ (𝑀 / 2) ∈ ℤ))
39	37, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((denom‘(𝑀 / 2)) = 1 ↔ (𝑀 / 2) ∈ ℤ))
40	35, 39	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (denom‘(𝑀 / 2)) = 1)
41	40	oveq2d 7383	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((𝑀 / 2) · (denom‘(𝑀 / 2))) = ((𝑀 / 2) · 1))
42		qmuldeneqnum 16717	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 / 2) ∈ ℚ → ((𝑀 / 2) · (denom‘(𝑀 / 2))) = (numer‘(𝑀 / 2)))
43	37, 42	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((𝑀 / 2) · (denom‘(𝑀 / 2))) = (numer‘(𝑀 / 2)))
44	35	zcnd 12634	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℂ)
45	44	mulridd 11162	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((𝑀 / 2) · 1) = (𝑀 / 2))
46	41, 43, 45	3eqtr3rd 2780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) = (numer‘(𝑀 / 2)))
47	3	nnred 12189	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
48		2re 12255	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
49	48	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℝ)
50	3	nngt0d 12226	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 𝑀)
51		2pos 12284	. . . . . . . . . . . . . . . 16 ⊢ 0 < 2
52	51	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 2)
53	47, 49, 50, 52	divgt0d 12091	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝑀 / 2))
54		qgt0numnn 16721	. . . . . . . . . . . . . 14 ⊢ (((𝑀 / 2) ∈ ℚ ∧ 0 < (𝑀 / 2)) → (numer‘(𝑀 / 2)) ∈ ℕ)
55	36, 53, 54	syl2anr 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (numer‘(𝑀 / 2)) ∈ ℕ)
56	46, 55	eqeltrd 2836	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℕ)
57	34, 56	mtand 816	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℤ)
58		evend2 16326	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (2 ∥ 𝑀 ↔ (𝑀 / 2) ∈ ℤ))
59	4, 58	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ∥ 𝑀 ↔ (𝑀 / 2) ∈ ℤ))
60	57, 59	mtbird 325	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝑀)
61		oexpneg 16314	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℕ ∧ ¬ 2 ∥ 𝑀) → (-𝑁↑𝑀) = -(𝑁↑𝑀))
62	33, 3, 60, 61	syl3anc 1374	. . . . . . . . 9 ⊢ (𝜑 → (-𝑁↑𝑀) = -(𝑁↑𝑀))
63	62	breq2d 5097	. . . . . . . 8 ⊢ (𝜑 → (0 < (-𝑁↑𝑀) ↔ 0 < -(𝑁↑𝑀)))
64	63	biimpd 229	. . . . . . 7 ⊢ (𝜑 → (0 < (-𝑁↑𝑀) → 0 < -(𝑁↑𝑀)))
65	3	nnnn0d 12498	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
66	1, 65	reexpcld 14125	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑𝑀) ∈ ℝ)
67	66	renegcld 11577	. . . . . . . . 9 ⊢ (𝜑 → -(𝑁↑𝑀) ∈ ℝ)
68	10, 67	lttrid 11284	. . . . . . . 8 ⊢ (𝜑 → (0 < -(𝑁↑𝑀) ↔ ¬ (0 = -(𝑁↑𝑀) ∨ -(𝑁↑𝑀) < 0)))
69		pm2.46 883	. . . . . . . 8 ⊢ (¬ (0 = -(𝑁↑𝑀) ∨ -(𝑁↑𝑀) < 0) → ¬ -(𝑁↑𝑀) < 0)
70	68, 69	biimtrdi 253	. . . . . . 7 ⊢ (𝜑 → (0 < -(𝑁↑𝑀) → ¬ -(𝑁↑𝑀) < 0))
71	32, 64, 70	3syld 60	. . . . . 6 ⊢ (𝜑 → (0 < -𝑁 → ¬ -(𝑁↑𝑀) < 0))
72	1	lt0neg1d 11719	. . . . . 6 ⊢ (𝜑 → (𝑁 < 0 ↔ 0 < -𝑁))
73	66	lt0neg2d 11720	. . . . . . 7 ⊢ (𝜑 → (0 < (𝑁↑𝑀) ↔ -(𝑁↑𝑀) < 0))
74	73	notbid 318	. . . . . 6 ⊢ (𝜑 → (¬ 0 < (𝑁↑𝑀) ↔ ¬ -(𝑁↑𝑀) < 0))
75	71, 72, 74	3imtr4d 294	. . . . 5 ⊢ (𝜑 → (𝑁 < 0 → ¬ 0 < (𝑁↑𝑀)))
76	25, 75	jaod 860	. . . 4 ⊢ (𝜑 → ((0 = 𝑁 ∨ 𝑁 < 0) → ¬ 0 < (𝑁↑𝑀)))
77	13, 76	syl5 34	. . 3 ⊢ (𝜑 → (¬ ¬ (0 = 𝑁 ∨ 𝑁 < 0) → ¬ 0 < (𝑁↑𝑀)))
78	12, 77	sylbid 240	. 2 ⊢ (𝜑 → (¬ 0 < 𝑁 → ¬ 0 < (𝑁↑𝑀)))
79	9, 78	impcon4bid 227	1 ⊢ (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁↑𝑀)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 < clt 11179 -cneg 11378 / cdiv 11807 ℕcn 12174 2c2 12236 ℤcz 12524 ℚcq 12898 ↑cexp 14023 ∥ cdvds 16221 numercnumer 16703 denomcdenom 16704

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 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 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 ax-pre-sup 11116

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 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 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-dvds 16222 df-gcd 16464 df-numer 16705 df-denom 16706

This theorem is referenced by: dffltz 43067

W3C validator