oexpreposd - Mathbox for Steven Nguyen

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Theorem oexpreposd 42598

Description: Lemma for dffltz 42898. For a more standard version, see expgt0b 32899. TODO-SN?: This can be used to show exp11d 42602 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 12516	. . . . 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 14020	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑁) → 0 < (𝑁↑𝑀))
8	2, 5, 6, 7	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑁) → 0 < (𝑁↑𝑀))
9	8	ex 412	. 2 ⊢ (𝜑 → (0 < 𝑁 → 0 < (𝑁↑𝑀)))
10		0red 11137	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
11	10, 1	lttrid 11273	. . . 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 11136	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
15	14	ltnri 11244	. . . . . . . . 9 ⊢ ¬ 0 < 0
16	3	0expd 14064	. . . . . . . . . 10 ⊢ (𝜑 → (0↑𝑀) = 0)
17	16	breq2d 5110	. . . . . . . . 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 7373	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 = 𝑁) → (𝑁↑𝑀) = (0↑𝑀))
23	22	breq2d 5110	. . . . . . 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 11566	. . . . . . . . . 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 14020	. . . . . . . . 9 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 0 < -𝑁) → 0 < (-𝑁↑𝑀))
31	27, 28, 29, 30	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < -𝑁) → 0 < (-𝑁↑𝑀))
32	31	ex 412	. . . . . . 7 ⊢ (𝜑 → (0 < -𝑁 → 0 < (-𝑁↑𝑀)))
33	1	recnd 11162	. . . . . . . . . 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 12869	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 / 2) ∈ ℤ → (𝑀 / 2) ∈ ℚ)
37	36	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℚ)
38		qden1elz 16686	. . . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((𝑀 / 2) · (denom‘(𝑀 / 2))) = ((𝑀 / 2) · 1))
42		qmuldeneqnum 16676	. . . . . . . . . . . . . . 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 12599	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℂ)
45	44	mulridd 11151	. . . . . . . . . . . . . 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 12162	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
48		2re 12221	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
49	48	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℝ)
50	3	nngt0d 12196	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 𝑀)
51		2pos 12250	. . . . . . . . . . . . . . . 16 ⊢ 0 < 2
52	51	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 2)
53	47, 49, 50, 52	divgt0d 12079	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝑀 / 2))
54		qgt0numnn 16680	. . . . . . . . . . . . . 14 ⊢ (((𝑀 / 2) ∈ ℚ ∧ 0 < (𝑀 / 2)) → (numer‘(𝑀 / 2)) ∈ ℕ)
55	36, 53, 54	syl2anr 597	. . . . . . . . . . . . 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 815	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℤ)
58		evend2 16286	. . . . . . . . . . . 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 16274	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℕ ∧ ¬ 2 ∥ 𝑀) → (-𝑁↑𝑀) = -(𝑁↑𝑀))
62	33, 3, 60, 61	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (-𝑁↑𝑀) = -(𝑁↑𝑀))
63	62	breq2d 5110	. . . . . . . 8 ⊢ (𝜑 → (0 < (-𝑁↑𝑀) ↔ 0 < -(𝑁↑𝑀)))
64	63	biimpd 229	. . . . . . 7 ⊢ (𝜑 → (0 < (-𝑁↑𝑀) → 0 < -(𝑁↑𝑀)))
65	3	nnnn0d 12464	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
66	1, 65	reexpcld 14088	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑𝑀) ∈ ℝ)
67	66	renegcld 11566	. . . . . . . . 9 ⊢ (𝜑 → -(𝑁↑𝑀) ∈ ℝ)
68	10, 67	lttrid 11273	. . . . . . . 8 ⊢ (𝜑 → (0 < -(𝑁↑𝑀) ↔ ¬ (0 = -(𝑁↑𝑀) ∨ -(𝑁↑𝑀) < 0)))
69		pm2.46 882	. . . . . . . 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 11708	. . . . . 6 ⊢ (𝜑 → (𝑁 < 0 ↔ 0 < -𝑁))
73	66	lt0neg2d 11709	. . . . . . 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 859	. . . 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 847 = wceq 1541 ∈ wcel 2113 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 · cmul 11033 < clt 11168 -cneg 11367 / cdiv 11796 ℕcn 12147 2c2 12202 ℤcz 12490 ℚcq 12863 ↑cexp 13986 ∥ cdvds 16181 numercnumer 16662 denomcdenom 16663

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 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 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8886 df-dom 8887 df-sdom 8888 df-sup 9347 df-inf 9348 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-z 12491 df-uz 12754 df-q 12864 df-rp 12908 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-dvds 16182 df-gcd 16424 df-numer 16664 df-denom 16665

This theorem is referenced by: dffltz 42898

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