oexpreposd - Mathbox for Steven Nguyen

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Theorem oexpreposd 42808

Description: Lemma for dffltz 43093. For a more standard version, see expgt0b 32910. TODO-SN?: This can be used to show exp11d 42812 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 481	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑁 ∈ ℝ)
3		oexpreposd.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4	3	nnzd 12542	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑀 ∈ ℤ)
6		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → 0 < 𝑁)
7		expgt0 14049	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑁) → 0 < (𝑁↑𝑀))
8	2, 5, 6, 7	syl3anc 1379	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑁) → 0 < (𝑁↑𝑀))
9	8	ex 413	. 2 ⊢ (𝜑 → (0 < 𝑁 → 0 < (𝑁↑𝑀)))
10		0red 11139	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
11	10, 1	lttrid 11276	. . . 4 ⊢ (𝜑 → (0 < 𝑁 ↔ ¬ (0 = 𝑁 ∨ 𝑁 < 0)))
12	11	notbid 319	. . 3 ⊢ (𝜑 → (¬ 0 < 𝑁 ↔ ¬ ¬ (0 = 𝑁 ∨ 𝑁 < 0)))
13		notnotr 130	. . . 4 ⊢ (¬ ¬ (0 = 𝑁 ∨ 𝑁 < 0) → (0 = 𝑁 ∨ 𝑁 < 0))
14		0re 11138	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
15	14	ltnri 11247	. . . . . . . . 9 ⊢ ¬ 0 < 0
16	3	0expd 14093	. . . . . . . . . 10 ⊢ (𝜑 → (0↑𝑀) = 0)
17	16	breq2d 5085	. . . . . . . . 9 ⊢ (𝜑 → (0 < (0↑𝑀) ↔ 0 < 0))
18	15, 17	mtbiri 328	. . . . . . . 8 ⊢ (𝜑 → ¬ 0 < (0↑𝑀))
19	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 = 𝑁) → ¬ 0 < (0↑𝑀))
20		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 = 𝑁) → 0 = 𝑁)
21	20	eqcomd 2745	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 = 𝑁) → 𝑁 = 0)
22	21	oveq1d 7372	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 = 𝑁) → (𝑁↑𝑀) = (0↑𝑀))
23	22	breq2d 5085	. . . . . . 7 ⊢ ((𝜑 ∧ 0 = 𝑁) → (0 < (𝑁↑𝑀) ↔ 0 < (0↑𝑀)))
24	19, 23	mtbird 326	. . . . . 6 ⊢ ((𝜑 ∧ 0 = 𝑁) → ¬ 0 < (𝑁↑𝑀))
25	24	ex 413	. . . . 5 ⊢ (𝜑 → (0 = 𝑁 → ¬ 0 < (𝑁↑𝑀)))
26	1	renegcld 11569	. . . . . . . . . 10 ⊢ (𝜑 → -𝑁 ∈ ℝ)
27	26	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < -𝑁) → -𝑁 ∈ ℝ)
28	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < -𝑁) → 𝑀 ∈ ℤ)
29		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < -𝑁) → 0 < -𝑁)
30		expgt0 14049	. . . . . . . . 9 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 0 < -𝑁) → 0 < (-𝑁↑𝑀))
31	27, 28, 29, 30	syl3anc 1379	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < -𝑁) → 0 < (-𝑁↑𝑀))
32	31	ex 413	. . . . . . 7 ⊢ (𝜑 → (0 < -𝑁 → 0 < (-𝑁↑𝑀)))
33	1	recnd 11165	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
34		oexpreposd.1	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℕ)
35		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℤ)
36		zq 12896	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 / 2) ∈ ℤ → (𝑀 / 2) ∈ ℚ)
37	36	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℚ)
38		qden1elz 16719	. . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (denom‘(𝑀 / 2)) = 1)
41	40	oveq2d 7373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((𝑀 / 2) · (denom‘(𝑀 / 2))) = ((𝑀 / 2) · 1))
42		qmuldeneqnum 16709	. . . . . . . . . . . . . . 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 12626	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℂ)
45	44	mulridd 11154	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((𝑀 / 2) · 1) = (𝑀 / 2))
46	41, 43, 45	3eqtr3rd 2783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) = (numer‘(𝑀 / 2)))
47	3	nnred 12181	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
48		2re 12247	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
49	48	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℝ)
50	3	nngt0d 12218	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 𝑀)
51		2pos 12276	. . . . . . . . . . . . . . . 16 ⊢ 0 < 2
52	51	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 2)
53	47, 49, 50, 52	divgt0d 12083	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝑀 / 2))
54		qgt0numnn 16713	. . . . . . . . . . . . . 14 ⊢ (((𝑀 / 2) ∈ ℚ ∧ 0 < (𝑀 / 2)) → (numer‘(𝑀 / 2)) ∈ ℕ)
55	36, 53, 54	syl2anr 603	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (numer‘(𝑀 / 2)) ∈ ℕ)
56	46, 55	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℕ)
57	34, 56	mtand 821	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℤ)
58		evend2 16318	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (2 ∥ 𝑀 ↔ (𝑀 / 2) ∈ ℤ))
59	4, 58	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ∥ 𝑀 ↔ (𝑀 / 2) ∈ ℤ))
60	57, 59	mtbird 326	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝑀)
61		oexpneg 16306	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℕ ∧ ¬ 2 ∥ 𝑀) → (-𝑁↑𝑀) = -(𝑁↑𝑀))
62	33, 3, 60, 61	syl3anc 1379	. . . . . . . . 9 ⊢ (𝜑 → (-𝑁↑𝑀) = -(𝑁↑𝑀))
63	62	breq2d 5085	. . . . . . . 8 ⊢ (𝜑 → (0 < (-𝑁↑𝑀) ↔ 0 < -(𝑁↑𝑀)))
64	63	biimpd 230	. . . . . . 7 ⊢ (𝜑 → (0 < (-𝑁↑𝑀) → 0 < -(𝑁↑𝑀)))
65	3	nnnn0d 12490	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
66	1, 65	reexpcld 14117	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑𝑀) ∈ ℝ)
67	66	renegcld 11569	. . . . . . . . 9 ⊢ (𝜑 → -(𝑁↑𝑀) ∈ ℝ)
68	10, 67	lttrid 11276	. . . . . . . 8 ⊢ (𝜑 → (0 < -(𝑁↑𝑀) ↔ ¬ (0 = -(𝑁↑𝑀) ∨ -(𝑁↑𝑀) < 0)))
69		pm2.46 888	. . . . . . . 8 ⊢ (¬ (0 = -(𝑁↑𝑀) ∨ -(𝑁↑𝑀) < 0) → ¬ -(𝑁↑𝑀) < 0)
70	68, 69	biimtrdi 254	. . . . . . 7 ⊢ (𝜑 → (0 < -(𝑁↑𝑀) → ¬ -(𝑁↑𝑀) < 0))
71	32, 64, 70	3syld 60	. . . . . 6 ⊢ (𝜑 → (0 < -𝑁 → ¬ -(𝑁↑𝑀) < 0))
72	1	lt0neg1d 11711	. . . . . 6 ⊢ (𝜑 → (𝑁 < 0 ↔ 0 < -𝑁))
73	66	lt0neg2d 11712	. . . . . . 7 ⊢ (𝜑 → (0 < (𝑁↑𝑀) ↔ -(𝑁↑𝑀) < 0))
74	73	notbid 319	. . . . . 6 ⊢ (𝜑 → (¬ 0 < (𝑁↑𝑀) ↔ ¬ -(𝑁↑𝑀) < 0))
75	71, 72, 74	3imtr4d 295	. . . . 5 ⊢ (𝜑 → (𝑁 < 0 → ¬ 0 < (𝑁↑𝑀)))
76	25, 75	jaod 865	. . . 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 241	. 2 ⊢ (𝜑 → (¬ 0 < 𝑁 → ¬ 0 < (𝑁↑𝑀)))
79	9, 78	impcon4bid 228	1 ⊢ (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁↑𝑀)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 class class class wbr 5073 ‘cfv 6486 (class class class)co 7357 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 · cmul 11035 < clt 11171 -cneg 11370 / cdiv 11799 ℕcn 12166 2c2 12228 ℤcz 12516 ℚcq 12890 ↑cexp 14015 ∥ cdvds 16213 numercnumer 16695 denomcdenom 16696

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7808 df-1st 7932 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9346 df-inf 9347 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-n0 12430 df-z 12517 df-uz 12781 df-q 12891 df-rp 12935 df-fl 13743 df-mod 13821 df-seq 13956 df-exp 14016 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16214 df-gcd 16456 df-numer 16697 df-denom 16698

This theorem is referenced by: dffltz 43093

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