oexpreposd - Mathbox for Steven Nguyen

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Theorem oexpreposd 40242

Description: Lemma for dffltz 40387. TODO-SN?: This can be used to show exp11d 40246 holds for all integers when the exponent is odd. The more standard ¬ 2 ∥ 𝑀 should be used. (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 12354	. . . . 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 13744	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑁) → 0 < (𝑁↑𝑀))
8	2, 5, 6, 7	syl3anc 1369	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑁) → 0 < (𝑁↑𝑀))
9	8	ex 412	. 2 ⊢ (𝜑 → (0 < 𝑁 → 0 < (𝑁↑𝑀)))
10		0red 10909	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
11	10, 1	lttrid 11043	. . . 4 ⊢ (𝜑 → (0 < 𝑁 ↔ ¬ (0 = 𝑁 ∨ 𝑁 < 0)))
12	11	notbid 317	. . 3 ⊢ (𝜑 → (¬ 0 < 𝑁 ↔ ¬ ¬ (0 = 𝑁 ∨ 𝑁 < 0)))
13		notnotr 130	. . . 4 ⊢ (¬ ¬ (0 = 𝑁 ∨ 𝑁 < 0) → (0 = 𝑁 ∨ 𝑁 < 0))
14		0re 10908	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
15	14	ltnri 11014	. . . . . . . . 9 ⊢ ¬ 0 < 0
16	3	0expd 13785	. . . . . . . . . 10 ⊢ (𝜑 → (0↑𝑀) = 0)
17	16	breq2d 5082	. . . . . . . . 9 ⊢ (𝜑 → (0 < (0↑𝑀) ↔ 0 < 0))
18	15, 17	mtbiri 326	. . . . . . . 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 2744	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 = 𝑁) → 𝑁 = 0)
22	21	oveq1d 7270	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 = 𝑁) → (𝑁↑𝑀) = (0↑𝑀))
23	22	breq2d 5082	. . . . . . 7 ⊢ ((𝜑 ∧ 0 = 𝑁) → (0 < (𝑁↑𝑀) ↔ 0 < (0↑𝑀)))
24	19, 23	mtbird 324	. . . . . 6 ⊢ ((𝜑 ∧ 0 = 𝑁) → ¬ 0 < (𝑁↑𝑀))
25	24	ex 412	. . . . 5 ⊢ (𝜑 → (0 = 𝑁 → ¬ 0 < (𝑁↑𝑀)))
26	1	renegcld 11332	. . . . . . . . . 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 13744	. . . . . . . . 9 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 0 < -𝑁) → 0 < (-𝑁↑𝑀))
31	27, 28, 29, 30	syl3anc 1369	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < -𝑁) → 0 < (-𝑁↑𝑀))
32	31	ex 412	. . . . . . 7 ⊢ (𝜑 → (0 < -𝑁 → 0 < (-𝑁↑𝑀)))
33	1	recnd 10934	. . . . . . . . . 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 12623	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 / 2) ∈ ℤ → (𝑀 / 2) ∈ ℚ)
37	36	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℚ)
38		qden1elz 16389	. . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (denom‘(𝑀 / 2)) = 1)
41	40	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((𝑀 / 2) · (denom‘(𝑀 / 2))) = ((𝑀 / 2) · 1))
42		qmuldeneqnum 16379	. . . . . . . . . . . . . . 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 12356	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℂ)
45	44	mulid1d 10923	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((𝑀 / 2) · 1) = (𝑀 / 2))
46	41, 43, 45	3eqtr3rd 2787	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) = (numer‘(𝑀 / 2)))
47	3	nnred 11918	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
48		2re 11977	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
49	48	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℝ)
50	3	nngt0d 11952	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 𝑀)
51		2pos 12006	. . . . . . . . . . . . . . . 16 ⊢ 0 < 2
52	51	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 2)
53	47, 49, 50, 52	divgt0d 11840	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝑀 / 2))
54		qgt0numnn 16383	. . . . . . . . . . . . . 14 ⊢ (((𝑀 / 2) ∈ ℚ ∧ 0 < (𝑀 / 2)) → (numer‘(𝑀 / 2)) ∈ ℕ)
55	36, 53, 54	syl2anr 596	. . . . . . . . . . . . 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 812	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℤ)
58		evend2 15994	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (2 ∥ 𝑀 ↔ (𝑀 / 2) ∈ ℤ))
59	4, 58	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ∥ 𝑀 ↔ (𝑀 / 2) ∈ ℤ))
60	57, 59	mtbird 324	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝑀)
61		oexpneg 15982	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℕ ∧ ¬ 2 ∥ 𝑀) → (-𝑁↑𝑀) = -(𝑁↑𝑀))
62	33, 3, 60, 61	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → (-𝑁↑𝑀) = -(𝑁↑𝑀))
63	62	breq2d 5082	. . . . . . . 8 ⊢ (𝜑 → (0 < (-𝑁↑𝑀) ↔ 0 < -(𝑁↑𝑀)))
64	63	biimpd 228	. . . . . . 7 ⊢ (𝜑 → (0 < (-𝑁↑𝑀) → 0 < -(𝑁↑𝑀)))
65	3	nnnn0d 12223	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
66	1, 65	reexpcld 13809	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑𝑀) ∈ ℝ)
67	66	renegcld 11332	. . . . . . . . 9 ⊢ (𝜑 → -(𝑁↑𝑀) ∈ ℝ)
68	10, 67	lttrid 11043	. . . . . . . 8 ⊢ (𝜑 → (0 < -(𝑁↑𝑀) ↔ ¬ (0 = -(𝑁↑𝑀) ∨ -(𝑁↑𝑀) < 0)))
69		pm2.46 879	. . . . . . . 8 ⊢ (¬ (0 = -(𝑁↑𝑀) ∨ -(𝑁↑𝑀) < 0) → ¬ -(𝑁↑𝑀) < 0)
70	68, 69	syl6bi 252	. . . . . . 7 ⊢ (𝜑 → (0 < -(𝑁↑𝑀) → ¬ -(𝑁↑𝑀) < 0))
71	32, 64, 70	3syld 60	. . . . . 6 ⊢ (𝜑 → (0 < -𝑁 → ¬ -(𝑁↑𝑀) < 0))
72	1	lt0neg1d 11474	. . . . . 6 ⊢ (𝜑 → (𝑁 < 0 ↔ 0 < -𝑁))
73	66	lt0neg2d 11475	. . . . . . 7 ⊢ (𝜑 → (0 < (𝑁↑𝑀) ↔ -(𝑁↑𝑀) < 0))
74	73	notbid 317	. . . . . 6 ⊢ (𝜑 → (¬ 0 < (𝑁↑𝑀) ↔ ¬ -(𝑁↑𝑀) < 0))
75	71, 72, 74	3imtr4d 293	. . . . 5 ⊢ (𝜑 → (𝑁 < 0 → ¬ 0 < (𝑁↑𝑀)))
76	25, 75	jaod 855	. . . 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 239	. 2 ⊢ (𝜑 → (¬ 0 < 𝑁 → ¬ 0 < (𝑁↑𝑀)))
79	9, 78	impcon4bid 226	1 ⊢ (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁↑𝑀)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 · cmul 10807 < clt 10940 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℤcz 12249 ℚcq 12617 ↑cexp 13710 ∥ cdvds 15891 numercnumer 16365 denomcdenom 16366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-numer 16367 df-denom 16368

This theorem is referenced by: dffltz 40387

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