oexpreposd - Mathbox for Steven Nguyen

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Theorem oexpreposd 40793

Description: Lemma for dffltz 40958. TODO-SN?: This can be used to show exp11d 40797 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 481	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑁 ∈ ℝ)
3		oexpreposd.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4	3	nnzd 12526	. . . . 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 14001	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑁) → 0 < (𝑁↑𝑀))
8	2, 5, 6, 7	syl3anc 1371	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑁) → 0 < (𝑁↑𝑀))
9	8	ex 413	. 2 ⊢ (𝜑 → (0 < 𝑁 → 0 < (𝑁↑𝑀)))
10		0red 11158	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
11	10, 1	lttrid 11293	. . . 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 11157	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
15	14	ltnri 11264	. . . . . . . . 9 ⊢ ¬ 0 < 0
16	3	0expd 14044	. . . . . . . . . 10 ⊢ (𝜑 → (0↑𝑀) = 0)
17	16	breq2d 5117	. . . . . . . . 9 ⊢ (𝜑 → (0 < (0↑𝑀) ↔ 0 < 0))
18	15, 17	mtbiri 326	. . . . . . . 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 2742	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 = 𝑁) → 𝑁 = 0)
22	21	oveq1d 7372	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 = 𝑁) → (𝑁↑𝑀) = (0↑𝑀))
23	22	breq2d 5117	. . . . . . 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 413	. . . . 5 ⊢ (𝜑 → (0 = 𝑁 → ¬ 0 < (𝑁↑𝑀)))
26	1	renegcld 11582	. . . . . . . . . 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 14001	. . . . . . . . 9 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 0 < -𝑁) → 0 < (-𝑁↑𝑀))
31	27, 28, 29, 30	syl3anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < -𝑁) → 0 < (-𝑁↑𝑀))
32	31	ex 413	. . . . . . 7 ⊢ (𝜑 → (0 < -𝑁 → 0 < (-𝑁↑𝑀)))
33	1	recnd 11183	. . . . . . . . . 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 12879	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 / 2) ∈ ℤ → (𝑀 / 2) ∈ ℚ)
37	36	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℚ)
38		qden1elz 16632	. . . . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((𝑀 / 2) · (denom‘(𝑀 / 2))) = ((𝑀 / 2) · 1))
42		qmuldeneqnum 16622	. . . . . . . . . . . . . . 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 12608	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℂ)
45	44	mulid1d 11172	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → ((𝑀 / 2) · 1) = (𝑀 / 2))
46	41, 43, 45	3eqtr3rd 2785	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) = (numer‘(𝑀 / 2)))
47	3	nnred 12168	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
48		2re 12227	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
49	48	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℝ)
50	3	nngt0d 12202	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 𝑀)
51		2pos 12256	. . . . . . . . . . . . . . . 16 ⊢ 0 < 2
52	51	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 2)
53	47, 49, 50, 52	divgt0d 12090	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝑀 / 2))
54		qgt0numnn 16626	. . . . . . . . . . . . . 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 2838	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 / 2) ∈ ℤ) → (𝑀 / 2) ∈ ℕ)
57	34, 56	mtand 814	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℤ)
58		evend2 16239	. . . . . . . . . . . 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 16227	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℕ ∧ ¬ 2 ∥ 𝑀) → (-𝑁↑𝑀) = -(𝑁↑𝑀))
62	33, 3, 60, 61	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → (-𝑁↑𝑀) = -(𝑁↑𝑀))
63	62	breq2d 5117	. . . . . . . 8 ⊢ (𝜑 → (0 < (-𝑁↑𝑀) ↔ 0 < -(𝑁↑𝑀)))
64	63	biimpd 228	. . . . . . 7 ⊢ (𝜑 → (0 < (-𝑁↑𝑀) → 0 < -(𝑁↑𝑀)))
65	3	nnnn0d 12473	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
66	1, 65	reexpcld 14068	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑𝑀) ∈ ℝ)
67	66	renegcld 11582	. . . . . . . . 9 ⊢ (𝜑 → -(𝑁↑𝑀) ∈ ℝ)
68	10, 67	lttrid 11293	. . . . . . . 8 ⊢ (𝜑 → (0 < -(𝑁↑𝑀) ↔ ¬ (0 = -(𝑁↑𝑀) ∨ -(𝑁↑𝑀) < 0)))
69		pm2.46 881	. . . . . . . 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 11724	. . . . . 6 ⊢ (𝜑 → (𝑁 < 0 ↔ 0 < -𝑁))
73	66	lt0neg2d 11725	. . . . . . 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 857	. . . 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 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 · cmul 11056 < clt 11189 -cneg 11386 / cdiv 11812 ℕcn 12153 2c2 12208 ℤcz 12499 ℚcq 12873 ↑cexp 13967 ∥ cdvds 16136 numercnumer 16608 denomcdenom 16609

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-gcd 16375 df-numer 16610 df-denom 16611

This theorem is referenced by: dffltz 40958

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