qexpz - Intuitionistic Logic Explorer

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Theorem qexpz 12350

Description: If a power of a rational number is an integer, then the number is an integer. (Contributed by Mario Carneiro, 10-Aug-2015.)

**Assertion**
Ref	Expression
qexpz	⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) → 𝐴 ∈ ℤ)

Proof of Theorem qexpz Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0z 9264	. . . 4 ⊢ 0 ∈ ℤ
2		eleq1 2240	. . . 4 ⊢ (𝐴 = 0 → (𝐴 ∈ ℤ ↔ 0 ∈ ℤ))
3	1, 2	mpbiri 168	. . 3 ⊢ (𝐴 = 0 → 𝐴 ∈ ℤ)
4	3	adantl 277	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 = 0) → 𝐴 ∈ ℤ)
5		simpll2 1037	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℕ)
6	5	nncnd 8933	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℂ)
7	6	mul01d 8350	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (𝑁 · 0) = 0)
8		simpr 110	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
9		simpll3 1038	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (𝐴↑𝑁) ∈ ℤ)
10		simpll1 1036	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℚ)
11		qcn 9634	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
12	10, 11	syl 14	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℂ)
13		simplr 528	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 𝐴 ≠ 0)
14		zq 9626	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
15	1, 14	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℚ
16		qapne 9639	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℚ ∧ 0 ∈ ℚ) → (𝐴 # 0 ↔ 𝐴 ≠ 0))
17	10, 15, 16	sylancl 413	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (𝐴 # 0 ↔ 𝐴 ≠ 0))
18	13, 17	mpbird 167	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 𝐴 # 0)
19	5	nnzd 9374	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℤ)
20	12, 18, 19	expap0d 10660	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (𝐴↑𝑁) # 0)
21		0zd 9265	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 0 ∈ ℤ)
22		zapne 9327	. . . . . . . . . . 11 ⊢ (((𝐴↑𝑁) ∈ ℤ ∧ 0 ∈ ℤ) → ((𝐴↑𝑁) # 0 ↔ (𝐴↑𝑁) ≠ 0))
23	9, 21, 22	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) # 0 ↔ (𝐴↑𝑁) ≠ 0))
24	20, 23	mpbid 147	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (𝐴↑𝑁) ≠ 0)
25		pczcl 12298	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ ((𝐴↑𝑁) ∈ ℤ ∧ (𝐴↑𝑁) ≠ 0)) → (𝑝 pCnt (𝐴↑𝑁)) ∈ ℕ₀)
26	8, 9, 24, 25	syl12anc 1236	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝐴↑𝑁)) ∈ ℕ₀)
27	26	nn0ge0d 9232	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 0 ≤ (𝑝 pCnt (𝐴↑𝑁)))
28		pcexp 12309	. . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ) → (𝑝 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑝 pCnt 𝐴)))
29	8, 10, 13, 19, 28	syl121anc 1243	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑝 pCnt 𝐴)))
30	27, 29	breqtrd 4030	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 0 ≤ (𝑁 · (𝑝 pCnt 𝐴)))
31	7, 30	eqbrtrd 4026	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (𝑁 · 0) ≤ (𝑁 · (𝑝 pCnt 𝐴)))
32		0red 7958	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 0 ∈ ℝ)
33		pcqcl 12306	. . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑝 pCnt 𝐴) ∈ ℤ)
34	8, 10, 13, 33	syl12anc 1236	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℤ)
35	34	zred 9375	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℝ)
36	5	nnred 8932	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℝ)
37	5	nngt0d 8963	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 0 < 𝑁)
38		lemul2 8814	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ (𝑝 pCnt 𝐴) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (0 ≤ (𝑝 pCnt 𝐴) ↔ (𝑁 · 0) ≤ (𝑁 · (𝑝 pCnt 𝐴))))
39	32, 35, 36, 37, 38	syl112anc 1242	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → (0 ≤ (𝑝 pCnt 𝐴) ↔ (𝑁 · 0) ≤ (𝑁 · (𝑝 pCnt 𝐴))))
40	31, 39	mpbird 167	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) ∧ 𝑝 ∈ ℙ) → 0 ≤ (𝑝 pCnt 𝐴))
41	40	ralrimiva 2550	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) → ∀𝑝 ∈ ℙ 0 ≤ (𝑝 pCnt 𝐴))
42		simpl1 1000	. . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℚ)
43		pcz 12331	. . . 4 ⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ ∀𝑝 ∈ ℙ 0 ≤ (𝑝 pCnt 𝐴)))
44	42, 43	syl 14	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) → (𝐴 ∈ ℤ ↔ ∀𝑝 ∈ ℙ 0 ≤ (𝑝 pCnt 𝐴)))
45	41, 44	mpbird 167	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℤ)
46		simp1 997	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) → 𝐴 ∈ ℚ)
47		qdceq 10247	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 0 ∈ ℚ) → DECID 𝐴 = 0)
48	46, 15, 47	sylancl 413	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) → DECID 𝐴 = 0)
49		dcne 2358	. . 3 ⊢ (DECID 𝐴 = 0 ↔ (𝐴 = 0 ∨ 𝐴 ≠ 0))
50	48, 49	sylib 122	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) → (𝐴 = 0 ∨ 𝐴 ≠ 0))
51	4, 45, 50	mpjaodan 798	1 ⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑁) ∈ ℤ) → 𝐴 ∈ ℤ)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 DECID wdc 834 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 class class class wbr 4004 (class class class)co 5875 ℂcc 7809 ℝcr 7810 0cc0 7811 · cmul 7816 < clt 7992 ≤ cle 7993 # cap 8538 ℕcn 8919 ℕ₀cn0 9176 ℤcz 9253 ℚcq 9619 ↑cexp 10519 ℙcprime 12107 pCnt cpc 12284

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-1o 6417 df-2o 6418 df-er 6535 df-en 6741 df-sup 6983 df-inf 6984 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-xnn0 9240 df-z 9254 df-uz 9529 df-q 9620 df-rp 9654 df-fz 10009 df-fzo 10143 df-fl 10270 df-mod 10323 df-seqfrec 10446 df-exp 10520 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-dvds 11795 df-gcd 11944 df-prm 12108 df-pc 12285

This theorem is referenced by: (None)

W3C validator