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Theorem pclemub 13010

Description: Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.)

**Hypothesis**
Ref	Expression
pclem.1	⊢ 𝐴 = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑁}

**Assertion**
Ref	Expression
pclemub	⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

Distinct variable groups: 𝑛,𝑁,𝑦 𝑥,𝑁,𝑦 𝑃,𝑛,𝑦 𝑥,𝑃 Allowed substitution hints: 𝐴(𝑥,𝑦,𝑛)

**Proof of Theorem pclemub**
Step	Hyp	Ref	Expression
1		nnssz 9611	. 2 ⊢ ℕ ⊆ ℤ
2		zcn 9599	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
3	2	abscld 11891	. . . . 5 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ)
4	3	ad2antrl 490	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (abs‘𝑁) ∈ ℝ)
5		eluzelre 9882	. . . . 5 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℝ)
6	5	adantr 276	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℝ)
7		eluz2gt1 9952	. . . . 5 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
8	7	adantr 276	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 1 < 𝑃)
9		expnbnd 11050	. . . 4 ⊢ (((abs‘𝑁) ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ 1 < 𝑃) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥))
10	4, 6, 8, 9	syl3anc 1274	. . 3 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥))
11		simprr 533	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
12		oveq2 6066	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑦 → (𝑃↑𝑛) = (𝑃↑𝑦))
13	12	breq1d 4124	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑦 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑𝑦) ∥ 𝑁))
14		pclem.1	. . . . . . . . . . . . 13 ⊢ 𝐴 = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑁}
15	13, 14	elrab2 2979	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℕ₀ ∧ (𝑃↑𝑦) ∥ 𝑁))
16	11, 15	sylib 122	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 ∈ ℕ₀ ∧ (𝑃↑𝑦) ∥ 𝑁))
17	16	simprd 114	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∥ 𝑁)
18		eluz2nn 9916	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℕ)
19	18	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℕ)
20	16	simpld 112	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℕ₀)
21	19, 20	nnexpcld 11082	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℕ)
22	21	nnzd 9717	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℤ)
23		simplrl 537	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ∈ ℤ)
24		simplrr 538	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ≠ 0)
25		dvdsleabs 12556	. . . . . . . . . . 11 ⊢ (((𝑃↑𝑦) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ((𝑃↑𝑦) ∥ 𝑁 → (𝑃↑𝑦) ≤ (abs‘𝑁)))
26	22, 23, 24, 25	syl3anc 1274	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((𝑃↑𝑦) ∥ 𝑁 → (𝑃↑𝑦) ≤ (abs‘𝑁)))
27	17, 26	mpd 13	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ≤ (abs‘𝑁))
28	21	nnred 9267	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℝ)
29	4	adantr 276	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (abs‘𝑁) ∈ ℝ)
30	5	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℝ)
31		nnnn0 9520	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
32	31	ad2antrl 490	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℕ₀)
33	30, 32	reexpcld 11077	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑥) ∈ ℝ)
34		lelttr 8378	. . . . . . . . . 10 ⊢ (((𝑃↑𝑦) ∈ ℝ ∧ (abs‘𝑁) ∈ ℝ ∧ (𝑃↑𝑥) ∈ ℝ) → (((𝑃↑𝑦) ≤ (abs‘𝑁) ∧ (abs‘𝑁) < (𝑃↑𝑥)) → (𝑃↑𝑦) < (𝑃↑𝑥)))
35	28, 29, 33, 34	syl3anc 1274	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (((𝑃↑𝑦) ≤ (abs‘𝑁) ∧ (abs‘𝑁) < (𝑃↑𝑥)) → (𝑃↑𝑦) < (𝑃↑𝑥)))
36	27, 35	mpand 429	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → (𝑃↑𝑦) < (𝑃↑𝑥)))
37	7	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 1 < 𝑃)
38		nn0ltexp2 11096	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℝ ∧ 𝑦 ∈ ℕ₀ ∧ 𝑥 ∈ ℕ₀) ∧ 1 < 𝑃) → (𝑦 < 𝑥 ↔ (𝑃↑𝑦) < (𝑃↑𝑥)))
39	30, 20, 32, 37, 38	syl31anc 1277	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 < 𝑥 ↔ (𝑃↑𝑦) < (𝑃↑𝑥)))
40	36, 39	sylibrd 169	. . . . . . 7 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 < 𝑥))
41	20	nn0red 9571	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℝ)
42		nnre 9261	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
43	42	ad2antrl 490	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ)
44		ltle 8377	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥))
45	41, 43, 44	syl2anc 411	. . . . . . 7 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥))
46	40, 45	syld 45	. . . . . 6 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 ≤ 𝑥))
47	46	anassrs 400	. . . . 5 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ 𝐴) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 ≤ 𝑥))
48	47	ralrimdva 2624	. . . 4 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) → ((abs‘𝑁) < (𝑃↑𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
49	48	reximdva 2646	. . 3 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
50	10, 49	mpd 13	. 2 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
51		ssrexv 3307	. 2 ⊢ (ℕ ⊆ ℤ → (∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
52	1, 50, 51	mpsyl 65	1 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1398 ∈ wcel 2205 ≠ wne 2414 ∀wral 2522 ∃wrex 2523 {crab 2526 ⊆ wss 3214 class class class wbr 4114 ‘cfv 5357 (class class class)co 6058 ℝcr 8142 0cc0 8143 1c1 8144 < clt 8324 ≤ cle 8325 ℕcn 9254 2c2 9305 ℕ₀cn0 9513 ℤcz 9594 ℤ_≥cuz 9871 ↑cexp 10924 abscabs 11707 ∥ cdvds 12498

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-seqfrec 10834 df-exp 10925 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-dvds 12499

This theorem is referenced by: pcprecl 13012 pcprendvds 13013 pcpremul 13016

W3C validator