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Theorem pclemub 12241

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 9229	. 2 ⊢ ℕ ⊆ ℤ
2		zcn 9217	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
3	2	abscld 11145	. . . . 5 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ)
4	3	ad2antrl 487	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (abs‘𝑁) ∈ ℝ)
5		eluzelre 9497	. . . . 5 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℝ)
6	5	adantr 274	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℝ)
7		eluz2gt1 9561	. . . . 5 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
8	7	adantr 274	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 1 < 𝑃)
9		expnbnd 10599	. . . 4 ⊢ (((abs‘𝑁) ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ 1 < 𝑃) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥))
10	4, 6, 8, 9	syl3anc 1233	. . 3 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥))
11		simprr 527	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
12		oveq2 5861	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑦 → (𝑃↑𝑛) = (𝑃↑𝑦))
13	12	breq1d 3999	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑦 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑𝑦) ∥ 𝑁))
14		pclem.1	. . . . . . . . . . . . 13 ⊢ 𝐴 = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑁}
15	13, 14	elrab2 2889	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℕ₀ ∧ (𝑃↑𝑦) ∥ 𝑁))
16	11, 15	sylib 121	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 ∈ ℕ₀ ∧ (𝑃↑𝑦) ∥ 𝑁))
17	16	simprd 113	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∥ 𝑁)
18		eluz2nn 9525	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℕ)
19	18	ad2antrr 485	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℕ)
20	16	simpld 111	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℕ₀)
21	19, 20	nnexpcld 10631	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℕ)
22	21	nnzd 9333	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℤ)
23		simplrl 530	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ∈ ℤ)
24		simplrr 531	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ≠ 0)
25		dvdsleabs 11805	. . . . . . . . . . 11 ⊢ (((𝑃↑𝑦) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ((𝑃↑𝑦) ∥ 𝑁 → (𝑃↑𝑦) ≤ (abs‘𝑁)))
26	22, 23, 24, 25	syl3anc 1233	. . . . . . . . . 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 8891	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℝ)
29	4	adantr 274	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (abs‘𝑁) ∈ ℝ)
30	5	ad2antrr 485	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℝ)
31		nnnn0 9142	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
32	31	ad2antrl 487	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℕ₀)
33	30, 32	reexpcld 10626	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑥) ∈ ℝ)
34		lelttr 8008	. . . . . . . . . 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 1233	. . . . . . . . 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 427	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → (𝑃↑𝑦) < (𝑃↑𝑥)))
37	7	ad2antrr 485	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 1 < 𝑃)
38		nn0ltexp2 10644	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℝ ∧ 𝑦 ∈ ℕ₀ ∧ 𝑥 ∈ ℕ₀) ∧ 1 < 𝑃) → (𝑦 < 𝑥 ↔ (𝑃↑𝑦) < (𝑃↑𝑥)))
39	30, 20, 32, 37, 38	syl31anc 1236	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 < 𝑥 ↔ (𝑃↑𝑦) < (𝑃↑𝑥)))
40	36, 39	sylibrd 168	. . . . . . 7 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 < 𝑥))
41	20	nn0red 9189	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℝ)
42		nnre 8885	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
43	42	ad2antrl 487	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ)
44		ltle 8007	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥))
45	41, 43, 44	syl2anc 409	. . . . . . 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 398	. . . . 5 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ 𝐴) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 ≤ 𝑥))
48	47	ralrimdva 2550	. . . 4 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) → ((abs‘𝑁) < (𝑃↑𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
49	48	reximdva 2572	. . 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 3212	. 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 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∀wral 2448 ∃wrex 2449 {crab 2452 ⊆ wss 3121 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 ℝcr 7773 0cc0 7774 1c1 7775 < clt 7954 ≤ cle 7955 ℕcn 8878 2c2 8929 ℕ₀cn0 9135 ℤcz 9212 ℤ_≥cuz 9487 ↑cexp 10475 abscabs 10961 ∥ cdvds 11749

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750

This theorem is referenced by: pcprecl 12243 pcprendvds 12244 pcpremul 12247

W3C validator