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Mirrors > Home > MPE Home > Th. List > dvdsprmpweqnn | Structured version Visualization version GIF version |
Description: If an integer greater than 1 divides a prime power, it is a (proper) prime power. (Contributed by AV, 13-Aug-2021.) |
Ref | Expression |
---|---|
dvdsprmpweqnn | ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2nn 12278 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ) | |
2 | dvdsprmpweq 16214 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛))) | |
3 | 1, 2 | syl3an2 1160 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛))) |
4 | 3 | imp 409 | . . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ∥ (𝑃↑𝑁)) → ∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛)) |
5 | df-n0 11892 | . . . . . 6 ⊢ ℕ0 = (ℕ ∪ {0}) | |
6 | 5 | rexeqi 3414 | . . . . 5 ⊢ (∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛) ↔ ∃𝑛 ∈ (ℕ ∪ {0})𝐴 = (𝑃↑𝑛)) |
7 | rexun 4165 | . . . . 5 ⊢ (∃𝑛 ∈ (ℕ ∪ {0})𝐴 = (𝑃↑𝑛) ↔ (∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛) ∨ ∃𝑛 ∈ {0}𝐴 = (𝑃↑𝑛))) | |
8 | 6, 7 | bitri 277 | . . . 4 ⊢ (∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛) ↔ (∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛) ∨ ∃𝑛 ∈ {0}𝐴 = (𝑃↑𝑛))) |
9 | 0z 11986 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
10 | oveq2 7158 | . . . . . . . . 9 ⊢ (𝑛 = 0 → (𝑃↑𝑛) = (𝑃↑0)) | |
11 | 10 | eqeq2d 2832 | . . . . . . . 8 ⊢ (𝑛 = 0 → (𝐴 = (𝑃↑𝑛) ↔ 𝐴 = (𝑃↑0))) |
12 | 11 | rexsng 4607 | . . . . . . 7 ⊢ (0 ∈ ℤ → (∃𝑛 ∈ {0}𝐴 = (𝑃↑𝑛) ↔ 𝐴 = (𝑃↑0))) |
13 | 9, 12 | ax-mp 5 | . . . . . 6 ⊢ (∃𝑛 ∈ {0}𝐴 = (𝑃↑𝑛) ↔ 𝐴 = (𝑃↑0)) |
14 | prmnn 16012 | . . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
15 | 14 | nncnd 11648 | . . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ) |
16 | 15 | exp0d 13498 | . . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → (𝑃↑0) = 1) |
17 | 16 | 3ad2ant1 1129 | . . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑃↑0) = 1) |
18 | 17 | eqeq2d 2832 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 = (𝑃↑0) ↔ 𝐴 = 1)) |
19 | eluz2b3 12316 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1)) | |
20 | eqneqall 3027 | . . . . . . . . . . . 12 ⊢ (𝐴 = 1 → (𝐴 ≠ 1 → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛)))) | |
21 | 20 | com12 32 | . . . . . . . . . . 11 ⊢ (𝐴 ≠ 1 → (𝐴 = 1 → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛)))) |
22 | 19, 21 | simplbiim 507 | . . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 = 1 → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛)))) |
23 | 22 | 3ad2ant2 1130 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 = 1 → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛)))) |
24 | 18, 23 | sylbid 242 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 = (𝑃↑0) → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛)))) |
25 | 24 | com12 32 | . . . . . . 7 ⊢ (𝐴 = (𝑃↑0) → ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛)))) |
26 | 25 | impd 413 | . . . . . 6 ⊢ (𝐴 = (𝑃↑0) → (((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ∥ (𝑃↑𝑁)) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛))) |
27 | 13, 26 | sylbi 219 | . . . . 5 ⊢ (∃𝑛 ∈ {0}𝐴 = (𝑃↑𝑛) → (((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ∥ (𝑃↑𝑁)) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛))) |
28 | 27 | jao1i 854 | . . . 4 ⊢ ((∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛) ∨ ∃𝑛 ∈ {0}𝐴 = (𝑃↑𝑛)) → (((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ∥ (𝑃↑𝑁)) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛))) |
29 | 8, 28 | sylbi 219 | . . 3 ⊢ (∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛) → (((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ∥ (𝑃↑𝑁)) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛))) |
30 | 4, 29 | mpcom 38 | . 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ∥ (𝑃↑𝑁)) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛)) |
31 | 30 | ex 415 | 1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 ∪ cun 3933 {csn 4560 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 ℕcn 11632 2c2 11686 ℕ0cn0 11891 ℤcz 11975 ℤ≥cuz 12237 ↑cexp 13423 ∥ cdvds 15601 ℙcprime 16009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-fz 12887 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-dvds 15602 df-gcd 15838 df-prm 16010 df-pc 16168 |
This theorem is referenced by: difsqpwdvds 16217 lighneallem4 43769 |
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