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Mirrors > Home > MPE Home > Th. List > isppw | Structured version Visualization version GIF version |
Description: Two ways to say that 𝐴 is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.) |
Ref | Expression |
---|---|
isppw | ⊢ (𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃!𝑝 ∈ ℙ 𝑝 ∥ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} | |
2 | 1 | vmaval 25617 | . . 3 ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0)) |
3 | 2 | neeq1d 3072 | . 2 ⊢ (𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0)) |
4 | reuen1 8566 | . . 3 ⊢ (∃!𝑝 ∈ ℙ 𝑝 ∥ 𝐴 ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) | |
5 | hash1 13753 | . . . . . . . . . 10 ⊢ (♯‘1o) = 1 | |
6 | 5 | eqeq2i 2831 | . . . . . . . . 9 ⊢ ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = (♯‘1o) ↔ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1) |
7 | prmdvdsfi 25611 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin) | |
8 | 1onn 8254 | . . . . . . . . . . 11 ⊢ 1o ∈ ω | |
9 | nnfi 8699 | . . . . . . . . . . 11 ⊢ (1o ∈ ω → 1o ∈ Fin) | |
10 | 8, 9 | ax-mp 5 | . . . . . . . . . 10 ⊢ 1o ∈ Fin |
11 | hashen 13695 | . . . . . . . . . 10 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin ∧ 1o ∈ Fin) → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = (♯‘1o) ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o)) | |
12 | 7, 10, 11 | sylancl 586 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = (♯‘1o) ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o)) |
13 | 6, 12 | syl5bbr 286 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1 ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o)) |
14 | 13 | biimpar 478 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1) |
15 | 14 | iftrued 4471 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) = (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) |
16 | simpr 485 | . . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) | |
17 | en1b 8565 | . . . . . . . . . . . . 13 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} = {∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}}) | |
18 | 16, 17 | sylib 219 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} = {∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}}) |
19 | ssrab2 4053 | . . . . . . . . . . . 12 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ⊆ ℙ | |
20 | 18, 19 | eqsstrrdi 4019 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → {∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}} ⊆ ℙ) |
21 | 7 | uniexd 7457 | . . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ V) |
22 | 21 | adantr 481 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ V) |
23 | snssg 4709 | . . . . . . . . . . . 12 ⊢ (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ V → (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℙ ↔ {∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}} ⊆ ℙ)) | |
24 | 22, 23 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℙ ↔ {∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}} ⊆ ℙ)) |
25 | 20, 24 | mpbird 258 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℙ) |
26 | prmuz2 16028 | . . . . . . . . . 10 ⊢ (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℙ → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ (ℤ≥‘2)) | |
27 | 25, 26 | syl 17 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ (ℤ≥‘2)) |
28 | eluzelre 12242 | . . . . . . . . 9 ⊢ (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ (ℤ≥‘2) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℝ) | |
29 | 27, 28 | syl 17 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℝ) |
30 | eluz2gt1 12308 | . . . . . . . . 9 ⊢ (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ (ℤ≥‘2) → 1 < ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) | |
31 | 27, 30 | syl 17 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → 1 < ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) |
32 | 29, 31 | rplogcld 25139 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℝ+) |
33 | 32 | rpne0d 12424 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ≠ 0) |
34 | 15, 33 | eqnetrd 3080 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0) |
35 | 34 | ex 413 | . . . 4 ⊢ (𝐴 ∈ ℕ → ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o → if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0)) |
36 | iffalse 4472 | . . . . . 6 ⊢ (¬ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1 → if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) = 0) | |
37 | 36 | necon1ai 3040 | . . . . 5 ⊢ (if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0 → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1) |
38 | 37, 13 | syl5ib 245 | . . . 4 ⊢ (𝐴 ∈ ℕ → (if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o)) |
39 | 35, 38 | impbid 213 | . . 3 ⊢ (𝐴 ∈ ℕ → ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o ↔ if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0)) |
40 | 4, 39 | syl5bb 284 | . 2 ⊢ (𝐴 ∈ ℕ → (∃!𝑝 ∈ ℙ 𝑝 ∥ 𝐴 ↔ if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0)) |
41 | 3, 40 | bitr4d 283 | 1 ⊢ (𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃!𝑝 ∈ ℙ 𝑝 ∥ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∃!wreu 3137 {crab 3139 Vcvv 3492 ⊆ wss 3933 ifcif 4463 {csn 4557 ∪ cuni 4830 class class class wbr 5057 ‘cfv 6348 ωcom 7569 1oc1o 8084 ≈ cen 8494 Fincfn 8497 ℝcr 10524 0cc0 10525 1c1 10526 < clt 10663 ℕcn 11626 2c2 11680 ℤ≥cuz 12231 ♯chash 13678 ∥ cdvds 15595 ℙcprime 16003 logclog 25065 Λcvma 25596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-dvds 15596 df-prm 16004 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-vma 25602 |
This theorem is referenced by: isppw2 25619 |
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