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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 2821 | . . . 4 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} | |
2 | 1 | vmaval 25618 | . . 3 ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0)) |
3 | 2 | neeq1d 3075 | . 2 ⊢ (𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0)) |
4 | reuen1 8567 | . . 3 ⊢ (∃!𝑝 ∈ ℙ 𝑝 ∥ 𝐴 ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) | |
5 | hash1 13755 | . . . . . . . . . 10 ⊢ (♯‘1o) = 1 | |
6 | 5 | eqeq2i 2834 | . . . . . . . . 9 ⊢ ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = (♯‘1o) ↔ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1) |
7 | prmdvdsfi 25612 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin) | |
8 | 1onn 8255 | . . . . . . . . . . 11 ⊢ 1o ∈ ω | |
9 | nnfi 8700 | . . . . . . . . . . 11 ⊢ (1o ∈ ω → 1o ∈ Fin) | |
10 | 8, 9 | ax-mp 5 | . . . . . . . . . 10 ⊢ 1o ∈ Fin |
11 | hashen 13697 | . . . . . . . . . 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 4473 | . . . . . 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 8566 | . . . . . . . . . . . . 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 4055 | . . . . . . . . . . . 12 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ⊆ ℙ | |
20 | 18, 19 | eqsstrrdi 4021 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → {∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}} ⊆ ℙ) |
21 | uniexg 7456 | . . . . . . . . . . . . . 14 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ V) | |
22 | 7, 21 | syl 17 | . . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ V) |
23 | 22 | adantr 481 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ V) |
24 | snssg 4711 | . . . . . . . . . . . 12 ⊢ (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ V → (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℙ ↔ {∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}} ⊆ ℙ)) | |
25 | 23, 24 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℙ ↔ {∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}} ⊆ ℙ)) |
26 | 20, 25 | mpbird 258 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℙ) |
27 | prmuz2 16030 | . . . . . . . . . 10 ⊢ (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℙ → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ (ℤ≥‘2)) | |
28 | 26, 27 | syl 17 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ (ℤ≥‘2)) |
29 | eluzelre 12243 | . . . . . . . . 9 ⊢ (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ (ℤ≥‘2) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℝ) | |
30 | 28, 29 | syl 17 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ ℝ) |
31 | eluz2gt1 12309 | . . . . . . . . 9 ⊢ (∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ (ℤ≥‘2) → 1 < ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) | |
32 | 28, 31 | syl 17 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → 1 < ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) |
33 | 30, 32 | rplogcld 25139 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℝ+) |
34 | 33 | rpne0d 12426 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ≠ 0) |
35 | 15, 34 | eqnetrd 3083 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o) → if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0) |
36 | 35 | ex 413 | . . . 4 ⊢ (𝐴 ∈ ℕ → ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o → if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0)) |
37 | iffalse 4474 | . . . . . 6 ⊢ (¬ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1 → if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) = 0) | |
38 | 37 | necon1ai 3043 | . . . . 5 ⊢ (if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0 → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1) |
39 | 38, 13 | syl5ib 245 | . . . 4 ⊢ (𝐴 ∈ ℕ → (if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o)) |
40 | 36, 39 | impbid 213 | . . 3 ⊢ (𝐴 ∈ ℕ → ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ≈ 1o ↔ if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0)) |
41 | 4, 40 | syl5bb 284 | . 2 ⊢ (𝐴 ∈ ℕ → (∃!𝑝 ∈ ℙ 𝑝 ∥ 𝐴 ↔ if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}), 0) ≠ 0)) |
42 | 3, 41 | 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 3016 ∃!wreu 3140 {crab 3142 Vcvv 3495 ⊆ wss 3935 ifcif 4465 {csn 4559 ∪ cuni 4832 class class class wbr 5058 ‘cfv 6349 ωcom 7568 1oc1o 8086 ≈ cen 8495 Fincfn 8498 ℝcr 10525 0cc0 10526 1c1 10527 < clt 10664 ℕcn 11627 2c2 11681 ℤ≥cuz 12232 ♯chash 13680 ∥ cdvds 15597 ℙcprime 16005 logclog 25065 Λcvma 25597 |
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 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
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 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 3497 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 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 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-se 5509 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-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-supp 7822 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fsupp 8823 df-fi 8864 df-sup 8895 df-inf 8896 df-oi 8963 df-dju 9319 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-ioo 12732 df-ioc 12733 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-fac 13624 df-bc 13653 df-hash 13681 df-shft 14416 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-limsup 14818 df-clim 14835 df-rlim 14836 df-sum 15033 df-ef 15411 df-sin 15413 df-cos 15414 df-pi 15416 df-dvds 15598 df-prm 16006 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-starv 16570 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-hom 16579 df-cco 16580 df-rest 16686 df-topn 16687 df-0g 16705 df-gsum 16706 df-topgen 16707 df-pt 16708 df-prds 16711 df-xrs 16765 df-qtop 16770 df-imas 16771 df-xps 16773 df-mre 16847 df-mrc 16848 df-acs 16850 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-submnd 17947 df-mulg 18165 df-cntz 18387 df-cmn 18839 df-psmet 20467 df-xmet 20468 df-met 20469 df-bl 20470 df-mopn 20471 df-fbas 20472 df-fg 20473 df-cnfld 20476 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cld 21557 df-ntr 21558 df-cls 21559 df-nei 21636 df-lp 21674 df-perf 21675 df-cn 21765 df-cnp 21766 df-haus 21853 df-tx 22100 df-hmeo 22293 df-fil 22384 df-fm 22476 df-flim 22477 df-flf 22478 df-xms 22859 df-ms 22860 df-tms 22861 df-cncf 23415 df-limc 24393 df-dv 24394 df-log 25067 df-vma 25603 |
This theorem is referenced by: isppw2 25620 |
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