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Mirrors > Home > MPE Home > Th. List > ablfac1a | Structured version Visualization version GIF version |
Description: The factors of ablfac1b 18469 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
ablfac1.b | ⊢ 𝐵 = (Base‘𝐺) |
ablfac1.o | ⊢ 𝑂 = (od‘𝐺) |
ablfac1.s | ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))}) |
ablfac1.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablfac1.f | ⊢ (𝜑 → 𝐵 ∈ Fin) |
ablfac1.1 | ⊢ (𝜑 → 𝐴 ⊆ ℙ) |
Ref | Expression |
---|---|
ablfac1a | ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (#‘(𝑆‘𝑃)) = (𝑃↑(𝑃 pCnt (#‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . . . 8 ⊢ (𝑝 = 𝑃 → 𝑝 = 𝑃) | |
2 | oveq1 6657 | . . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝 pCnt (#‘𝐵)) = (𝑃 pCnt (#‘𝐵))) | |
3 | 1, 2 | oveq12d 6668 | . . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝↑(𝑝 pCnt (#‘𝐵))) = (𝑃↑(𝑃 pCnt (#‘𝐵)))) |
4 | 3 | breq2d 4665 | . . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (#‘𝐵))))) |
5 | 4 | rabbidv 3189 | . . . . 5 ⊢ (𝑝 = 𝑃 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (#‘𝐵)))}) |
6 | ablfac1.s | . . . . 5 ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))}) | |
7 | ablfac1.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
8 | fvex 6201 | . . . . . . 7 ⊢ (Base‘𝐺) ∈ V | |
9 | 7, 8 | eqeltri 2697 | . . . . . 6 ⊢ 𝐵 ∈ V |
10 | 9 | rabex 4813 | . . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ∈ V |
11 | 5, 6, 10 | fvmpt3i 6287 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → (𝑆‘𝑃) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (#‘𝐵)))}) |
12 | 11 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑆‘𝑃) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (#‘𝐵)))}) |
13 | 12 | fveq2d 6195 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (#‘(𝑆‘𝑃)) = (#‘{𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (#‘𝐵)))})) |
14 | ablfac1.o | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
15 | eqid 2622 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (#‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (#‘𝐵)))} | |
16 | eqid 2622 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵))))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵))))} | |
17 | ablfac1.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
18 | 17 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝐺 ∈ Abel) |
19 | ablfac1.f | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
20 | ablfac1.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℙ) | |
21 | eqid 2622 | . . . . . . 7 ⊢ (𝑃↑(𝑃 pCnt (#‘𝐵))) = (𝑃↑(𝑃 pCnt (#‘𝐵))) | |
22 | eqid 2622 | . . . . . . 7 ⊢ ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵)))) = ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵)))) | |
23 | 7, 14, 6, 17, 19, 20, 21, 22 | ablfac1lem 18467 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (((𝑃↑(𝑃 pCnt (#‘𝐵))) ∈ ℕ ∧ ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵)))) ∈ ℕ) ∧ ((𝑃↑(𝑃 pCnt (#‘𝐵))) gcd ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵))))) = 1 ∧ (#‘𝐵) = ((𝑃↑(𝑃 pCnt (#‘𝐵))) · ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵))))))) |
24 | 23 | simp1d 1073 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑃↑(𝑃 pCnt (#‘𝐵))) ∈ ℕ ∧ ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵)))) ∈ ℕ)) |
25 | 24 | simpld 475 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (#‘𝐵))) ∈ ℕ) |
26 | 24 | simprd 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵)))) ∈ ℕ) |
27 | 23 | simp2d 1074 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑃↑(𝑃 pCnt (#‘𝐵))) gcd ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵))))) = 1) |
28 | 23 | simp3d 1075 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (#‘𝐵) = ((𝑃↑(𝑃 pCnt (#‘𝐵))) · ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵)))))) |
29 | 7, 14, 15, 16, 18, 25, 26, 27, 28 | ablfacrp2 18466 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((#‘{𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (#‘𝐵)))}) = (𝑃↑(𝑃 pCnt (#‘𝐵))) ∧ (#‘{𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵))))}) = ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵)))))) |
30 | 29 | simpld 475 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (#‘{𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (#‘𝐵)))}) = (𝑃↑(𝑃 pCnt (#‘𝐵)))) |
31 | 13, 30 | eqtrd 2656 | 1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (#‘(𝑆‘𝑃)) = (𝑃↑(𝑃 pCnt (#‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 1c1 9937 · cmul 9941 / cdiv 10684 ℕcn 11020 ↑cexp 12860 #chash 13117 ∥ cdvds 14983 gcd cgcd 15216 ℙcprime 15385 pCnt cpc 15541 Basecbs 15857 odcod 17944 Abelcabl 18194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-eqg 17593 df-ga 17723 df-cntz 17750 df-od 17948 df-lsm 18051 df-pj1 18052 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: ablfac1c 18470 ablfac1eu 18472 ablfaclem3 18486 |
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