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Mirrors > Home > MPE Home > Th. List > ablfac1a | Structured version Visualization version GIF version |
Description: The factors of ablfac1b 18640 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 6808 | . . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝 pCnt (♯‘𝐵)) = (𝑃 pCnt (♯‘𝐵))) | |
3 | 1, 2 | oveq12d 6819 | . . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
4 | 3 | breq2d 4804 | . . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵))))) |
5 | 4 | rabbidv 3317 | . . . . 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 6350 | . . . . . . 7 ⊢ (Base‘𝐺) ∈ V | |
9 | 7, 8 | eqeltri 2823 | . . . . . 6 ⊢ 𝐵 ∈ V |
10 | 9 | rabex 4952 | . . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V |
11 | 5, 6, 10 | fvmpt3i 6437 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → (𝑆‘𝑃) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))}) |
12 | 11 | adantl 473 | . . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑆‘𝑃) = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))}) |
13 | 12 | fveq2d 6344 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘(𝑆‘𝑃)) = (♯‘{𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))})) |
14 | ablfac1.o | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
15 | eqid 2748 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))} = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))} | |
16 | eqid 2748 | . . . 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 472 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝐺 ∈ Abel) |
19 | ablfac1.f | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
20 | ablfac1.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℙ) | |
21 | eqid 2748 | . . . . . . 7 ⊢ (𝑃↑(𝑃 pCnt (♯‘𝐵))) = (𝑃↑(𝑃 pCnt (♯‘𝐵))) | |
22 | eqid 2748 | . . . . . . 7 ⊢ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) | |
23 | 7, 14, 6, 17, 19, 20, 21, 22 | ablfac1lem 18638 | . . . . . 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 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑃↑(𝑃 pCnt (♯‘𝐵))) ∈ ℕ ∧ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) ∈ ℕ)) |
25 | 24 | simpld 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∈ ℕ) |
26 | 24 | simprd 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))) ∈ ℕ) |
27 | 23 | simp2d 1135 | . . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))) = 1) |
28 | 23 | simp3d 1136 | . . . 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 18637 | . . 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 477 | . 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘{𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑃↑(𝑃 pCnt (♯‘𝐵)))}) = (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
31 | 13, 30 | eqtrd 2782 | 1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘(𝑆‘𝑃)) = (𝑃↑(𝑃 pCnt (♯‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 {crab 3042 Vcvv 3328 ⊆ wss 3703 class class class wbr 4792 ↦ cmpt 4869 ‘cfv 6037 (class class class)co 6801 Fincfn 8109 1c1 10100 · cmul 10104 / cdiv 10847 ℕcn 11183 ↑cexp 13025 ♯chash 13282 ∥ cdvds 15153 gcd cgcd 15389 ℙcprime 15558 pCnt cpc 15714 Basecbs 16030 odcod 18115 Abelcabl 18365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-inf2 8699 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-pre-sup 10177 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-disj 4761 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-se 5214 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-isom 6046 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-1st 7321 df-2nd 7322 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-omul 7722 df-er 7899 df-ec 7901 df-qs 7905 df-map 8013 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8501 df-inf 8502 df-oi 8568 df-card 8926 df-acn 8929 df-cda 9153 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-div 10848 df-nn 11184 df-2 11242 df-3 11243 df-n0 11456 df-xnn0 11527 df-z 11541 df-uz 11851 df-q 11953 df-rp 11997 df-fz 12491 df-fzo 12631 df-fl 12758 df-mod 12834 df-seq 12967 df-exp 13026 df-fac 13226 df-bc 13255 df-hash 13283 df-cj 14009 df-re 14010 df-im 14011 df-sqrt 14145 df-abs 14146 df-clim 14389 df-sum 14587 df-dvds 15154 df-gcd 15390 df-prm 15559 df-pc 15715 df-ndx 16033 df-slot 16034 df-base 16036 df-sets 16037 df-ress 16038 df-plusg 16127 df-0g 16275 df-mgm 17414 df-sgrp 17456 df-mnd 17467 df-submnd 17508 df-grp 17597 df-minusg 17598 df-sbg 17599 df-mulg 17713 df-subg 17763 df-eqg 17765 df-ga 17894 df-cntz 17921 df-od 18119 df-lsm 18222 df-pj1 18223 df-cmn 18366 df-abl 18367 |
This theorem is referenced by: ablfac1c 18641 ablfac1eu 18643 ablfaclem3 18657 |
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