aks6d1c7lem4 - Mathbox for metakunt

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Theorem aks6d1c7lem4 42437

Description: In the AKS algorithm there exists a unique prime number 𝑝 that divides 𝑁. (Contributed by metakunt, 16-May-2025.)

**Hypotheses**
Ref	Expression
aks6d1c7.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly₁‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘𝑓)‘𝑦)) = (((eval₁‘𝐾)‘𝑓)‘(𝑒(._g‘(mulGrp‘𝐾))𝑦)))}
aks6d1c7.2	⊢ 𝑃 = (chr‘𝐾)
aks6d1c7.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c7.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c7.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c7.6	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
aks6d1c7.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c7.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c7.9	⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
aks6d1c7.10	⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
aks6d1c7.11	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c7.12	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c7.13	⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
aks6d1c7.14	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))

**Assertion**
Ref	Expression
aks6d1c7lem4	⊢ (𝜑 → ∃!𝑝 ∈ ℙ 𝑝 ∥ 𝑁)

Distinct variable groups: ∼ ,𝑎 𝐴,𝑎 𝐴,𝑏 𝐴,𝑒,𝑓,𝑦 𝑥,𝐴,𝑦 𝐾,𝑎 𝐾,𝑏 𝑒,𝐾,𝑓,𝑦 𝑥,𝐾 𝑀,𝑎 𝑀,𝑏 𝑥,𝑀,𝑦 𝑁,𝑎,𝑝 𝑁,𝑏,𝑝 𝑒,𝑁,𝑓,𝑦 𝑥,𝑁,𝑝,𝑦 𝑃,𝑎,𝑝 𝑃,𝑏 𝑃,𝑒,𝑓,𝑦 𝑥,𝑃 𝑅,𝑎 𝑅,𝑒,𝑓,𝑦 𝑥,𝑅 𝜑,𝑎,𝑝 𝜑,𝑏 𝜑,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑒,𝑓) 𝐴(𝑝) ∼ (𝑥,𝑦,𝑒,𝑓,𝑝,𝑏) 𝑅(𝑝,𝑏) 𝐾(𝑝) 𝑀(𝑒,𝑓,𝑝)

**Proof of Theorem aks6d1c7lem4**
Step	Hyp	Ref	Expression
1		aks6d1c7.4	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
2		aks6d1c7.7	. . 3 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
3		aks6d1c7.1	. . . . . . 7 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly₁‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘𝑓)‘𝑦)) = (((eval₁‘𝐾)‘𝑓)‘(𝑒(._g‘(mulGrp‘𝐾))𝑦)))}
4		aks6d1c7.2	. . . . . . 7 ⊢ 𝑃 = (chr‘𝐾)
5		aks6d1c7.3	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Field)
6	5	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝐾 ∈ Field)
7	1	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑃 ∈ ℙ)
8		aks6d1c7.5	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℕ)
9	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑅 ∈ ℕ)
10		aks6d1c7.6	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
11	10	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑁 ∈ (ℤ_≥‘3))
12	2	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑃 ∥ 𝑁)
13		aks6d1c7.8	. . . . . . . 8 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
14	13	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → (𝑁 gcd 𝑅) = 1)
15		aks6d1c7.9	. . . . . . 7 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
16		aks6d1c7.10	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
17	16	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
18		aks6d1c7.11	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
19	18	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
20		aks6d1c7.12	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
21	20	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
22		aks6d1c7.13	. . . . . . . 8 ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
23	22	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
24		aks6d1c7.14	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
25	24	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
26		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑝 ∈ ℙ)
27		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑝 ∥ 𝑁)
28	26, 27	jca 511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑁))
29	3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 19, 21, 23, 25, 28	aks6d1c7lem3 42436	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑃 = 𝑝)
30	29	eqcomd 2742	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑝 = 𝑃)
31	30	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ 𝑁 → 𝑝 = 𝑃))
32	31	ralrimiva 3128	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑁 → 𝑝 = 𝑃))
33	1, 2, 32	3jca 1128	. 2 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑃 ∥ 𝑁 ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑁 → 𝑝 = 𝑃)))
34		breq1 5101	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑁 ↔ 𝑃 ∥ 𝑁))
35	34	eqreu 3687	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ 𝑁 ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑁 → 𝑝 = 𝑃)) → ∃!𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
36	33, 35	syl 17	1 ⊢ (𝜑 → ∃!𝑝 ∈ ℙ 𝑝 ∥ 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ∃!wreu 3348 class class class wbr 5098 {copab 5160 ↦ cmpt 5179 ‘cfv 6492 (class class class)co 7358 1c1 11027 · cmul 11031 < clt 11166 ℕcn 12145 2c2 12200 3c3 12201 ℤ_≥cuz 12751 ...cfz 13423 ⌊cfl 13710 ↑cexp 13984 √csqrt 15156 ∥ cdvds 16179 gcd cgcd 16421 ℙcprime 16598 od_ℤcodz 16690 ϕcphi 16691 Basecbs 17136 +_gcplusg 17177 ._gcmg 18997 mulGrpcmgp 20075 RingIso crs 20406 Fieldcfield 20663 ℤRHomczrh 21454 chrcchr 21456 algSccascl 21807 var₁cv1 22116 Poly₁cpl1 22117 eval₁ce1 22258 log_b clogb 26730 PrimRoots cprimroots 42345

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-ec 8637 df-qs 8641 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-prod 15827 df-fallfac 15930 df-ef 15990 df-sin 15992 df-cos 15993 df-pi 15995 df-dvds 16180 df-gcd 16422 df-prm 16599 df-odz 16692 df-phi 16693 df-pc 16765 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-pws 17369 df-xrs 17423 df-qtop 17428 df-imas 17429 df-qus 17430 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-nsg 19054 df-eqg 19055 df-ghm 19142 df-gim 19188 df-cntz 19246 df-od 19457 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-srg 20122 df-ring 20170 df-cring 20171 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-rhm 20408 df-rim 20409 df-nzr 20446 df-subrng 20479 df-subrg 20503 df-rlreg 20627 df-domn 20628 df-idom 20629 df-drng 20664 df-field 20665 df-lmod 20813 df-lss 20883 df-lsp 20923 df-sra 21125 df-rgmod 21126 df-lidl 21163 df-rsp 21164 df-2idl 21205 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-zring 21402 df-zrh 21458 df-chr 21460 df-zn 21461 df-assa 21808 df-asp 21809 df-ascl 21810 df-psr 21865 df-mvr 21866 df-mpl 21867 df-opsr 21869 df-evls 22029 df-evl 22030 df-psr1 22120 df-vr1 22121 df-ply1 22122 df-coe1 22123 df-evl1 22260 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824 df-mdeg 26016 df-deg1 26017 df-mon1 26092 df-uc1p 26093 df-q1p 26094 df-r1p 26095 df-log 26521 df-cxp 26522 df-logb 26731 df-primroots 42346

This theorem is referenced by: aks6d1c7 42438

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