aks6d1c7lem4 - Mathbox for metakunt

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Theorem aks6d1c7lem4 42166

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 42165	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑃 = 𝑝)
30	29	eqcomd 2736	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑝 = 𝑃)
31	30	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ 𝑁 → 𝑝 = 𝑃))
32	31	ralrimiva 3126	. . 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 5112	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑁 ↔ 𝑃 ∥ 𝑁))
35	34	eqreu 3702	. 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 1540 ∈ wcel 2109 ∀wral 3045 ∃!wreu 3354 class class class wbr 5109 {copab 5171 ↦ cmpt 5190 ‘cfv 6513 (class class class)co 7389 1c1 11075 · cmul 11079 < clt 11214 ℕcn 12187 2c2 12242 3c3 12243 ℤ_≥cuz 12799 ...cfz 13474 ⌊cfl 13758 ↑cexp 14032 √csqrt 15205 ∥ cdvds 16228 gcd cgcd 16470 ℙcprime 16647 od_ℤcodz 16739 ϕcphi 16740 Basecbs 17185 +_gcplusg 17226 ._gcmg 19005 mulGrpcmgp 20055 RingIso crs 20385 Fieldcfield 20645 ℤRHomczrh 21415 chrcchr 21417 algSccascl 21767 var₁cv1 22066 Poly₁cpl1 22067 eval₁ce1 22207 log_b clogb 26680 PrimRoots cprimroots 42074

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-inf2 9600 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153 ax-mulf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-ofr 7656 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-oadd 8440 df-er 8673 df-ec 8675 df-qs 8679 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-fi 9368 df-sup 9399 df-inf 9400 df-oi 9469 df-dju 9860 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-xnn0 12522 df-z 12536 df-dec 12656 df-uz 12800 df-q 12914 df-rp 12958 df-xneg 13078 df-xadd 13079 df-xmul 13080 df-ioo 13316 df-ioc 13317 df-ico 13318 df-icc 13319 df-fz 13475 df-fzo 13622 df-fl 13760 df-mod 13838 df-seq 13973 df-exp 14033 df-fac 14245 df-bc 14274 df-hash 14302 df-shft 15039 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-limsup 15443 df-clim 15460 df-rlim 15461 df-sum 15659 df-prod 15876 df-fallfac 15979 df-ef 16039 df-sin 16041 df-cos 16042 df-pi 16044 df-dvds 16229 df-gcd 16471 df-prm 16648 df-odz 16741 df-phi 16742 df-pc 16814 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17391 df-topn 17392 df-0g 17410 df-gsum 17411 df-topgen 17412 df-pt 17413 df-prds 17416 df-pws 17418 df-xrs 17471 df-qtop 17476 df-imas 17477 df-qus 17478 df-xps 17479 df-mre 17553 df-mrc 17554 df-acs 17556 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18716 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-mulg 19006 df-subg 19061 df-nsg 19062 df-eqg 19063 df-ghm 19151 df-gim 19197 df-cntz 19255 df-od 19464 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-srg 20102 df-ring 20150 df-cring 20151 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-dvr 20316 df-rhm 20387 df-rim 20388 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-rlreg 20609 df-domn 20610 df-idom 20611 df-drng 20646 df-field 20647 df-lmod 20774 df-lss 20844 df-lsp 20884 df-sra 21086 df-rgmod 21087 df-lidl 21124 df-rsp 21125 df-2idl 21166 df-psmet 21262 df-xmet 21263 df-met 21264 df-bl 21265 df-mopn 21266 df-fbas 21267 df-fg 21268 df-cnfld 21271 df-zring 21363 df-zrh 21419 df-chr 21421 df-zn 21422 df-assa 21768 df-asp 21769 df-ascl 21770 df-psr 21824 df-mvr 21825 df-mpl 21826 df-opsr 21828 df-evls 21987 df-evl 21988 df-psr1 22070 df-vr1 22071 df-ply1 22072 df-coe1 22073 df-evl1 22209 df-top 22787 df-topon 22804 df-topsp 22826 df-bases 22839 df-cld 22912 df-ntr 22913 df-cls 22914 df-nei 22991 df-lp 23029 df-perf 23030 df-cn 23120 df-cnp 23121 df-haus 23208 df-tx 23455 df-hmeo 23648 df-fil 23739 df-fm 23831 df-flim 23832 df-flf 23833 df-xms 24214 df-ms 24215 df-tms 24216 df-cncf 24777 df-limc 25773 df-dv 25774 df-mdeg 25966 df-deg1 25967 df-mon1 26042 df-uc1p 26043 df-q1p 26044 df-r1p 26045 df-log 26471 df-cxp 26472 df-logb 26681 df-primroots 42075

This theorem is referenced by: aks6d1c7 42167

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