aks6d1c7lem4 - Mathbox for metakunt

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Theorem aks6d1c7lem4 42179

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 769	. . . . . . . 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 42178	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑃 = 𝑝)
30	29	eqcomd 2743	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ 𝑁) → 𝑝 = 𝑃)
31	30	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ 𝑁 → 𝑝 = 𝑃))
32	31	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑁 → 𝑝 = 𝑃))
33	1, 2, 32	3jca 1129	. 2 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑃 ∥ 𝑁 ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑁 → 𝑝 = 𝑃)))
34		breq1 5154	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑁 ↔ 𝑃 ∥ 𝑁))
35	34	eqreu 3741	. 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 1087 = wceq 1539 ∈ wcel 2108 ∀wral 3061 ∃!wreu 3378 class class class wbr 5151 {copab 5213 ↦ cmpt 5234 ‘cfv 6569 (class class class)co 7438 1c1 11163 · cmul 11167 < clt 11302 ℕcn 12273 2c2 12328 3c3 12329 ℤ_≥cuz 12885 ...cfz 13553 ⌊cfl 13836 ↑cexp 14108 √csqrt 15278 ∥ cdvds 16296 gcd cgcd 16537 ℙcprime 16714 od_ℤcodz 16806 ϕcphi 16807 Basecbs 17254 +_gcplusg 17307 ._gcmg 19107 mulGrpcmgp 20161 RingIso crs 20496 Fieldcfield 20756 ℤRHomczrh 21537 chrcchr 21539 algSccascl 21899 var₁cv1 22202 Poly₁cpl1 22203 eval₁ce1 22343 log_b clogb 26833 PrimRoots cprimroots 42087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-inf2 9688 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239 ax-pre-sup 11240 ax-addf 11241 ax-mulf 11242

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-int 4955 df-iun 5001 df-iin 5002 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-se 5646 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-isom 6578 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-of 7704 df-ofr 7705 df-om 7895 df-1st 8022 df-2nd 8023 df-supp 8194 df-tpos 8259 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8753 df-ec 8755 df-qs 8759 df-map 8876 df-pm 8877 df-ixp 8946 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-fsupp 9409 df-fi 9458 df-sup 9489 df-inf 9490 df-oi 9557 df-dju 9948 df-card 9986 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-div 11928 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-7 12341 df-8 12342 df-9 12343 df-n0 12534 df-xnn0 12607 df-z 12621 df-dec 12741 df-uz 12886 df-q 12998 df-rp 13042 df-xneg 13161 df-xadd 13162 df-xmul 13163 df-ioo 13397 df-ioc 13398 df-ico 13399 df-icc 13400 df-fz 13554 df-fzo 13701 df-fl 13838 df-mod 13916 df-seq 14049 df-exp 14109 df-fac 14319 df-bc 14348 df-hash 14376 df-shft 15112 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-limsup 15513 df-clim 15530 df-rlim 15531 df-sum 15729 df-prod 15946 df-fallfac 16049 df-ef 16109 df-sin 16111 df-cos 16112 df-pi 16114 df-dvds 16297 df-gcd 16538 df-prm 16715 df-odz 16808 df-phi 16809 df-pc 16880 df-struct 17190 df-sets 17207 df-slot 17225 df-ndx 17237 df-base 17255 df-ress 17284 df-plusg 17320 df-mulr 17321 df-starv 17322 df-sca 17323 df-vsca 17324 df-ip 17325 df-tset 17326 df-ple 17327 df-ds 17329 df-unif 17330 df-hom 17331 df-cco 17332 df-rest 17478 df-topn 17479 df-0g 17497 df-gsum 17498 df-topgen 17499 df-pt 17500 df-prds 17503 df-pws 17505 df-xrs 17558 df-qtop 17563 df-imas 17564 df-qus 17565 df-xps 17566 df-mre 17640 df-mrc 17641 df-acs 17643 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-gim 19299 df-cntz 19357 df-od 19570 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-srg 20214 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-rim 20499 df-nzr 20539 df-subrng 20572 df-subrg 20596 df-rlreg 20720 df-domn 20721 df-idom 20722 df-drng 20757 df-field 20758 df-lmod 20886 df-lss 20957 df-lsp 20997 df-sra 21199 df-rgmod 21200 df-lidl 21245 df-rsp 21246 df-2idl 21287 df-psmet 21383 df-xmet 21384 df-met 21385 df-bl 21386 df-mopn 21387 df-fbas 21388 df-fg 21389 df-cnfld 21392 df-zring 21485 df-zrh 21541 df-chr 21543 df-zn 21544 df-assa 21900 df-asp 21901 df-ascl 21902 df-psr 21956 df-mvr 21957 df-mpl 21958 df-opsr 21960 df-evls 22125 df-evl 22126 df-psr1 22206 df-vr1 22207 df-ply1 22208 df-coe1 22209 df-evl1 22345 df-top 22925 df-topon 22942 df-topsp 22964 df-bases 22978 df-cld 23052 df-ntr 23053 df-cls 23054 df-nei 23131 df-lp 23169 df-perf 23170 df-cn 23260 df-cnp 23261 df-haus 23348 df-tx 23595 df-hmeo 23788 df-fil 23879 df-fm 23971 df-flim 23972 df-flf 23973 df-xms 24355 df-ms 24356 df-tms 24357 df-cncf 24929 df-limc 25927 df-dv 25928 df-mdeg 26120 df-deg1 26121 df-mon1 26196 df-uc1p 26197 df-q1p 26198 df-r1p 26199 df-log 26624 df-cxp 26625 df-logb 26834 df-primroots 42088

This theorem is referenced by: aks6d1c7 42180

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