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| Mirrors > Home > MPE Home > Th. List > Mathboxes > aks6d1c1rh | Structured version Visualization version GIF version | ||
| Description: Claim 1 of AKS primality proof with collapsed definitions since their ease of use is no longer needed. (Contributed by metakunt, 1-May-2025.) | 
| Ref | Expression | 
|---|---|
| aks6d1c1rh.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)))} | 
| aks6d1c1rh.2 | ⊢ 𝑃 = (chr‘𝐾) | 
| aks6d1c1rh.3 | ⊢ (𝜑 → 𝐾 ∈ Field) | 
| aks6d1c1rh.4 | ⊢ (𝜑 → 𝑃 ∈ ℙ) | 
| aks6d1c1rh.5 | ⊢ (𝜑 → 𝑅 ∈ ℕ) | 
| aks6d1c1rh.6 | ⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| aks6d1c1rh.7 | ⊢ (𝜑 → 𝑃 ∥ 𝑁) | 
| aks6d1c1rh.8 | ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) | 
| aks6d1c1rh.9 | ⊢ (𝜑 → 𝐹:(0...𝐴)⟶ℕ0) | 
| aks6d1c1rh.10 | ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) | 
| aks6d1c1rh.11 | ⊢ (𝜑 → 𝐴 ∈ ℕ0) | 
| aks6d1c1rh.12 | ⊢ (𝜑 → 𝑈 ∈ ℕ0) | 
| aks6d1c1rh.13 | ⊢ (𝜑 → 𝐿 ∈ ℕ0) | 
| aks6d1c1rh.14 | ⊢ 𝐸 = ((𝑃↑𝑈) · ((𝑁 / 𝑃)↑𝐿)) | 
| aks6d1c1rh.15 | ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)))) | 
| aks6d1c1rh.16 | ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾)) | 
| Ref | Expression | 
|---|---|
| aks6d1c1rh | ⊢ (𝜑 → 𝐸 ∼ (𝐺‘𝐹)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | aks6d1c1rh.1 | . . 3 ⊢ ∼ = {〈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)))} | |
| 2 | nfv 1913 | . . . . . 6 ⊢ Ⅎ𝑧(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)) | |
| 3 | nfv 1913 | . . . . . 6 ⊢ Ⅎ𝑦(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑧)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑧)) | |
| 4 | fveq2 6905 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (((eval1‘𝐾)‘𝑓)‘𝑦) = (((eval1‘𝐾)‘𝑓)‘𝑧)) | |
| 5 | 4 | oveq2d 7448 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑧))) | 
| 6 | oveq2 7440 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑒(.g‘(mulGrp‘𝐾))𝑦) = (𝑒(.g‘(mulGrp‘𝐾))𝑧)) | |
| 7 | 6 | fveq2d 6909 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑧))) | 
| 8 | 5, 7 | eqeq12d 2752 | . . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)) ↔ (𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑧)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑧)))) | 
| 9 | 2, 3, 8 | cbvralw 3305 | . . . . 5 ⊢ (∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)) ↔ ∀𝑧 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑧)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑧))) | 
| 10 | 9 | 3anbi3i 1159 | . . . 4 ⊢ ((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))) ↔ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑧 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑧)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑧)))) | 
| 11 | 10 | opabbii 5209 | . . 3 ⊢ {〈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)))} = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑧 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑧)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑧)))} | 
| 12 | 1, 11 | eqtri 2764 | . 2 ⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑧 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑧)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑧)))} | 
| 13 | eqid 2736 | . 2 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾) | |
| 14 | eqid 2736 | . 2 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾)) | |
| 15 | eqid 2736 | . 2 ⊢ (var1‘𝐾) = (var1‘𝐾) | |
| 16 | eqid 2736 | . 2 ⊢ (mulGrp‘(Poly1‘𝐾)) = (mulGrp‘(Poly1‘𝐾)) | |
| 17 | eqid 2736 | . 2 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
| 18 | eqid 2736 | . 2 ⊢ (.g‘(mulGrp‘𝐾)) = (.g‘(mulGrp‘𝐾)) | |
| 19 | eqid 2736 | . 2 ⊢ (algSc‘(Poly1‘𝐾)) = (algSc‘(Poly1‘𝐾)) | |
| 20 | eqid 2736 | . 2 ⊢ (.g‘(mulGrp‘(Poly1‘𝐾))) = (.g‘(mulGrp‘(Poly1‘𝐾))) | |
| 21 | aks6d1c1rh.2 | . 2 ⊢ 𝑃 = (chr‘𝐾) | |
| 22 | eqid 2736 | . 2 ⊢ (eval1‘𝐾) = (eval1‘𝐾) | |
| 23 | eqid 2736 | . 2 ⊢ (+g‘(Poly1‘𝐾)) = (+g‘(Poly1‘𝐾)) | |
| 24 | aks6d1c1rh.3 | . 2 ⊢ (𝜑 → 𝐾 ∈ Field) | |
| 25 | aks6d1c1rh.4 | . 2 ⊢ (𝜑 → 𝑃 ∈ ℙ) | |
| 26 | aks6d1c1rh.5 | . 2 ⊢ (𝜑 → 𝑅 ∈ ℕ) | |
| 27 | aks6d1c1rh.6 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 28 | aks6d1c1rh.7 | . 2 ⊢ (𝜑 → 𝑃 ∥ 𝑁) | |
| 29 | aks6d1c1rh.8 | . 2 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) | |
| 30 | aks6d1c1rh.9 | . 2 ⊢ (𝜑 → 𝐹:(0...𝐴)⟶ℕ0) | |
| 31 | aks6d1c1rh.10 | . 2 ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) | |
| 32 | aks6d1c1rh.11 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
| 33 | aks6d1c1rh.12 | . 2 ⊢ (𝜑 → 𝑈 ∈ ℕ0) | |
| 34 | aks6d1c1rh.13 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
| 35 | aks6d1c1rh.14 | . 2 ⊢ 𝐸 = ((𝑃↑𝑈) · ((𝑁 / 𝑃)↑𝐿)) | |
| 36 | aks6d1c1rh.15 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)))) | |
| 37 | aks6d1c1rh.16 | . 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾)) | |
| 38 | 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37 | aks6d1c1 42118 | 1 ⊢ (𝜑 → 𝐸 ∼ (𝐺‘𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 class class class wbr 5142 {copab 5204 ↦ cmpt 5224 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 0cc0 11156 1c1 11157 · cmul 11161 / cdiv 11921 ℕcn 12267 ℕ0cn0 12528 ...cfz 13548 ↑cexp 14103 ∥ cdvds 16291 gcd cgcd 16532 ℙcprime 16709 Basecbs 17248 +gcplusg 17298 Σg cgsu 17486 .gcmg 19086 mulGrpcmgp 20138 RingIso crs 20471 Fieldcfield 20731 ℤRHomczrh 21511 chrcchr 21513 algSccascl 21873 var1cv1 22178 Poly1cpl1 22179 eval1ce1 22319 PrimRoots cprimroots 42093 | 
| 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 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-dvds 16292 df-gcd 16533 df-prm 16710 df-phi 16804 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-od 19547 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-srg 20185 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-rhm 20473 df-rim 20474 df-subrng 20547 df-subrg 20571 df-drng 20732 df-field 20733 df-lmod 20861 df-lss 20931 df-lsp 20971 df-cnfld 21366 df-zring 21459 df-zrh 21515 df-chr 21517 df-assa 21874 df-asp 21875 df-ascl 21876 df-psr 21930 df-mvr 21931 df-mpl 21932 df-opsr 21934 df-evls 22099 df-evl 22100 df-psr1 22182 df-vr1 22183 df-ply1 22184 df-coe1 22185 df-evl1 22321 df-primroots 42094 | 
| This theorem is referenced by: aks6d1c2lem3 42128 aks6d1c2lem4 42129 aks6d1c6lem2 42173 | 
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