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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 1912 | . . . . . 6 ⊢ Ⅎ𝑧(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)) | |
3 | nfv 1912 | . . . . . 6 ⊢ Ⅎ𝑦(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑧)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑧)) | |
4 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (((eval1‘𝐾)‘𝑓)‘𝑦) = (((eval1‘𝐾)‘𝑓)‘𝑧)) | |
5 | 4 | oveq2d 7447 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑧))) |
6 | oveq2 7439 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑒(.g‘(mulGrp‘𝐾))𝑦) = (𝑒(.g‘(mulGrp‘𝐾))𝑧)) | |
7 | 6 | fveq2d 6911 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑧))) |
8 | 5, 7 | eqeq12d 2751 | . . . . . 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 3304 | . . . . 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 1158 | . . . 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 5215 | . . 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 2763 | . 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 2735 | . 2 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾) | |
14 | eqid 2735 | . 2 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾)) | |
15 | eqid 2735 | . 2 ⊢ (var1‘𝐾) = (var1‘𝐾) | |
16 | eqid 2735 | . 2 ⊢ (mulGrp‘(Poly1‘𝐾)) = (mulGrp‘(Poly1‘𝐾)) | |
17 | eqid 2735 | . 2 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
18 | eqid 2735 | . 2 ⊢ (.g‘(mulGrp‘𝐾)) = (.g‘(mulGrp‘𝐾)) | |
19 | eqid 2735 | . 2 ⊢ (algSc‘(Poly1‘𝐾)) = (algSc‘(Poly1‘𝐾)) | |
20 | eqid 2735 | . 2 ⊢ (.g‘(mulGrp‘(Poly1‘𝐾))) = (.g‘(mulGrp‘(Poly1‘𝐾))) | |
21 | aks6d1c1rh.2 | . 2 ⊢ 𝑃 = (chr‘𝐾) | |
22 | eqid 2735 | . 2 ⊢ (eval1‘𝐾) = (eval1‘𝐾) | |
23 | eqid 2735 | . 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 42098 | 1 ⊢ (𝜑 → 𝐸 ∼ (𝐺‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 {copab 5210 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 0cc0 11153 1c1 11154 · cmul 11158 / cdiv 11918 ℕcn 12264 ℕ0cn0 12524 ...cfz 13544 ↑cexp 14099 ∥ cdvds 16287 gcd cgcd 16528 ℙcprime 16705 Basecbs 17245 +gcplusg 17298 Σg cgsu 17487 .gcmg 19098 mulGrpcmgp 20152 RingIso crs 20487 Fieldcfield 20747 ℤRHomczrh 21528 chrcchr 21530 algSccascl 21890 var1cv1 22193 Poly1cpl1 22194 eval1ce1 22334 PrimRoots cprimroots 42073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 df-phi 16800 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 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 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-od 19561 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-rim 20490 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-lmod 20877 df-lss 20948 df-lsp 20988 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-chr 21534 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-evl1 22336 df-primroots 42074 |
This theorem is referenced by: aks6d1c2lem3 42108 aks6d1c2lem4 42109 aks6d1c6lem2 42153 |
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