aks6d1c1p7 - Mathbox for metakunt

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Theorem aks6d1c1p7 42572

Description: 𝑋 is introspective to all positive integers. (Contributed by metakunt, 30-Apr-2025.)

**Hypotheses**
Ref	Expression
aks6d1c1p7.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
aks6d1c1p7.2	⊢ 𝑆 = (Poly₁‘𝐾)
aks6d1c1p7.3	⊢ 𝐵 = (Base‘𝑆)
aks6d1c1p7.4	⊢ 𝑋 = (var₁‘𝐾)
aks6d1c1p7.5	⊢ 𝑉 = (mulGrp‘𝐾)
aks6d1c1p7.6	⊢ ↑ = (._g‘𝑉)
aks6d1c1p7.7	⊢ 𝑃 = (chr‘𝐾)
aks6d1c1p7.8	⊢ 𝑂 = (eval₁‘𝐾)
aks6d1c1p7.9	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c1p7.10	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c1p7.11	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c1p7.12	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks6d1c1p7.13	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c1p7.14	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c1p7.15	⊢ (𝜑 → 𝐿 ∈ ℕ)

**Assertion**
Ref	Expression
aks6d1c1p7	⊢ (𝜑 → 𝐿 ∼ 𝑋)

Distinct variable groups: ↑ ,𝑒,𝑓 𝐵,𝑒,𝑓 𝑒,𝐿,𝑓,𝑦 𝑒,𝑂,𝑓 𝑅,𝑒,𝑓 𝑒,𝑉,𝑓 𝑒,𝑋,𝑓,𝑦 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑒,𝑓) 𝐵(𝑦) 𝑃(𝑦,𝑒,𝑓) ∼ (𝑦,𝑒,𝑓) 𝑅(𝑦) 𝑆(𝑦,𝑒,𝑓) ↑ (𝑦) 𝐾(𝑦,𝑒,𝑓) 𝑁(𝑦,𝑒,𝑓) 𝑂(𝑦) 𝑉(𝑦)

Proof of Theorem aks6d1c1p7 Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c1p7.8	. . . . . . 7 ⊢ 𝑂 = (eval₁‘𝐾)
2		aks6d1c1p7.4	. . . . . . 7 ⊢ 𝑋 = (var₁‘𝐾)
3		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		aks6d1c1p7.2	. . . . . . 7 ⊢ 𝑆 = (Poly₁‘𝐾)
5		aks6d1c1p7.3	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
6		aks6d1c1p7.9	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Field)
7	6	fldcrngd 20714	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ CRing)
8	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
9		aks6d1c1p7.5	. . . . . . . . . . . . . 14 ⊢ 𝑉 = (mulGrp‘𝐾)
10	9	crngmgp 20217	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
11	7, 10	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ CMnd)
12		aks6d1c1p7.11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13	12	nnnn0d 12493	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
14		aks6d1c1p7.6	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘𝑉)
15	11, 13, 14	isprimroot 42552	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) ↔ (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙))))
16	15	biimpd 229	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙))))
17	16	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 ↑ 𝑦) = (0_g‘𝑉) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙 ↑ 𝑦) = (0_g‘𝑉) → 𝑅 ∥ 𝑙)))
18	17	simp1d 1143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
19	9, 3	mgpbas 20121	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝑉)
20	18, 19	eleqtrrdi 2848	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
21	1, 2, 3, 4, 5, 8, 20	evl1vard 22316	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑦) = 𝑦))
22	21	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝑋)‘𝑦) = 𝑦)
23	22	oveq2d 7378	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = (𝐿 ↑ 𝑦))
24	11	cmnmndd 19774	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ Mnd)
25	24	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ Mnd)
26		aks6d1c1p7.15	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℕ)
27	26	nnnn0d 12493	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℕ₀)
28	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐿 ∈ ℕ₀)
29	20, 19	eleqtrdi 2847	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
30		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑉) = (Base‘𝑉)
31	30, 14	mulgnn0cl 19061	. . . . . . . . 9 ⊢ ((𝑉 ∈ Mnd ∧ 𝐿 ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘𝑉)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝑉))
32	25, 28, 29, 31	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝑉))
33	32, 19	eleqtrrdi 2848	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝐾))
34	1, 2, 3, 4, 5, 8, 33	evl1vard 22316	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)) = (𝐿 ↑ 𝑦)))
35	34	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)) = (𝐿 ↑ 𝑦))
36		eqidd 2738	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) = (𝐿 ↑ 𝑦))
37	35, 36	eqtr2d 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
38	23, 37	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
39	38	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
40		aks6d1c1p7.1	. . 3 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}
41		crngring 20221	. . . . . 6 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
42	7, 41	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
43		eqid 2737	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
44	2, 4, 43	vr1cl 22195	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘𝑆))
45	42, 44	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑆))
46	45, 5	eleqtrrdi 2848	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
47	40, 46, 26	aks6d1c1p1 42566	. 2 ⊢ (𝜑 → (𝐿 ∼ 𝑋 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦))))
48	39, 47	mpbird 257	1 ⊢ (𝜑 → 𝐿 ∼ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 class class class wbr 5086 {copab 5148 ‘cfv 6494 (class class class)co 7362 1c1 11034 ℕcn 12169 ℕ₀cn0 12432 ∥ cdvds 16216 gcd cgcd 16458 ℙcprime 16635 Basecbs 17174 0_gc0g 17397 Mndcmnd 18697 ._gcmg 19038 CMndccmn 19750 mulGrpcmgp 20116 Ringcrg 20209 CRingccrg 20210 Fieldcfield 20702 chrcchr 21495 var₁cv1 22153 Poly₁cpl1 22154 eval₁ce1 22293 PrimRoots cprimroots 42550

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-ofr 7627 df-om 7813 df-1st 7937 df-2nd 7938 df-supp 8106 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-er 8638 df-map 8770 df-pm 8771 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-sup 9350 df-oi 9420 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-fzo 13604 df-seq 13959 df-hash 14288 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-hom 17239 df-cco 17240 df-0g 17399 df-gsum 17400 df-prds 17405 df-pws 17407 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-ghm 19183 df-cntz 19287 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-srg 20163 df-ring 20211 df-cring 20212 df-rhm 20447 df-subrng 20518 df-subrg 20542 df-field 20704 df-lmod 20852 df-lss 20922 df-lsp 20962 df-assa 21847 df-asp 21848 df-ascl 21849 df-psr 21903 df-mvr 21904 df-mpl 21905 df-opsr 21907 df-evls 22066 df-evl 22067 df-psr1 22157 df-vr1 22158 df-ply1 22159 df-evl1 22295 df-primroots 42551

This theorem is referenced by: aks6d1c1 42575

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