aks6d1c1p7 - Mathbox for metakunt

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Theorem aks6d1c1p7 42094

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 2734	. . . . . . 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 20758	. . . . . . . 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 20258	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
11	7, 10	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ CMnd)
12		aks6d1c1p7.11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13	12	nnnn0d 12584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
14		aks6d1c1p7.6	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘𝑉)
15	11, 13, 14	isprimroot 42074	. . . . . . . . . . 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 1141	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
19	9, 3	mgpbas 20157	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝑉)
20	18, 19	eleqtrrdi 2849	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
21	1, 2, 3, 4, 5, 8, 20	evl1vard 22356	. . . . . 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 7446	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = (𝐿 ↑ 𝑦))
24	11	cmnmndd 19836	. . . . . . . . . 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 12584	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℕ₀)
28	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐿 ∈ ℕ₀)
29	20, 19	eleqtrdi 2848	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
30		eqid 2734	. . . . . . . . . 10 ⊢ (Base‘𝑉) = (Base‘𝑉)
31	30, 14	mulgnn0cl 19120	. . . . . . . . 9 ⊢ ((𝑉 ∈ Mnd ∧ 𝐿 ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘𝑉)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝑉))
32	25, 28, 29, 31	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝑉))
33	32, 19	eleqtrrdi 2849	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝐾))
34	1, 2, 3, 4, 5, 8, 33	evl1vard 22356	. . . . . 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 2735	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) = (𝐿 ↑ 𝑦))
37	35, 36	eqtr2d 2775	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
38	23, 37	eqtrd 2774	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
39	38	ralrimiva 3143	. 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 20262	. . . . . 6 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
42	7, 41	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
43		eqid 2734	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
44	2, 4, 43	vr1cl 22234	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘𝑆))
45	42, 44	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑆))
46	45, 5	eleqtrrdi 2849	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
47	40, 46, 26	aks6d1c1p1 42088	. 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 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 class class class wbr 5147 {copab 5209 ‘cfv 6562 (class class class)co 7430 1c1 11153 ℕcn 12263 ℕ₀cn0 12523 ∥ cdvds 16286 gcd cgcd 16527 ℙcprime 16704 Basecbs 17244 0_gc0g 17485 Mndcmnd 18759 ._gcmg 19097 CMndccmn 19812 mulGrpcmgp 20151 Ringcrg 20250 CRingccrg 20251 Fieldcfield 20746 chrcchr 21529 var₁cv1 22192 Poly₁cpl1 22193 eval₁ce1 22333 PrimRoots cprimroots 42072

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-ofr 7697 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-srg 20204 df-ring 20252 df-cring 20253 df-rhm 20488 df-subrng 20562 df-subrg 20586 df-field 20748 df-lmod 20876 df-lss 20947 df-lsp 20987 df-assa 21890 df-asp 21891 df-ascl 21892 df-psr 21946 df-mvr 21947 df-mpl 21948 df-opsr 21950 df-evls 22115 df-evl 22116 df-psr1 22196 df-vr1 22197 df-ply1 22198 df-evl1 22335 df-primroots 42073

This theorem is referenced by: aks6d1c1 42097

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