aks6d1c1p7 - Mathbox for metakunt

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Theorem aks6d1c1p7 43131

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 2761	. . . . . . 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 20975	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ CRing)
8	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
9		aks6d1c1p7.5	. . . . . . . . . . . . . 14 ⊢ 𝑉 = (mulGrp‘𝐾)
10	9	crngmgp 20447	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
11	7, 10	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ CMnd)
12		aks6d1c1p7.11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13	12	nnnn0d 12648	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
14		aks6d1c1p7.6	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘𝑉)
15	11, 13, 14	isprimroot 43111	. . . . . . . . . . 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 232	. . . . . . . . . 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 412	. . . . . . . . 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 1160	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
19	9, 3	mgpbas 20345	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝑉)
20	18, 19	eleqtrrdi 2872	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
21	1, 2, 3, 4, 5, 8, 20	evl1vard 22635	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑦) = 𝑦))
22	21	simprd 501	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝑋)‘𝑦) = 𝑦)
23	22	oveq2d 7428	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = (𝐿 ↑ 𝑦))
24	11	cmnmndd 19998	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ Mnd)
25	24	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ Mnd)
26		aks6d1c1p7.15	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℕ)
27	26	nnnn0d 12648	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℕ₀)
28	27	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐿 ∈ ℕ₀)
29	20, 19	eleqtrdi 2871	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
30		eqid 2761	. . . . . . . . . 10 ⊢ (Base‘𝑉) = (Base‘𝑉)
31	30, 14	mulgnn0cl 19280	. . . . . . . . 9 ⊢ ((𝑉 ∈ Mnd ∧ 𝐿 ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘𝑉)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝑉))
32	25, 28, 29, 31	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝑉))
33	32, 19	eleqtrrdi 2872	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝐾))
34	1, 2, 3, 4, 5, 8, 33	evl1vard 22635	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)) = (𝐿 ↑ 𝑦)))
35	34	simprd 501	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)) = (𝐿 ↑ 𝑦))
36		eqidd 2762	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) = (𝐿 ↑ 𝑦))
37	35, 36	eqtr2d 2797	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
38	23, 37	eqtrd 2796	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
39	38	ralrimiva 3155	. 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 20452	. . . . . 6 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
42	7, 41	syl 18	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
43		eqid 2761	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
44	2, 4, 43	vr1cl 22515	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘𝑆))
45	42, 44	syl 18	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑆))
46	45, 5	eleqtrrdi 2872	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
47	40, 46, 26	aks6d1c1p1 43125	. 2 ⊢ (𝜑 → (𝐿 ∼ 𝑋 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦))))
48	39, 47	mpbird 260	1 ⊢ (𝜑 → 𝐿 ∼ 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 class class class wbr 5103 {copab 5167 ‘cfv 6531 (class class class)co 7412 1c1 11182 ℕcn 12316 ℕ₀cn0 12587 ∥ cdvds 16402 gcd cgcd 16644 ℙcprime 16826 Basecbs 17367 0_gc0g 17590 Mndcmnd 18903 ._gcmg 19257 CMndccmn 19974 mulGrpcmgp 20340 Ringcrg 20439 CRingccrg 20440 Fieldcfield 20961 chrcchr 21787 var₁cv1 22474 Poly₁cpl1 22475 eval₁ce1 22612 PrimRoots cprimroots 43109

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-ofr 7683 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-sup 9418 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-fz 13621 df-fzo 13769 df-seq 14125 df-hash 14455 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-hom 17432 df-cco 17433 df-0g 17592 df-gsum 17593 df-prds 17598 df-pws 17600 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-mhm 18958 df-submnd 18959 df-grp 19127 df-minusg 19128 df-sbg 19129 df-mulg 19258 df-subg 19313 df-ghm 19408 df-cntz 19511 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-srg 20393 df-ring 20441 df-cring 20442 df-rhm 20682 df-subrng 20778 df-subrg 20802 df-field 20963 df-lmod 21117 df-lss 21187 df-lsp 21227 df-assa 22141 df-asp 22142 df-ascl 22143 df-psr 22197 df-mvr 22198 df-mpl 22199 df-opsr 22201 df-evls 22363 df-evl 22364 df-psr1 22478 df-vr1 22479 df-ply1 22480 df-evl1 22614 df-primroots 43110

This theorem is used by: aks6d1c1 43134

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