aks6d1c1p7 - Mathbox for metakunt

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Theorem aks6d1c1p7 42735

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 2764	. . . . . . 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 20794	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ CRing)
8	7	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
9		aks6d1c1p7.5	. . . . . . . . . . . . . 14 ⊢ 𝑉 = (mulGrp‘𝐾)
10	9	crngmgp 20293	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
11	7, 10	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ CMnd)
12		aks6d1c1p7.11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13	12	nnnn0d 12544	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
14		aks6d1c1p7.6	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘𝑉)
15	11, 13, 14	isprimroot 42715	. . . . . . . . . . 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 231	. . . . . . . . . 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 410	. . . . . . . . 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 1156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
19	9, 3	mgpbas 20193	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝑉)
20	18, 19	eleqtrrdi 2875	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
21	1, 2, 3, 4, 5, 8, 20	evl1vard 22402	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑦) = 𝑦))
22	21	simprd 499	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝑋)‘𝑦) = 𝑦)
23	22	oveq2d 7414	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = (𝐿 ↑ 𝑦))
24	11	cmnmndd 19846	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ Mnd)
25	24	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ Mnd)
26		aks6d1c1p7.15	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℕ)
27	26	nnnn0d 12544	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℕ₀)
28	27	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐿 ∈ ℕ₀)
29	20, 19	eleqtrdi 2874	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
30		eqid 2764	. . . . . . . . . 10 ⊢ (Base‘𝑉) = (Base‘𝑉)
31	30, 14	mulgnn0cl 19134	. . . . . . . . 9 ⊢ ((𝑉 ∈ Mnd ∧ 𝐿 ∈ ℕ₀ ∧ 𝑦 ∈ (Base‘𝑉)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝑉))
32	25, 28, 29, 31	syl3anc 1392	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝑉))
33	32, 19	eleqtrrdi 2875	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝐾))
34	1, 2, 3, 4, 5, 8, 33	evl1vard 22402	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)) = (𝐿 ↑ 𝑦)))
35	34	simprd 499	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)) = (𝐿 ↑ 𝑦))
36		eqidd 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) = (𝐿 ↑ 𝑦))
37	35, 36	eqtr2d 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
38	23, 37	eqtrd 2799	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
39	38	ralrimiva 3156	. 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 20297	. . . . . 6 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
42	7, 41	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
43		eqid 2764	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
44	2, 4, 43	vr1cl 22281	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘𝑆))
45	42, 44	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑆))
46	45, 5	eleqtrrdi 2875	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
47	40, 46, 26	aks6d1c1p1 42729	. 2 ⊢ (𝜑 → (𝐿 ∼ 𝑋 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦))))
48	39, 47	mpbird 259	1 ⊢ (𝜑 → 𝐿 ∼ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1099 = wceq 1562 ∈ wcel 2144 ∀wral 3078 class class class wbr 5102 {copab 5164 ‘cfv 6523 (class class class)co 7398 1c1 11076 ℕcn 12212 ℕ₀cn0 12483 ∥ cdvds 16288 gcd cgcd 16530 ℙcprime 16707 Basecbs 17247 0_gc0g 17470 Mndcmnd 18770 ._gcmg 19111 CMndccmn 19822 mulGrpcmgp 20188 Ringcrg 20285 CRingccrg 20286 Fieldcfield 20782 chrcchr 21555 var₁cv1 22240 Poly₁cpl1 22241 eval₁ce1 22379 PrimRoots cprimroots 42713

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 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 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-ofr 7663 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-sup 9390 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-fz 13515 df-fzo 13662 df-seq 14017 df-hash 14346 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-hom 17312 df-cco 17313 df-0g 17472 df-gsum 17473 df-prds 17478 df-pws 17480 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-mhm 18819 df-submnd 18820 df-grp 18980 df-minusg 18981 df-sbg 18982 df-mulg 19112 df-subg 19167 df-ghm 19256 df-cntz 19359 df-cmn 19824 df-abl 19825 df-mgp 20189 df-rng 20201 df-ur 20234 df-srg 20239 df-ring 20287 df-cring 20288 df-rhm 20523 df-subrng 20598 df-subrg 20622 df-field 20784 df-lmod 20931 df-lss 21001 df-lsp 21041 df-assa 21907 df-asp 21908 df-ascl 21909 df-psr 21963 df-mvr 21964 df-mpl 21965 df-opsr 21967 df-evls 22129 df-evl 22130 df-psr1 22244 df-vr1 22245 df-ply1 22246 df-evl1 22381 df-primroots 42714

This theorem is referenced by: aks6d1c1 42738

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