aks6d1c1p7 - Mathbox for metakunt

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Theorem aks6d1c1p7 43256

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 2760	. . . . . . 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 21000	. . . . . . . 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 20472	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
11	7, 10	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ CMnd)
12		aks6d1c1p7.11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13	12	nnnn0d 12670	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
14		aks6d1c1p7.6	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘𝑉)
15	11, 13, 14	isprimroot 43236	. . . . . . . . . . 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 20370	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝑉)
20	18, 19	eleqtrrdi 2871	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
21	1, 2, 3, 4, 5, 8, 20	evl1vard 22660	. . . . . 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 7436	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = (𝐿 ↑ 𝑦))
24	11	cmnmndd 20023	. . . . . . . . . 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 12670	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℕ₀)
28	27	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐿 ∈ ℕ₀)
29	20, 19	eleqtrdi 2870	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
30		eqid 2760	. . . . . . . . . 10 ⊢ (Base‘𝑉) = (Base‘𝑉)
31	30, 14	mulgnn0cl 19305	. . . . . . . . 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 2871	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) ∈ (Base‘𝐾))
34	1, 2, 3, 4, 5, 8, 33	evl1vard 22660	. . . . . 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 2761	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) = (𝐿 ↑ 𝑦))
37	35, 36	eqtr2d 2796	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ 𝑦) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
38	23, 37	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐿 ↑ ((𝑂‘𝑋)‘𝑦)) = ((𝑂‘𝑋)‘(𝐿 ↑ 𝑦)))
39	38	ralrimiva 3154	. 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 20477	. . . . . 6 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
42	7, 41	syl 18	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring)
43		eqid 2760	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
44	2, 4, 43	vr1cl 22540	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘𝑆))
45	42, 44	syl 18	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑆))
46	45, 5	eleqtrrdi 2871	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
47	40, 46, 26	aks6d1c1p1 43250	. 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 1569 ∈ wcel 2144 ∀wral 3076 class class class wbr 5102 {copab 5166 ‘cfv 6538 (class class class)co 7420 1c1 11204 ℕcn 12338 ℕ₀cn0 12609 ∥ cdvds 16427 gcd cgcd 16669 ℙcprime 16851 Basecbs 17392 0_gc0g 17615 Mndcmnd 18928 ._gcmg 19282 CMndccmn 19999 mulGrpcmgp 20365 Ringcrg 20464 CRingccrg 20465 Fieldcfield 20986 chrcchr 21812 var₁cv1 22499 Poly₁cpl1 22500 eval₁ce1 22637 PrimRoots cprimroots 43234

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-sup 9437 df-oi 9507 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-fz 13643 df-fzo 13792 df-seq 14148 df-hash 14478 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-hom 17457 df-cco 17458 df-0g 17617 df-gsum 17618 df-prds 17623 df-pws 17625 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-mhm 18983 df-submnd 18984 df-grp 19152 df-minusg 19153 df-sbg 19154 df-mulg 19283 df-subg 19338 df-ghm 19433 df-cntz 19536 df-cmn 20001 df-abl 20002 df-mgp 20366 df-rng 20380 df-ur 20413 df-srg 20418 df-ring 20466 df-cring 20467 df-rhm 20707 df-subrng 20803 df-subrg 20827 df-field 20988 df-lmod 21142 df-lss 21212 df-lsp 21252 df-assa 22166 df-asp 22167 df-ascl 22168 df-psr 22222 df-mvr 22223 df-mpl 22224 df-opsr 22226 df-evls 22388 df-evl 22389 df-psr1 22503 df-vr1 22504 df-ply1 22505 df-evl1 22639 df-primroots 43235

This theorem is used by: aks6d1c1 43259

W3C validator