Theorem aks6d1c1p1 42089

Description: Definition of the introspective relation. (Contributed by metakunt, 25-Apr-2025.)

Hypotheses

Ref	Expression
aks6d1c1p1.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}
aks6d1c1p1.2	⊢ (𝜑 → 𝐹 ∈ 𝐵)
aks6d1c1p1.3	⊢ (𝜑 → 𝐸 ∈ ℕ)

Assertion

Ref	Expression
aks6d1c1p1	⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))

Distinct variable groups:   ↑ ,𝑒,𝑓   𝐵,𝑒,𝑓   𝐷,𝑒,𝑓   𝑒,𝐸,𝑓,𝑦   𝑒,𝐹,𝑓,𝑦   𝑒,𝐾,𝑓   𝑒,𝑂,𝑓   𝑅,𝑒,𝑓

Allowed substitution hints:   𝜑(𝑦,𝑒,𝑓)   𝐵(𝑦)   𝐷(𝑦)   ∼ (𝑦,𝑒,𝑓)   𝑅(𝑦)   ↑ (𝑦)   𝐾(𝑦)   𝑂(𝑦)

Proof of Theorem aks6d1c1p1

Step	Hyp	Ref	Expression
1		aks6d1c1p1.3	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℕ)
2		aks6d1c1p1.2	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
3		simpl 482	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸)
4	3	eleq1d 2824	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ∈ ℕ ↔ 𝐸 ∈ ℕ))
5		simpr 484	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
6	5	eleq1d 2824	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑓 ∈ 𝐵 ↔ 𝐹 ∈ 𝐵))
7	5	fveq2d 6911	. . . . . . . . . . . . 13 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑂‘𝑓) = (𝑂‘𝐹))
8	7	fveq1d 6909	. . . . . . . . . . . 12 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑂‘𝑓)‘𝑦) = ((𝑂‘𝐹)‘𝑦))
9	3, 8	oveq12d 7449	. . . . . . . . . . 11 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)))
10	3	oveq1d 7446	. . . . . . . . . . . 12 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒𝐷𝑦) = (𝐸𝐷𝑦))
11	7, 10	fveq12d 6914	. . . . . . . . . . 11 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑂‘𝑓)‘(𝑒𝐷𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))
12	9, 11	eqeq12d 2751	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)) ↔ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
13	12	ralbidv 3176	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)) ↔ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
14	4, 6, 13	3anbi123d 1435	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦))) ↔ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
15		aks6d1c1p1.1	. . . . . . . 8 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}
16	14, 15	brabga 5544	. . . . . . 7 ⊢ ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) → (𝐸 ∼ 𝐹 ↔ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
17	1, 2, 16	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
18	17	biimpd 229	. . . . 5 ⊢ (𝜑 → (𝐸 ∼ 𝐹 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
19	18	imp 406	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∼ 𝐹) → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
20	19	simp3d 1143	. . 3 ⊢ ((𝜑 ∧ 𝐸 ∼ 𝐹) → ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))
21	20	ex 412	. 2 ⊢ (𝜑 → (𝐸 ∼ 𝐹 → ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
22	1, 2	jca 511	. . 3 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
23		df-3an 1088	. . . . . . . . 9 ⊢ ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) ↔ ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
24	23	bicomi 224	. . . . . . . 8 ⊢ (((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) ↔ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → (((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) ↔ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
26	17	biimprd 248	. . . . . . 7 ⊢ (𝜑 → ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹))
27	25, 26	sylbid 240	. . . . . 6 ⊢ (𝜑 → (((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹))
28	27	imp 406	. . . . 5 ⊢ ((𝜑 ∧ ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))) → 𝐸 ∼ 𝐹)
29	28	anassrs 467	. . . 4 ⊢ (((𝜑 ∧ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵)) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹)
30	29	ex 412	. . 3 ⊢ ((𝜑 ∧ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵)) → (∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)) → 𝐸 ∼ 𝐹))
31	22, 30	mpdan 687	. 2 ⊢ (𝜑 → (∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)) → 𝐸 ∼ 𝐹))
32	21, 31	impbid 212	1 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059   class class class wbr 5148  {copab 5210  ‘cfv 6563  (class class class)co 7431  ℕcn 12264   PrimRoots cprimroots 42073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571  df-ov 7434

This theorem is referenced by:  aks6d1c1p2  42091  aks6d1c1p3  42092  aks6d1c1p4  42093  aks6d1c1p5  42094  aks6d1c1p7  42095  aks6d1c1p6  42096  aks6d1c1p8  42097  aks6d1c2lem3  42108  aks6d1c6lem2  42153  aks5lem5a  42173