Theorem aks6d1c1p1 42729

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 486	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸)
4	3	eleq1d 2849	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ∈ ℕ ↔ 𝐸 ∈ ℕ))
5		simpr 488	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
6	5	eleq1d 2849	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑓 ∈ 𝐵 ↔ 𝐹 ∈ 𝐵))
7	5	fveq2d 6873	. . . . . . . . . . . . 13 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑂‘𝑓) = (𝑂‘𝐹))
8	7	fveq1d 6871	. . . . . . . . . . . 12 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑂‘𝑓)‘𝑦) = ((𝑂‘𝐹)‘𝑦))
9	3, 8	oveq12d 7416	. . . . . . . . . . 11 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)))
10	3	oveq1d 7413	. . . . . . . . . . . 12 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒𝐷𝑦) = (𝐸𝐷𝑦))
11	7, 10	fveq12d 6876	. . . . . . . . . . 11 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑂‘𝑓)‘(𝑒𝐷𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))
12	9, 11	eqeq12d 2780	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)) ↔ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
13	12	ralbidv 3187	. . . . . . . . 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 1459	. . . . . . . 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 5506	. . . . . . 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 593	. . . . . 6 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
18	17	biimpd 231	. . . . 5 ⊢ (𝜑 → (𝐸 ∼ 𝐹 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
19	18	imp 410	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∼ 𝐹) → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
20	19	simp3d 1158	. . 3 ⊢ ((𝜑 ∧ 𝐸 ∼ 𝐹) → ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))
21	20	ex 416	. 2 ⊢ (𝜑 → (𝐸 ∼ 𝐹 → ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
22	1, 2	jca 519	. . 3 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
23		df-3an 1101	. . . . . . . . 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 226	. . . . . . . 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 250	. . . . . . 7 ⊢ (𝜑 → ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹))
27	25, 26	sylbid 242	. . . . . 6 ⊢ (𝜑 → (((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹))
28	27	imp 410	. . . . 5 ⊢ ((𝜑 ∧ ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))) → 𝐸 ∼ 𝐹)
29	28	anassrs 471	. . . 4 ⊢ (((𝜑 ∧ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵)) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹)
30	29	ex 416	. . 3 ⊢ ((𝜑 ∧ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵)) → (∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)) → 𝐸 ∼ 𝐹))
31	22, 30	mpdan 697	. 2 ⊢ (𝜑 → (∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)) → 𝐸 ∼ 𝐹))
32	21, 31	impbid 214	1 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078   class class class wbr 5102  {copab 5164  ‘cfv 6523  (class class class)co 7398  ℕcn 12212   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-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-iota 6479  df-fv 6531  df-ov 7401

This theorem is referenced by:  aks6d1c1p2  42731  aks6d1c1p3  42732  aks6d1c1p4  42733  aks6d1c1p5  42734  aks6d1c1p7  42735  aks6d1c1p6  42736  aks6d1c1p8  42737  aks6d1c2lem3  42748  aks6d1c6lem2  42793  aks5lem5a  42813