Theorem aks6d1c1p1 43158

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 488	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸)
4	3	eleq1d 2845	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ∈ ℕ ↔ 𝐸 ∈ ℕ))
5		simpr 490	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
6	5	eleq1d 2845	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑓 ∈ 𝐵 ↔ 𝐹 ∈ 𝐵))
7	5	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑂‘𝑓) = (𝑂‘𝐹))
8	7	fveq1d 6887	. . . . . . . . . . . 12 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑂‘𝑓)‘𝑦) = ((𝑂‘𝐹)‘𝑦))
9	3, 8	oveq12d 7438	. . . . . . . . . . 11 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)))
10	3	oveq1d 7435	. . . . . . . . . . . 12 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒𝐷𝑦) = (𝐸𝐷𝑦))
11	7, 10	fveq12d 6892	. . . . . . . . . . 11 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑂‘𝑓)‘(𝑒𝐷𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))
12	9, 11	eqeq12d 2776	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)) ↔ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
13	12	ralbidv 3185	. . . . . . . . 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 1464	. . . . . . . 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 5507	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
18	17	biimpd 232	. . . . 5 ⊢ (𝜑 → (𝐸 ∼ 𝐹 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
19	18	imp 412	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∼ 𝐹) → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
20	19	simp3d 1162	. . 3 ⊢ ((𝜑 ∧ 𝐸 ∼ 𝐹) → ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))
21	20	ex 418	. 2 ⊢ (𝜑 → (𝐸 ∼ 𝐹 → ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
22	1, 2	jca 521	. . 3 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
23		df-3an 1105	. . . . . . . . 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 227	. . . . . . . 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 251	. . . . . . 7 ⊢ (𝜑 → ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹))
27	25, 26	sylbid 243	. . . . . 6 ⊢ (𝜑 → (((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹))
28	27	imp 412	. . . . 5 ⊢ ((𝜑 ∧ ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))) → 𝐸 ∼ 𝐹)
29	28	anassrs 473	. . . 4 ⊢ (((𝜑 ∧ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵)) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹)
30	29	ex 418	. . 3 ⊢ ((𝜑 ∧ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵)) → (∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)) → 𝐸 ∼ 𝐹))
31	22, 30	mpdan 700	. 2 ⊢ (𝜑 → (∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)) → 𝐸 ∼ 𝐹))
32	21, 31	impbid 215	1 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102  {copab 5166  ‘cfv 6538  (class class class)co 7420  ℕcn 12335   PrimRoots cprimroots 43142

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-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-iota 6494  df-fv 6546  df-ov 7423

This theorem is used by:  aks6d1c1p2  43160  aks6d1c1p3  43161  aks6d1c1p4  43162  aks6d1c1p5  43163  aks6d1c1p7  43164  aks6d1c1p6  43165  aks6d1c1p8  43166  aks6d1c2lem3  43177  aks6d1c6lem2  43222  aks5lem5a  43242