Theorem aks6d1c1p1 41610

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 481	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸)
4	3	eleq1d 2814	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ∈ ℕ ↔ 𝐸 ∈ ℕ))
5		simpr 483	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
6	5	eleq1d 2814	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑓 ∈ 𝐵 ↔ 𝐹 ∈ 𝐵))
7	5	fveq2d 6906	. . . . . . . . . . . . 13 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑂‘𝑓) = (𝑂‘𝐹))
8	7	fveq1d 6904	. . . . . . . . . . . 12 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑂‘𝑓)‘𝑦) = ((𝑂‘𝐹)‘𝑦))
9	3, 8	oveq12d 7444	. . . . . . . . . . 11 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)))
10	3	oveq1d 7441	. . . . . . . . . . . 12 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒𝐷𝑦) = (𝐸𝐷𝑦))
11	7, 10	fveq12d 6909	. . . . . . . . . . 11 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑂‘𝑓)‘(𝑒𝐷𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))
12	9, 11	eqeq12d 2744	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)) ↔ (𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
13	12	ralbidv 3175	. . . . . . . . 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 1432	. . . . . . . 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 5540	. . . . . . 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 582	. . . . . 6 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
18	17	biimpd 228	. . . . 5 ⊢ (𝜑 → (𝐸 ∼ 𝐹 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))))
19	18	imp 405	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∼ 𝐹) → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
20	19	simp3d 1141	. . 3 ⊢ ((𝜑 ∧ 𝐸 ∼ 𝐹) → ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))
21	20	ex 411	. 2 ⊢ (𝜑 → (𝐸 ∼ 𝐹 → ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
22	1, 2	jca 510	. . 3 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
23		df-3an 1086	. . . . . . . . 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 223	. . . . . . . 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 247	. . . . . . 7 ⊢ (𝜑 → ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹))
27	25, 26	sylbid 239	. . . . . 6 ⊢ (𝜑 → (((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹))
28	27	imp 405	. . . . 5 ⊢ ((𝜑 ∧ ((𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))) → 𝐸 ∼ 𝐹)
29	28	anassrs 466	. . . 4 ⊢ (((𝜑 ∧ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵)) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))) → 𝐸 ∼ 𝐹)
30	29	ex 411	. . 3 ⊢ ((𝜑 ∧ (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵)) → (∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)) → 𝐸 ∼ 𝐹))
31	22, 30	mpdan 685	. 2 ⊢ (𝜑 → (∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)) → 𝐸 ∼ 𝐹))
32	21, 31	impbid 211	1 ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3058   class class class wbr 5152  {copab 5214  ‘cfv 6553  (class class class)co 7426  ℕcn 12250   PrimRoots cprimroots 41594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-iota 6505  df-fv 6561  df-ov 7429

This theorem is referenced by:  aks6d1c1p2  41612  aks6d1c1p3  41613  aks6d1c1p4  41614  aks6d1c1p5  41615  aks6d1c1p7  41616  aks6d1c1p6  41617  aks6d1c1p8  41618  aks6d1c2lem3  41629  aks6d1c6lem2  41675