Theorem prjsprel 40443

Description: Utility theorem regarding the relation used in ℙ𝕣𝕠𝕛. (Contributed by Steven Nguyen, 29-Apr-2023.)

Hypothesis

Ref	Expression
prjsprel.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}

Assertion

Ref	Expression
prjsprel	⊢ (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑌)))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦,𝑙,𝑚   𝑥,𝑌,𝑦,𝑙,𝑚   𝑥,𝐾,𝑦,𝑙,𝑚   𝑥, · ,𝑦,𝑙,𝑚

Allowed substitution hints:   𝐵(𝑚,𝑙)   ∼ (𝑥,𝑦,𝑚,𝑙)

Proof of Theorem prjsprel

Step	Hyp	Ref	Expression
1		simpll 764	. . . 4 ⊢ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑥 = 𝑋)
2		simpr 485	. . . . 5 ⊢ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑙 = 𝑚)
3		simplr 766	. . . . 5 ⊢ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑦 = 𝑌)
4	2, 3	oveq12d 7293	. . . 4 ⊢ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → (𝑙 · 𝑦) = (𝑚 · 𝑌))
5	1, 4	eqeq12d 2754	. . 3 ⊢ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → (𝑥 = (𝑙 · 𝑦) ↔ 𝑋 = (𝑚 · 𝑌)))
6	5	cbvrexdva 3395	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦) ↔ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑌)))
7		prjsprel.1	. 2 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}
8	6, 7	brab2a 5680	1 ⊢ (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑌)))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   class class class wbr 5074  {copab 5136  (class class class)co 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  prjspertr  40444  prjsperref  40445  prjspersym  40446  prjspreln0  40448  prjspvs  40449  prjspner1  40463  0prjspnrel  40464