Theorem modfsummodslem1 15713

Description: Lemma 1 for modfsummods 15714. (Contributed by Alexander van der Vekens, 1-Sep-2018.)

Assertion

Ref	Expression
modfsummodslem1	⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)

Distinct variable groups:   𝐴,𝑘   𝑧,𝑘

Allowed substitution hints:   𝐴(𝑧)   𝐵(𝑧,𝑘)

Proof of Theorem modfsummodslem1

Step	Hyp	Ref	Expression
1		vsnid 4618	. . 3 ⊢ 𝑧 ∈ {𝑧}
2		elun2 4133	. . 3 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧}))
3	1, 2	ax-mp 5	. 2 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧})
4		rspcsbela 4388	. 2 ⊢ ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)
5	3, 4	mpan 690	1 ⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2113  ∀wral 3049  ⦋csb 3847   ∪ cun 3897  {csn 4578  ℤcz 12486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-sn 4579

This theorem is referenced by:  modfsummods  15714