Theorem modfsummodslem1 15750

Description: Lemma 1 for modfsummods 15751. (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 4598	. . 3 ⊢ 𝑧 ∈ {𝑧}
2		elun2 4115	. . 3 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧}))
3	1, 2	ax-mp 5	. 2 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧})
4		rspcsbela 4369	. 2 ⊢ ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)
5	3, 4	mpan 697	1 ⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2121  ∀wral 3055  ⦋csb 3833   ∪ cun 3883  {csn 4558  ℤcz 12519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-sn 4559

This theorem is referenced by:  modfsummods  15751