Theorem modfsummodlem1 11933

Description: Lemma for modfsummod 11935. (Contributed by Alexander van der Vekens, 1-Sep-2018.)

Assertion

Ref	Expression
modfsummodlem1	⊢ (∀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 modfsummodlem1

Step	Hyp	Ref	Expression
1		vsnid 3678	. . 3 ⊢ 𝑧 ∈ {𝑧}
2		elun2 3352	. . 3 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧}))
3	1, 2	ax-mp 5	. 2 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧})
4		rspcsbela 3164	. 2 ⊢ ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)
5	3, 4	mpan 424	1 ⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2180  ∀wral 2488  ⦋csb 3104   ∪ cun 3175  {csn 3646  ℤcz 9414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-sn 3652

This theorem is referenced by:  modfsummodlemstep  11934