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Theorem modfsummodslem1 14568
Description: Lemma 1 for modfsummods 14569. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
Assertion
Ref Expression
modfsummodslem1 (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → 𝑧 / 𝑘𝐵 ∈ ℤ)
Distinct variable groups:   𝐴,𝑘   𝑧,𝑘
Allowed substitution hints:   𝐴(𝑧)   𝐵(𝑧,𝑘)

Proof of Theorem modfsummodslem1
StepHypRef Expression
1 vsnid 4242 . . 3 𝑧 ∈ {𝑧}
2 elun2 3814 . . 3 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧}))
31, 2ax-mp 5 . 2 𝑧 ∈ (𝐴 ∪ {𝑧})
4 rspcsbela 4039 . 2 ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → 𝑧 / 𝑘𝐵 ∈ ℤ)
53, 4mpan 706 1 (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → 𝑧 / 𝑘𝐵 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wral 2941  csb 3566  cun 3605  {csn 4210  cz 11415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211
This theorem is referenced by:  modfsummods  14569
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