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Theorem modfsummodslem1 14446
Description: Lemma 1 for modfsummods 14447. (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 4185 . . 3 𝑧 ∈ {𝑧}
2 elun2 3764 . . 3 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧}))
31, 2ax-mp 5 . 2 𝑧 ∈ (𝐴 ∪ {𝑧})
4 rspcsbela 3983 . 2 ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → 𝑧 / 𝑘𝐵 ∈ ℤ)
53, 4mpan 705 1 (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → 𝑧 / 𝑘𝐵 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1992  wral 2912  csb 3519  cun 3558  {csn 4153  cz 11322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-sn 4154
This theorem is referenced by:  modfsummods  14447
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