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Theorem modfsummodlem1 11599
Description: Lemma for modfsummod 11601. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
Assertion
Ref Expression
modfsummodlem1 (∀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 modfsummodlem1
StepHypRef Expression
1 vsnid 3650 . . 3 𝑧 ∈ {𝑧}
2 elun2 3327 . . 3 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧}))
31, 2ax-mp 5 . 2 𝑧 ∈ (𝐴 ∪ {𝑧})
4 rspcsbela 3140 . 2 ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → 𝑧 / 𝑘𝐵 ∈ ℤ)
53, 4mpan 424 1 (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → 𝑧 / 𝑘𝐵 ∈ ℤ)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2164  wral 2472  csb 3080  cun 3151  {csn 3618  cz 9317
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624
This theorem is referenced by:  modfsummodlemstep  11600
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