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Theorem modfsummodlem1 11459
Description: Lemma for modfsummod 11461. (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 3624 . . 3 𝑧 ∈ {𝑧}
2 elun2 3303 . . 3 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧}))
31, 2ax-mp 5 . 2 𝑧 ∈ (𝐴 ∪ {𝑧})
4 rspcsbela 3116 . 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 2148  wral 2455  csb 3057  cun 3127  {csn 3592  cz 9251
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598
This theorem is referenced by:  modfsummodlemstep  11460
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