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Theorem modfsummodslem1 15836
Description: Lemma 1 for modfsummods 15837. (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 4685 . . 3 𝑧 ∈ {𝑧}
2 elun2 4200 . . 3 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝐴 ∪ {𝑧}))
31, 2ax-mp 5 . 2 𝑧 ∈ (𝐴 ∪ {𝑧})
4 rspcsbela 4457 . 2 ((𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → 𝑧 / 𝑘𝐵 ∈ ℤ)
53, 4mpan 689 1 (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → 𝑧 / 𝑘𝐵 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2103  wral 3063  csb 3915  cun 3968  {csn 4648  cz 12635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-sn 4649
This theorem is referenced by:  modfsummods  15837
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