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Theorem modfsummodslem1 13901
Description: Lemma 1 for modfsummods 13902. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
Assertion
Ref Expression
modfsummodslem1  |-  ( A. k  e.  ( A  u.  { z } ) B  e.  ZZ  ->  [_ z  /  k ]_ B  e.  ZZ )
Distinct variable groups:    A, k    z, k
Allowed substitution hints:    A( z)    B( z, k)

Proof of Theorem modfsummodslem1
StepHypRef Expression
1 ssnid 4009 . . 3  |-  z  e. 
{ z }
2 elun2 3614 . . 3  |-  ( z  e.  { z }  ->  z  e.  ( A  u.  { z } ) )
31, 2ax-mp 5 . 2  |-  z  e.  ( A  u.  {
z } )
4 rspcsbela 3807 . 2  |-  ( ( z  e.  ( A  u.  { z } )  /\  A. k  e.  ( A  u.  {
z } ) B  e.  ZZ )  ->  [_ z  /  k ]_ B  e.  ZZ )
53, 4mpan 681 1  |-  ( A. k  e.  ( A  u.  { z } ) B  e.  ZZ  ->  [_ z  /  k ]_ B  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1898   A.wral 2749   [_csb 3375    u. cun 3414   {csn 3980   ZZcz 10966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-sn 3981
This theorem is referenced by:  modfsummods  13902
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