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Theorem modfsummodslem1 13608
Description: Lemma 1 for modfsummods 13609. (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 3973 . . 3  |-  z  e. 
{ z }
2 elun2 3586 . . 3  |-  ( z  e.  { z }  ->  z  e.  ( A  u.  { z } ) )
31, 2ax-mp 5 . 2  |-  z  e.  ( A  u.  {
z } )
4 rspcsbela 3773 . 2  |-  ( ( z  e.  ( A  u.  { z } )  /\  A. k  e.  ( A  u.  {
z } ) B  e.  ZZ )  ->  [_ z  /  k ]_ B  e.  ZZ )
53, 4mpan 668 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 1826   A.wral 2732   [_csb 3348    u. cun 3387   {csn 3944   ZZcz 10781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-sn 3945
This theorem is referenced by:  modfsummods  13609
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