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Theorem gsumcllem 15517
Description: Lemma for gsumcl 15522 and related theorems. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
gsumcllem.f  |-  ( ph  ->  F : A --> B )
gsumcllem.s  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  W )
Assertion
Ref Expression
gsumcllem  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
Distinct variable groups:    A, k    k, F    ph, k    k, W
Allowed substitution hints:    B( k)    .0. ( k)

Proof of Theorem gsumcllem
StepHypRef Expression
1 gsumcllem.f . . . 4  |-  ( ph  ->  F : A --> B )
21feqmptd 5780 . . 3  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
32adantr 453 . 2  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( k  e.  A  |->  ( F `  k
) ) )
4 difeq2 3460 . . . . . . . 8  |-  ( W  =  (/)  ->  ( A 
\  W )  =  ( A  \  (/) ) )
5 dif0 3699 . . . . . . . 8  |-  ( A 
\  (/) )  =  A
64, 5syl6eq 2485 . . . . . . 7  |-  ( W  =  (/)  ->  ( A 
\  W )  =  A )
76eleq2d 2504 . . . . . 6  |-  ( W  =  (/)  ->  ( k  e.  ( A  \  W )  <->  k  e.  A ) )
87biimpar 473 . . . . 5  |-  ( ( W  =  (/)  /\  k  e.  A )  ->  k  e.  ( A  \  W
) )
9 gsumcllem.s . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  W )
101, 9suppssr 5865 . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  .0.  )
118, 10sylan2 462 . . . 4  |-  ( (
ph  /\  ( W  =  (/)  /\  k  e.  A ) )  -> 
( F `  k
)  =  .0.  )
1211anassrs 631 . . 3  |-  ( ( ( ph  /\  W  =  (/) )  /\  k  e.  A )  ->  ( F `  k )  =  .0.  )
1312mpteq2dva 4296 . 2  |-  ( (
ph  /\  W  =  (/) )  ->  ( k  e.  A  |->  ( F `
 k ) )  =  ( k  e.  A  |->  .0.  ) )
143, 13eqtrd 2469 1  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2957    \ cdif 3318    C_ wss 3321   (/)c0 3629   {csn 3815    e. cmpt 4267   `'ccnv 4878   "cima 4882   -->wf 5451   ` cfv 5455
This theorem is referenced by:  gsumzres  15518  gsumzcl  15519  gsumzf1o  15520  gsumzaddlem  15527  gsumzmhm  15534  gsumzoppg  15540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463
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