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Theorem gsumcllem 15193
Description: Lemma for gsumcl 15198 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 5575 . . 3  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
32adantr 451 . 2  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( k  e.  A  |->  ( F `  k
) ) )
4 difeq2 3288 . . . . . . . 8  |-  ( W  =  (/)  ->  ( A 
\  W )  =  ( A  \  (/) ) )
5 dif0 3524 . . . . . . . 8  |-  ( A 
\  (/) )  =  A
64, 5syl6eq 2331 . . . . . . 7  |-  ( W  =  (/)  ->  ( A 
\  W )  =  A )
76eleq2d 2350 . . . . . 6  |-  ( W  =  (/)  ->  ( k  e.  ( A  \  W )  <->  k  e.  A ) )
87biimpar 471 . . . . 5  |-  ( ( W  =  (/)  /\  k  e.  A )  ->  k  e.  ( A  \  W
) )
9 gsumcllem.s . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  W )
101, 9suppssr 5659 . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  W ) )  ->  ( F `  k )  =  .0.  )
118, 10sylan2 460 . . . 4  |-  ( (
ph  /\  ( W  =  (/)  /\  k  e.  A ) )  -> 
( F `  k
)  =  .0.  )
1211anassrs 629 . . 3  |-  ( ( ( ph  /\  W  =  (/) )  /\  k  e.  A )  ->  ( F `  k )  =  .0.  )
1312mpteq2dva 4106 . 2  |-  ( (
ph  /\  W  =  (/) )  ->  ( k  e.  A  |->  ( F `
 k ) )  =  ( k  e.  A  |->  .0.  ) )
143, 13eqtrd 2315 1  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692   -->wf 5251   ` cfv 5255
This theorem is referenced by:  gsumzres  15194  gsumzcl  15195  gsumzf1o  15196  gsumzaddlem  15203  gsumzmhm  15210  gsumzoppg  15216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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