Theorem gsumcllem 18309

Description: Lemma for gsumcl 18316 and related theorems. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)

Hypotheses

Ref	Expression
gsumcllem.f	|- ( ph -> F : A --> B )
gsumcllem.a	|- ( ph -> A e. V )
gsumcllem.z	|- ( ph -> Z e. U )
gsumcllem.s	|- ( ph -> ( F supp Z ) C_ W )

Assertion

Ref	Expression
gsumcllem	|- ( ( ph /\ W = (/) ) -> F = ( k e. A |-> Z ) )

Distinct variable groups:   A, k   k, F   ph, k   k, W

Allowed substitution hints:   B( k)   U( k)   V( k)   Z( k)

Proof of Theorem gsumcllem

Step	Hyp	Ref	Expression
1		gsumcllem.f	. . . 4 |- ( ph -> F : A --> B )
2	1	feqmptd 6249	. . 3 |- ( ph -> F = ( k e. A |-> ( F ` k ) ) )
3	2	adantr 481	. 2 |- ( ( ph /\ W = (/) ) -> F = ( k e. A |-> ( F ` k ) ) )
4		difeq2 3722	. . . . . . . 8 |- ( W = (/) -> ( A \ W ) = ( A \ (/) ) )
5		dif0 3950	. . . . . . . 8 |- ( A \ (/) ) = A
6	4, 5	syl6eq 2672	. . . . . . 7 |- ( W = (/) -> ( A \ W ) = A )
7	6	eleq2d 2687	. . . . . 6 |- ( W = (/) -> ( k e. ( A \ W ) <-> k e. A ) )
8	7	biimpar 502	. . . . 5 |- ( ( W = (/) /\ k e. A ) -> k e. ( A \ W ) )
9		gsumcllem.s	. . . . . 6 |- ( ph -> ( F supp Z ) C_ W )
10		gsumcllem.a	. . . . . 6 |- ( ph -> A e. V )
11		gsumcllem.z	. . . . . 6 |- ( ph -> Z e. U )
12	1, 9, 10, 11	suppssr 7326	. . . . 5 |- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )
13	8, 12	sylan2 491	. . . 4 |- ( ( ph /\ ( W = (/) /\ k e. A ) ) -> ( F ` k ) = Z )
14	13	anassrs 680	. . 3 |- ( ( ( ph /\ W = (/) ) /\ k e. A ) -> ( F ` k ) = Z )
15	14	mpteq2dva 4744	. 2 |- ( ( ph /\ W = (/) ) -> ( k e. A |-> ( F ` k ) ) = ( k e. A |-> Z ) )
16	3, 15	eqtrd 2656	1 |- ( ( ph /\ W = (/) ) -> F = ( k e. A |-> Z ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  gsumzres  18310  gsumzcl2  18311  gsumzf1o  18313  gsumzaddlem  18321  gsumzmhm  18337  gsumzoppg  18344