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Metamath Proof Explorer

Theorem fsumcllem

Description: - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005) (Revised by Mario Carneiro, 3-Jun-2014)

Ref Expression
Hypotheses fsumcllem.1 
|- ( ph -> S C_ CC )
fsumcllem.2 
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
fsumcllem.3 
|- ( ph -> A e. Fin )
fsumcllem.4 
|- ( ( ph /\ k e. A ) -> B e. S )
fsumcllem.5 
|- ( ph -> 0 e. S )
Assertion fsumcllem 
|- ( ph -> sum_ k e. A B e. S )

Proof

Step Hyp Ref Expression
1 fsumcllem.1 
 |-  ( ph -> S C_ CC )
2 fsumcllem.2 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
3 fsumcllem.3 
 |-  ( ph -> A e. Fin )
4 fsumcllem.4 
 |-  ( ( ph /\ k e. A ) -> B e. S )
5 fsumcllem.5 
 |-  ( ph -> 0 e. S )
6 simpr 
 |-  ( ( ph /\ A = (/) ) -> A = (/) )
7 6 sumeq1d 
 |-  ( ( ph /\ A = (/) ) -> sum_ k e. A B = sum_ k e. (/) B )
8 sum0 
 |-  sum_ k e. (/) B = 0
9 7 8 eqtrdi 
 |-  ( ( ph /\ A = (/) ) -> sum_ k e. A B = 0 )
10 5 adantr 
 |-  ( ( ph /\ A = (/) ) -> 0 e. S )
11 9 10 eqeltrd 
 |-  ( ( ph /\ A = (/) ) -> sum_ k e. A B e. S )
12 1 adantr 
 |-  ( ( ph /\ A =/= (/) ) -> S C_ CC )
13 2 adantlr 
 |-  ( ( ( ph /\ A =/= (/) ) /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
14 3 adantr 
 |-  ( ( ph /\ A =/= (/) ) -> A e. Fin )
15 4 adantlr 
 |-  ( ( ( ph /\ A =/= (/) ) /\ k e. A ) -> B e. S )
16 simpr 
 |-  ( ( ph /\ A =/= (/) ) -> A =/= (/) )
17 12 13 14 15 16 fsumcl2lem 
 |-  ( ( ph /\ A =/= (/) ) -> sum_ k e. A B e. S )
18 11 17 pm2.61dane 
 |-  ( ph -> sum_ k e. A B e. S )