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Metamath Proof Explorer

Theorem fprodcllem

Description: Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017)

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

Proof

Step Hyp Ref Expression
1 fprodcllem.1 
 |-  ( ph -> S C_ CC )
2 fprodcllem.2 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
3 fprodcllem.3 
 |-  ( ph -> A e. Fin )
4 fprodcllem.4 
 |-  ( ( ph /\ k e. A ) -> B e. S )
5 fprodcllem.5 
 |-  ( ph -> 1 e. S )
6 prodeq1 
 |-  ( A = (/) -> prod_ k e. A B = prod_ k e. (/) B )
7 prod0 
 |-  prod_ k e. (/) B = 1
8 6 7 eqtrdi 
 |-  ( A = (/) -> prod_ k e. A B = 1 )
9 8 adantl 
 |-  ( ( ph /\ A = (/) ) -> prod_ k e. A B = 1 )
10 5 adantr 
 |-  ( ( ph /\ A = (/) ) -> 1 e. S )
11 9 10 eqeltrd 
 |-  ( ( ph /\ A = (/) ) -> prod_ 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 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 fprodcl2lem 
 |-  ( ( ph /\ A =/= (/) ) -> prod_ k e. A B e. S )
18 11 17 pm2.61dane 
 |-  ( ph -> prod_ k e. A B e. S )