Metamath Proof Explorer


Theorem prdsmslem1

Description: Lemma for prdsms . The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015)

Ref Expression
Hypotheses prdsxms.y 𝑌 = ( 𝑆 Xs 𝑅 )
prdsxms.s ( 𝜑𝑆𝑊 )
prdsxms.i ( 𝜑𝐼 ∈ Fin )
prdsxms.d 𝐷 = ( dist ‘ 𝑌 )
prdsxms.b 𝐵 = ( Base ‘ 𝑌 )
prdsms.r ( 𝜑𝑅 : 𝐼 ⟶ MetSp )
Assertion prdsmslem1 ( 𝜑𝐷 ∈ ( Met ‘ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 prdsxms.y 𝑌 = ( 𝑆 Xs 𝑅 )
2 prdsxms.s ( 𝜑𝑆𝑊 )
3 prdsxms.i ( 𝜑𝐼 ∈ Fin )
4 prdsxms.d 𝐷 = ( dist ‘ 𝑌 )
5 prdsxms.b 𝐵 = ( Base ‘ 𝑌 )
6 prdsms.r ( 𝜑𝑅 : 𝐼 ⟶ MetSp )
7 eqid ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) = ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) )
8 eqid ( Base ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) ) = ( Base ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) )
9 eqid ( Base ‘ ( 𝑅𝑘 ) ) = ( Base ‘ ( 𝑅𝑘 ) )
10 eqid ( ( dist ‘ ( 𝑅𝑘 ) ) ↾ ( ( Base ‘ ( 𝑅𝑘 ) ) × ( Base ‘ ( 𝑅𝑘 ) ) ) ) = ( ( dist ‘ ( 𝑅𝑘 ) ) ↾ ( ( Base ‘ ( 𝑅𝑘 ) ) × ( Base ‘ ( 𝑅𝑘 ) ) ) )
11 eqid ( dist ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) ) = ( dist ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) )
12 6 ffvelrnda ( ( 𝜑𝑘𝐼 ) → ( 𝑅𝑘 ) ∈ MetSp )
13 9 10 msmet ( ( 𝑅𝑘 ) ∈ MetSp → ( ( dist ‘ ( 𝑅𝑘 ) ) ↾ ( ( Base ‘ ( 𝑅𝑘 ) ) × ( Base ‘ ( 𝑅𝑘 ) ) ) ) ∈ ( Met ‘ ( Base ‘ ( 𝑅𝑘 ) ) ) )
14 12 13 syl ( ( 𝜑𝑘𝐼 ) → ( ( dist ‘ ( 𝑅𝑘 ) ) ↾ ( ( Base ‘ ( 𝑅𝑘 ) ) × ( Base ‘ ( 𝑅𝑘 ) ) ) ) ∈ ( Met ‘ ( Base ‘ ( 𝑅𝑘 ) ) ) )
15 7 8 9 10 11 2 3 12 14 prdsmet ( 𝜑 → ( dist ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) ) ∈ ( Met ‘ ( Base ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) ) ) )
16 6 feqmptd ( 𝜑𝑅 = ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) )
17 16 oveq2d ( 𝜑 → ( 𝑆 Xs 𝑅 ) = ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) )
18 1 17 syl5eq ( 𝜑𝑌 = ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) )
19 18 fveq2d ( 𝜑 → ( dist ‘ 𝑌 ) = ( dist ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) ) )
20 4 19 syl5eq ( 𝜑𝐷 = ( dist ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) ) )
21 18 fveq2d ( 𝜑 → ( Base ‘ 𝑌 ) = ( Base ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) ) )
22 5 21 syl5eq ( 𝜑𝐵 = ( Base ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) ) )
23 22 fveq2d ( 𝜑 → ( Met ‘ 𝐵 ) = ( Met ‘ ( Base ‘ ( 𝑆 Xs ( 𝑘𝐼 ↦ ( 𝑅𝑘 ) ) ) ) ) )
24 15 20 23 3eltr4d ( 𝜑𝐷 ∈ ( Met ‘ 𝐵 ) )