Metamath Proof Explorer


Theorem prdsxmet

Description: The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem . (Contributed by Mario Carneiro, 26-Sep-2015)

Ref Expression
Hypotheses prdsdsf.y 𝑌 = ( 𝑆 Xs ( 𝑥𝐼𝑅 ) )
prdsdsf.b 𝐵 = ( Base ‘ 𝑌 )
prdsdsf.v 𝑉 = ( Base ‘ 𝑅 )
prdsdsf.e 𝐸 = ( ( dist ‘ 𝑅 ) ↾ ( 𝑉 × 𝑉 ) )
prdsdsf.d 𝐷 = ( dist ‘ 𝑌 )
prdsdsf.s ( 𝜑𝑆𝑊 )
prdsdsf.i ( 𝜑𝐼𝑋 )
prdsdsf.r ( ( 𝜑𝑥𝐼 ) → 𝑅𝑍 )
prdsdsf.m ( ( 𝜑𝑥𝐼 ) → 𝐸 ∈ ( ∞Met ‘ 𝑉 ) )
Assertion prdsxmet ( 𝜑𝐷 ∈ ( ∞Met ‘ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 prdsdsf.y 𝑌 = ( 𝑆 Xs ( 𝑥𝐼𝑅 ) )
2 prdsdsf.b 𝐵 = ( Base ‘ 𝑌 )
3 prdsdsf.v 𝑉 = ( Base ‘ 𝑅 )
4 prdsdsf.e 𝐸 = ( ( dist ‘ 𝑅 ) ↾ ( 𝑉 × 𝑉 ) )
5 prdsdsf.d 𝐷 = ( dist ‘ 𝑌 )
6 prdsdsf.s ( 𝜑𝑆𝑊 )
7 prdsdsf.i ( 𝜑𝐼𝑋 )
8 prdsdsf.r ( ( 𝜑𝑥𝐼 ) → 𝑅𝑍 )
9 prdsdsf.m ( ( 𝜑𝑥𝐼 ) → 𝐸 ∈ ( ∞Met ‘ 𝑉 ) )
10 nfcv 𝑦 𝑅
11 nfcsb1v 𝑥 𝑦 / 𝑥 𝑅
12 csbeq1a ( 𝑥 = 𝑦𝑅 = 𝑦 / 𝑥 𝑅 )
13 10 11 12 cbvmpt ( 𝑥𝐼𝑅 ) = ( 𝑦𝐼 𝑦 / 𝑥 𝑅 )
14 13 oveq2i ( 𝑆 Xs ( 𝑥𝐼𝑅 ) ) = ( 𝑆 Xs ( 𝑦𝐼 𝑦 / 𝑥 𝑅 ) )
15 1 14 eqtri 𝑌 = ( 𝑆 Xs ( 𝑦𝐼 𝑦 / 𝑥 𝑅 ) )
16 eqid ( Base ‘ 𝑦 / 𝑥 𝑅 ) = ( Base ‘ 𝑦 / 𝑥 𝑅 )
17 eqid ( ( dist ‘ 𝑦 / 𝑥 𝑅 ) ↾ ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) ) = ( ( dist ‘ 𝑦 / 𝑥 𝑅 ) ↾ ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) )
18 8 elexd ( ( 𝜑𝑥𝐼 ) → 𝑅 ∈ V )
19 18 ralrimiva ( 𝜑 → ∀ 𝑥𝐼 𝑅 ∈ V )
20 11 nfel1 𝑥 𝑦 / 𝑥 𝑅 ∈ V
21 12 eleq1d ( 𝑥 = 𝑦 → ( 𝑅 ∈ V ↔ 𝑦 / 𝑥 𝑅 ∈ V ) )
22 20 21 rspc ( 𝑦𝐼 → ( ∀ 𝑥𝐼 𝑅 ∈ V → 𝑦 / 𝑥 𝑅 ∈ V ) )
23 19 22 mpan9 ( ( 𝜑𝑦𝐼 ) → 𝑦 / 𝑥 𝑅 ∈ V )
24 9 ralrimiva ( 𝜑 → ∀ 𝑥𝐼 𝐸 ∈ ( ∞Met ‘ 𝑉 ) )
25 nfcv 𝑥 dist
26 25 11 nffv 𝑥 ( dist ‘ 𝑦 / 𝑥 𝑅 )
27 nfcv 𝑥 Base
28 27 11 nffv 𝑥 ( Base ‘ 𝑦 / 𝑥 𝑅 )
29 28 28 nfxp 𝑥 ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) )
30 26 29 nfres 𝑥 ( ( dist ‘ 𝑦 / 𝑥 𝑅 ) ↾ ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) )
31 nfcv 𝑥 ∞Met
32 31 28 nffv 𝑥 ( ∞Met ‘ ( Base ‘ 𝑦 / 𝑥 𝑅 ) )
33 30 32 nfel 𝑥 ( ( dist ‘ 𝑦 / 𝑥 𝑅 ) ↾ ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑦 / 𝑥 𝑅 ) )
34 12 fveq2d ( 𝑥 = 𝑦 → ( dist ‘ 𝑅 ) = ( dist ‘ 𝑦 / 𝑥 𝑅 ) )
35 12 fveq2d ( 𝑥 = 𝑦 → ( Base ‘ 𝑅 ) = ( Base ‘ 𝑦 / 𝑥 𝑅 ) )
36 3 35 eqtrid ( 𝑥 = 𝑦𝑉 = ( Base ‘ 𝑦 / 𝑥 𝑅 ) )
37 36 sqxpeqd ( 𝑥 = 𝑦 → ( 𝑉 × 𝑉 ) = ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) )
38 34 37 reseq12d ( 𝑥 = 𝑦 → ( ( dist ‘ 𝑅 ) ↾ ( 𝑉 × 𝑉 ) ) = ( ( dist ‘ 𝑦 / 𝑥 𝑅 ) ↾ ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) ) )
39 4 38 eqtrid ( 𝑥 = 𝑦𝐸 = ( ( dist ‘ 𝑦 / 𝑥 𝑅 ) ↾ ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) ) )
40 36 fveq2d ( 𝑥 = 𝑦 → ( ∞Met ‘ 𝑉 ) = ( ∞Met ‘ ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) )
41 39 40 eleq12d ( 𝑥 = 𝑦 → ( 𝐸 ∈ ( ∞Met ‘ 𝑉 ) ↔ ( ( dist ‘ 𝑦 / 𝑥 𝑅 ) ↾ ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) ) )
42 33 41 rspc ( 𝑦𝐼 → ( ∀ 𝑥𝐼 𝐸 ∈ ( ∞Met ‘ 𝑉 ) → ( ( dist ‘ 𝑦 / 𝑥 𝑅 ) ↾ ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) ) )
43 24 42 mpan9 ( ( 𝜑𝑦𝐼 ) → ( ( dist ‘ 𝑦 / 𝑥 𝑅 ) ↾ ( ( Base ‘ 𝑦 / 𝑥 𝑅 ) × ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑦 / 𝑥 𝑅 ) ) )
44 15 2 16 17 5 6 7 23 43 prdsxmetlem ( 𝜑𝐷 ∈ ( ∞Met ‘ 𝐵 ) )