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Theorem prdsdsf 22365
Description: The product metric is a function into the nonnegative extended reals. In general this means that it is not a metric but rather an *extended* metric (even when all the factors are metrics), but it will be a metric when restricted to regions where it does not take infinite values. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
prdsdsf.y 𝑌 = (𝑆Xs(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
Ref Expression
prdsdsf (𝜑𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))
Distinct variable groups:   𝑥,𝐼   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)   𝑅(𝑥)   𝑆(𝑥)   𝐸(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)   𝑌(𝑥)   𝑍(𝑥)

Proof of Theorem prdsdsf
Dummy variables 𝑓 𝑔 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → 𝑦𝐼)
2 prdsdsf.r . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐼) → 𝑅𝑍)
3 elex 3344 . . . . . . . . . . . . . . . . . 18 (𝑅𝑍𝑅 ∈ V)
42, 3syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐼) → 𝑅 ∈ V)
54ralrimiva 3096 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥𝐼 𝑅 ∈ V)
65adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 𝑅 ∈ V)
7 nfcsb1v 3682 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥𝑅
87nfel1 2909 . . . . . . . . . . . . . . . 16 𝑥𝑦 / 𝑥𝑅 ∈ V
9 csbeq1a 3675 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
109eleq1d 2816 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑅 ∈ V ↔ 𝑦 / 𝑥𝑅 ∈ V))
118, 10rspc 3435 . . . . . . . . . . . . . . 15 (𝑦𝐼 → (∀𝑥𝐼 𝑅 ∈ V → 𝑦 / 𝑥𝑅 ∈ V))
126, 11mpan9 487 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → 𝑦 / 𝑥𝑅 ∈ V)
13 eqid 2752 . . . . . . . . . . . . . . 15 (𝑥𝐼𝑅) = (𝑥𝐼𝑅)
1413fvmpts 6439 . . . . . . . . . . . . . 14 ((𝑦𝐼𝑦 / 𝑥𝑅 ∈ V) → ((𝑥𝐼𝑅)‘𝑦) = 𝑦 / 𝑥𝑅)
151, 12, 14syl2anc 696 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑥𝐼𝑅)‘𝑦) = 𝑦 / 𝑥𝑅)
1615fveq2d 6348 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (dist‘((𝑥𝐼𝑅)‘𝑦)) = (dist‘𝑦 / 𝑥𝑅))
1716oveqd 6822 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦)) = ((𝑓𝑦)(dist‘𝑦 / 𝑥𝑅)(𝑔𝑦)))
18 prdsdsf.y . . . . . . . . . . . . . 14 𝑌 = (𝑆Xs(𝑥𝐼𝑅))
19 prdsdsf.b . . . . . . . . . . . . . 14 𝐵 = (Base‘𝑌)
20 prdsdsf.s . . . . . . . . . . . . . . 15 (𝜑𝑆𝑊)
2120adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑆𝑊)
22 prdsdsf.i . . . . . . . . . . . . . . 15 (𝜑𝐼𝑋)
2322adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝐼𝑋)
24 prdsdsf.v . . . . . . . . . . . . . 14 𝑉 = (Base‘𝑅)
25 simprl 811 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑓𝐵)
2618, 19, 21, 23, 6, 24, 25prdsbascl 16337 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 (𝑓𝑥) ∈ 𝑉)
27 nfcsb1v 3682 . . . . . . . . . . . . . . 15 𝑥𝑦 / 𝑥𝑉
2827nfel2 2911 . . . . . . . . . . . . . 14 𝑥(𝑓𝑦) ∈ 𝑦 / 𝑥𝑉
29 fveq2 6344 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
30 csbeq1a 3675 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦𝑉 = 𝑦 / 𝑥𝑉)
3129, 30eleq12d 2825 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝑉 ↔ (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉))
3228, 31rspc 3435 . . . . . . . . . . . . 13 (𝑦𝐼 → (∀𝑥𝐼 (𝑓𝑥) ∈ 𝑉 → (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉))
3326, 32mpan9 487 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉)
34 simprr 813 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑔𝐵)
3518, 19, 21, 23, 6, 24, 34prdsbascl 16337 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉)
3627nfel2 2911 . . . . . . . . . . . . . 14 𝑥(𝑔𝑦) ∈ 𝑦 / 𝑥𝑉
37 fveq2 6344 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑔𝑥) = (𝑔𝑦))
3837, 30eleq12d 2825 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑔𝑥) ∈ 𝑉 ↔ (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉))
3936, 38rspc 3435 . . . . . . . . . . . . 13 (𝑦𝐼 → (∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉 → (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉))
4035, 39mpan9 487 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉)
4133, 40ovresd 6958 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) = ((𝑓𝑦)(dist‘𝑦 / 𝑥𝑅)(𝑔𝑦)))
4217, 41eqtr4d 2789 . . . . . . . . . 10 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦)) = ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)))
43 prdsdsf.m . . . . . . . . . . . . . 14 ((𝜑𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
4443ralrimiva 3096 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉))
4544adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉))
46 nfcv 2894 . . . . . . . . . . . . . . . 16 𝑥dist
4746, 7nffv 6351 . . . . . . . . . . . . . . 15 𝑥(dist‘𝑦 / 𝑥𝑅)
4827, 27nfxp 5291 . . . . . . . . . . . . . . 15 𝑥(𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)
4947, 48nfres 5545 . . . . . . . . . . . . . 14 𝑥((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))
50 nfcv 2894 . . . . . . . . . . . . . . 15 𝑥∞Met
5150, 27nffv 6351 . . . . . . . . . . . . . 14 𝑥(∞Met‘𝑦 / 𝑥𝑉)
5249, 51nfel 2907 . . . . . . . . . . . . 13 𝑥((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)
53 prdsdsf.e . . . . . . . . . . . . . . 15 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
549fveq2d 6348 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘𝑦 / 𝑥𝑅))
5530sqxpeqd 5290 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑉 × 𝑉) = (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))
5654, 55reseq12d 5544 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)))
5753, 56syl5eq 2798 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐸 = ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)))
5830fveq2d 6348 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘𝑦 / 𝑥𝑉))
5957, 58eleq12d 2825 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐸 ∈ (∞Met‘𝑉) ↔ ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)))
6052, 59rspc 3435 . . . . . . . . . . . 12 (𝑦𝐼 → (∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉) → ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)))
6145, 60mpan9 487 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉))
62 xmetcl 22329 . . . . . . . . . . 11 ((((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉) ∧ (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉 ∧ (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) ∈ ℝ*)
6361, 33, 40, 62syl3anc 1473 . . . . . . . . . 10 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) ∈ ℝ*)
6442, 63eqeltrd 2831 . . . . . . . . 9 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦)) ∈ ℝ*)
65 eqid 2752 . . . . . . . . 9 (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) = (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦)))
6664, 65fmptd 6540 . . . . . . . 8 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))):𝐼⟶ℝ*)
67 frn 6206 . . . . . . . 8 ((𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))):𝐼⟶ℝ* → ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ⊆ ℝ*)
6866, 67syl 17 . . . . . . 7 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ⊆ ℝ*)
69 0xr 10270 . . . . . . . . 9 0 ∈ ℝ*
7069a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 0 ∈ ℝ*)
7170snssd 4477 . . . . . . 7 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → {0} ⊆ ℝ*)
7268, 71unssd 3924 . . . . . 6 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ⊆ ℝ*)
73 supxrcl 12330 . . . . . 6 ((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ⊆ ℝ* → sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ ℝ*)
7472, 73syl 17 . . . . 5 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ ℝ*)
75 ssun2 3912 . . . . . . 7 {0} ⊆ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0})
76 c0ex 10218 . . . . . . . 8 0 ∈ V
7776snss 4452 . . . . . . 7 (0 ∈ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ↔ {0} ⊆ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}))
7875, 77mpbir 221 . . . . . 6 0 ∈ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0})
79 supxrub 12339 . . . . . 6 (((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ⊆ ℝ* ∧ 0 ∈ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0})) → 0 ≤ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ))
8072, 78, 79sylancl 697 . . . . 5 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 0 ≤ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ))
81 elxrge0 12466 . . . . 5 (sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞) ↔ (sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )))
8274, 80, 81sylanbrc 701 . . . 4 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞))
8382ralrimivva 3101 . . 3 (𝜑 → ∀𝑓𝐵𝑔𝐵 sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞))
84 eqid 2752 . . . 4 (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )) = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ))
8584fmpt2 7397 . . 3 (∀𝑓𝐵𝑔𝐵 sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞) ↔ (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞))
8683, 85sylib 208 . 2 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞))
87 mptexg 6640 . . . . 5 (𝐼𝑋 → (𝑥𝐼𝑅) ∈ V)
8822, 87syl 17 . . . 4 (𝜑 → (𝑥𝐼𝑅) ∈ V)
892ralrimiva 3096 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑅𝑍)
90 dmmptg 5785 . . . . 5 (∀𝑥𝐼 𝑅𝑍 → dom (𝑥𝐼𝑅) = 𝐼)
9189, 90syl 17 . . . 4 (𝜑 → dom (𝑥𝐼𝑅) = 𝐼)
92 prdsdsf.d . . . 4 𝐷 = (dist‘𝑌)
9318, 20, 88, 19, 91, 92prdsds 16318 . . 3 (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )))
9493feq1d 6183 . 2 (𝜑 → (𝐷:(𝐵 × 𝐵)⟶(0[,]+∞) ↔ (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞)))
9586, 94mpbird 247 1 (𝜑𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  csb 3666  cun 3705  wss 3707  {csn 4313   class class class wbr 4796  cmpt 4873   × cxp 5256  dom cdm 5258  ran crn 5259  cres 5260  wf 6037  cfv 6041  (class class class)co 6805  cmpt2 6807  supcsup 8503  0cc0 10120  +∞cpnf 10255  *cxr 10257   < clt 10258  cle 10259  [,]cicc 12363  Basecbs 16051  distcds 16144  Xscprds 16300  ∞Metcxmt 19925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8505  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-icc 12367  df-fz 12512  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-plusg 16148  df-mulr 16149  df-sca 16151  df-vsca 16152  df-ip 16153  df-tset 16154  df-ple 16155  df-ds 16158  df-hom 16160  df-cco 16161  df-prds 16302  df-xmet 19933
This theorem is referenced by:  prdsxmetlem  22366  prdsmet  22368
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