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Theorem prdsdsf 23015
 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 488 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → 𝑦𝐼)
2 prdsdsf.r . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐼) → 𝑅𝑍)
32elexd 3462 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐼) → 𝑅 ∈ V)
43ralrimiva 3149 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥𝐼 𝑅 ∈ V)
54adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 𝑅 ∈ V)
6 nfcsb1v 3854 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥𝑅
76nfel1 2971 . . . . . . . . . . . . . . . 16 𝑥𝑦 / 𝑥𝑅 ∈ V
8 csbeq1a 3844 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
98eleq1d 2874 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑅 ∈ V ↔ 𝑦 / 𝑥𝑅 ∈ V))
107, 9rspc 3560 . . . . . . . . . . . . . . 15 (𝑦𝐼 → (∀𝑥𝐼 𝑅 ∈ V → 𝑦 / 𝑥𝑅 ∈ V))
115, 10mpan9 510 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → 𝑦 / 𝑥𝑅 ∈ V)
12 eqid 2798 . . . . . . . . . . . . . . 15 (𝑥𝐼𝑅) = (𝑥𝐼𝑅)
1312fvmpts 6758 . . . . . . . . . . . . . 14 ((𝑦𝐼𝑦 / 𝑥𝑅 ∈ V) → ((𝑥𝐼𝑅)‘𝑦) = 𝑦 / 𝑥𝑅)
141, 11, 13syl2anc 587 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑥𝐼𝑅)‘𝑦) = 𝑦 / 𝑥𝑅)
1514fveq2d 6659 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (dist‘((𝑥𝐼𝑅)‘𝑦)) = (dist‘𝑦 / 𝑥𝑅))
1615oveqd 7162 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦)) = ((𝑓𝑦)(dist‘𝑦 / 𝑥𝑅)(𝑔𝑦)))
17 prdsdsf.y . . . . . . . . . . . . . 14 𝑌 = (𝑆Xs(𝑥𝐼𝑅))
18 prdsdsf.b . . . . . . . . . . . . . 14 𝐵 = (Base‘𝑌)
19 prdsdsf.s . . . . . . . . . . . . . . 15 (𝜑𝑆𝑊)
2019adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑆𝑊)
21 prdsdsf.i . . . . . . . . . . . . . . 15 (𝜑𝐼𝑋)
2221adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝐼𝑋)
23 prdsdsf.v . . . . . . . . . . . . . 14 𝑉 = (Base‘𝑅)
24 simprl 770 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑓𝐵)
2517, 18, 20, 22, 5, 23, 24prdsbascl 16768 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 (𝑓𝑥) ∈ 𝑉)
26 nfcsb1v 3854 . . . . . . . . . . . . . . 15 𝑥𝑦 / 𝑥𝑉
2726nfel2 2973 . . . . . . . . . . . . . 14 𝑥(𝑓𝑦) ∈ 𝑦 / 𝑥𝑉
28 fveq2 6655 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
29 csbeq1a 3844 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦𝑉 = 𝑦 / 𝑥𝑉)
3028, 29eleq12d 2884 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝑉 ↔ (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉))
3127, 30rspc 3560 . . . . . . . . . . . . 13 (𝑦𝐼 → (∀𝑥𝐼 (𝑓𝑥) ∈ 𝑉 → (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉))
3225, 31mpan9 510 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉)
33 simprr 772 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑔𝐵)
3417, 18, 20, 22, 5, 23, 33prdsbascl 16768 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉)
3526nfel2 2973 . . . . . . . . . . . . . 14 𝑥(𝑔𝑦) ∈ 𝑦 / 𝑥𝑉
36 fveq2 6655 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑔𝑥) = (𝑔𝑦))
3736, 29eleq12d 2884 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑔𝑥) ∈ 𝑉 ↔ (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉))
3835, 37rspc 3560 . . . . . . . . . . . . 13 (𝑦𝐼 → (∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉 → (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉))
3934, 38mpan9 510 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉)
4032, 39ovresd 7306 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) = ((𝑓𝑦)(dist‘𝑦 / 𝑥𝑅)(𝑔𝑦)))
4116, 40eqtr4d 2836 . . . . . . . . . 10 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦)) = ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)))
42 prdsdsf.m . . . . . . . . . . . . . 14 ((𝜑𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
4342ralrimiva 3149 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉))
4443adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉))
45 nfcv 2955 . . . . . . . . . . . . . . . 16 𝑥dist
4645, 6nffv 6665 . . . . . . . . . . . . . . 15 𝑥(dist‘𝑦 / 𝑥𝑅)
4726, 26nfxp 5556 . . . . . . . . . . . . . . 15 𝑥(𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)
4846, 47nfres 5824 . . . . . . . . . . . . . 14 𝑥((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))
49 nfcv 2955 . . . . . . . . . . . . . . 15 𝑥∞Met
5049, 26nffv 6665 . . . . . . . . . . . . . 14 𝑥(∞Met‘𝑦 / 𝑥𝑉)
5148, 50nfel 2969 . . . . . . . . . . . . 13 𝑥((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)
52 prdsdsf.e . . . . . . . . . . . . . . 15 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
538fveq2d 6659 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘𝑦 / 𝑥𝑅))
5429sqxpeqd 5555 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑉 × 𝑉) = (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))
5553, 54reseq12d 5823 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)))
5652, 55syl5eq 2845 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐸 = ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)))
5729fveq2d 6659 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘𝑦 / 𝑥𝑉))
5856, 57eleq12d 2884 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐸 ∈ (∞Met‘𝑉) ↔ ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)))
5951, 58rspc 3560 . . . . . . . . . . . 12 (𝑦𝐼 → (∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉) → ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)))
6044, 59mpan9 510 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉))
61 xmetcl 22979 . . . . . . . . . . 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*)
6260, 32, 39, 61syl3anc 1368 . . . . . . . . . 10 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) ∈ ℝ*)
6341, 62eqeltrd 2890 . . . . . . . . 9 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦)) ∈ ℝ*)
6463fmpttd 6866 . . . . . . . 8 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))):𝐼⟶ℝ*)
6564frnd 6502 . . . . . . 7 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ⊆ ℝ*)
66 0xr 10695 . . . . . . . . 9 0 ∈ ℝ*
6766a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 0 ∈ ℝ*)
6867snssd 4705 . . . . . . 7 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → {0} ⊆ ℝ*)
6965, 68unssd 4116 . . . . . 6 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ⊆ ℝ*)
70 supxrcl 12716 . . . . . 6 ((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ⊆ ℝ* → sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ ℝ*)
7169, 70syl 17 . . . . 5 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ ℝ*)
72 ssun2 4103 . . . . . . 7 {0} ⊆ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0})
73 c0ex 10642 . . . . . . . 8 0 ∈ V
7473snss 4682 . . . . . . 7 (0 ∈ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ↔ {0} ⊆ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}))
7572, 74mpbir 234 . . . . . 6 0 ∈ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0})
76 supxrub 12725 . . . . . 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*, < ))
7769, 75, 76sylancl 589 . . . . 5 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 0 ≤ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ))
78 elxrge0 12855 . . . . 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*, < )))
7971, 77, 78sylanbrc 586 . . . 4 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞))
8079ralrimivva 3156 . . 3 (𝜑 → ∀𝑓𝐵𝑔𝐵 sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞))
81 eqid 2798 . . . 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*, < ))
8281fmpo 7761 . . 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[,]+∞))
8380, 82sylib 221 . 2 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞))
8421mptexd 6974 . . . 4 (𝜑 → (𝑥𝐼𝑅) ∈ V)
852ralrimiva 3149 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑅𝑍)
86 dmmptg 6068 . . . . 5 (∀𝑥𝐼 𝑅𝑍 → dom (𝑥𝐼𝑅) = 𝐼)
8785, 86syl 17 . . . 4 (𝜑 → dom (𝑥𝐼𝑅) = 𝐼)
88 prdsdsf.d . . . 4 𝐷 = (dist‘𝑌)
8917, 19, 84, 18, 87, 88prdsds 16749 . . 3 (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )))
9089feq1d 6480 . 2 (𝜑 → (𝐷:(𝐵 × 𝐵)⟶(0[,]+∞) ↔ (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞)))
9183, 90mpbird 260 1 (𝜑𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3442  ⦋csb 3830   ∪ cun 3881   ⊆ wss 3883  {csn 4528   class class class wbr 5034   ↦ cmpt 5114   × cxp 5521  dom cdm 5523  ran crn 5524   ↾ cres 5525  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145   ∈ cmpo 7147  supcsup 8906  0cc0 10544  +∞cpnf 10679  ℝ*cxr 10681   < clt 10682   ≤ cle 10683  [,]cicc 12749  Basecbs 16495  distcds 16586  Xscprds 16731  ∞Metcxmet 20097 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-pre-sup 10622 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-oadd 8107  df-er 8290  df-map 8409  df-ixp 8463  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-sup 8908  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-2 11706  df-3 11707  df-4 11708  df-5 11709  df-6 11710  df-7 11711  df-8 11712  df-9 11713  df-n0 11904  df-z 11990  df-dec 12107  df-uz 12252  df-icc 12753  df-fz 12906  df-struct 16497  df-ndx 16498  df-slot 16499  df-base 16501  df-plusg 16590  df-mulr 16591  df-sca 16593  df-vsca 16594  df-ip 16595  df-tset 16596  df-ple 16597  df-ds 16599  df-hom 16601  df-cco 16602  df-prds 16733  df-xmet 20105 This theorem is referenced by:  prdsxmetlem  23016  prdsmet  23018
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