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Theorem prdsdsf 22904
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 485 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → 𝑦𝐼)
2 prdsdsf.r . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐼) → 𝑅𝑍)
32elexd 3512 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐼) → 𝑅 ∈ V)
43ralrimiva 3179 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥𝐼 𝑅 ∈ V)
54adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 𝑅 ∈ V)
6 nfcsb1v 3904 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥𝑅
76nfel1 2991 . . . . . . . . . . . . . . . 16 𝑥𝑦 / 𝑥𝑅 ∈ V
8 csbeq1a 3894 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
98eleq1d 2894 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑅 ∈ V ↔ 𝑦 / 𝑥𝑅 ∈ V))
107, 9rspc 3608 . . . . . . . . . . . . . . 15 (𝑦𝐼 → (∀𝑥𝐼 𝑅 ∈ V → 𝑦 / 𝑥𝑅 ∈ V))
115, 10mpan9 507 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → 𝑦 / 𝑥𝑅 ∈ V)
12 eqid 2818 . . . . . . . . . . . . . . 15 (𝑥𝐼𝑅) = (𝑥𝐼𝑅)
1312fvmpts 6764 . . . . . . . . . . . . . 14 ((𝑦𝐼𝑦 / 𝑥𝑅 ∈ V) → ((𝑥𝐼𝑅)‘𝑦) = 𝑦 / 𝑥𝑅)
141, 11, 13syl2anc 584 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑥𝐼𝑅)‘𝑦) = 𝑦 / 𝑥𝑅)
1514fveq2d 6667 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑆𝑊)
21 prdsdsf.i . . . . . . . . . . . . . . 15 (𝜑𝐼𝑋)
2221adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝐼𝑋)
23 prdsdsf.v . . . . . . . . . . . . . 14 𝑉 = (Base‘𝑅)
24 simprl 767 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑓𝐵)
2517, 18, 20, 22, 5, 23, 24prdsbascl 16744 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 (𝑓𝑥) ∈ 𝑉)
26 nfcsb1v 3904 . . . . . . . . . . . . . . 15 𝑥𝑦 / 𝑥𝑉
2726nfel2 2993 . . . . . . . . . . . . . 14 𝑥(𝑓𝑦) ∈ 𝑦 / 𝑥𝑉
28 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
29 csbeq1a 3894 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦𝑉 = 𝑦 / 𝑥𝑉)
3028, 29eleq12d 2904 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝑉 ↔ (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉))
3127, 30rspc 3608 . . . . . . . . . . . . 13 (𝑦𝐼 → (∀𝑥𝐼 (𝑓𝑥) ∈ 𝑉 → (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉))
3225, 31mpan9 507 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉)
33 simprr 769 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑔𝐵)
3417, 18, 20, 22, 5, 23, 33prdsbascl 16744 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉)
3526nfel2 2993 . . . . . . . . . . . . . 14 𝑥(𝑔𝑦) ∈ 𝑦 / 𝑥𝑉
36 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑔𝑥) = (𝑔𝑦))
3736, 29eleq12d 2904 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑔𝑥) ∈ 𝑉 ↔ (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉))
3835, 37rspc 3608 . . . . . . . . . . . . 13 (𝑦𝐼 → (∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉 → (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉))
3934, 38mpan9 507 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉)
4032, 39ovresd 7304 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) = ((𝑓𝑦)(dist‘𝑦 / 𝑥𝑅)(𝑔𝑦)))
4116, 40eqtr4d 2856 . . . . . . . . . 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 3179 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉))
4443adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉))
45 nfcv 2974 . . . . . . . . . . . . . . . 16 𝑥dist
4645, 6nffv 6673 . . . . . . . . . . . . . . 15 𝑥(dist‘𝑦 / 𝑥𝑅)
4726, 26nfxp 5581 . . . . . . . . . . . . . . 15 𝑥(𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)
4846, 47nfres 5848 . . . . . . . . . . . . . 14 𝑥((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))
49 nfcv 2974 . . . . . . . . . . . . . . 15 𝑥∞Met
5049, 26nffv 6673 . . . . . . . . . . . . . 14 𝑥(∞Met‘𝑦 / 𝑥𝑉)
5148, 50nfel 2989 . . . . . . . . . . . . 13 𝑥((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)
52 prdsdsf.e . . . . . . . . . . . . . . 15 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
538fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘𝑦 / 𝑥𝑅))
5429sqxpeqd 5580 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑉 × 𝑉) = (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))
5553, 54reseq12d 5847 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)))
5652, 55syl5eq 2865 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐸 = ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)))
5729fveq2d 6667 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘𝑦 / 𝑥𝑉))
5856, 57eleq12d 2904 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐸 ∈ (∞Met‘𝑉) ↔ ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)))
5951, 58rspc 3608 . . . . . . . . . . . 12 (𝑦𝐼 → (∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉) → ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)))
6044, 59mpan9 507 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉))
61 xmetcl 22868 . . . . . . . . . . 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 1363 . . . . . . . . . 10 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) ∈ ℝ*)
6341, 62eqeltrd 2910 . . . . . . . . 9 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦)) ∈ ℝ*)
6463fmpttd 6871 . . . . . . . 8 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))):𝐼⟶ℝ*)
6564frnd 6514 . . . . . . 7 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ⊆ ℝ*)
66 0xr 10676 . . . . . . . . 9 0 ∈ ℝ*
6766a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 0 ∈ ℝ*)
6867snssd 4734 . . . . . . 7 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → {0} ⊆ ℝ*)
6965, 68unssd 4159 . . . . . 6 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ⊆ ℝ*)
70 supxrcl 12696 . . . . . 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 4146 . . . . . . 7 {0} ⊆ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0})
73 c0ex 10623 . . . . . . . 8 0 ∈ V
7473snss 4710 . . . . . . 7 (0 ∈ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ↔ {0} ⊆ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}))
7572, 74mpbir 232 . . . . . 6 0 ∈ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0})
76 supxrub 12705 . . . . . 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 586 . . . . 5 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 0 ≤ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ))
78 elxrge0 12833 . . . . 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 583 . . . 4 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞))
8079ralrimivva 3188 . . 3 (𝜑 → ∀𝑓𝐵𝑔𝐵 sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞))
81 eqid 2818 . . . 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 7755 . . 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 219 . 2 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞))
8421mptexd 6978 . . . 4 (𝜑 → (𝑥𝐼𝑅) ∈ V)
852ralrimiva 3179 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑅𝑍)
86 dmmptg 6089 . . . . 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 16725 . . 3 (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )))
9089feq1d 6492 . 2 (𝜑 → (𝐷:(𝐵 × 𝐵)⟶(0[,]+∞) ↔ (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞)))
9183, 90mpbird 258 1 (𝜑𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  csb 3880  cun 3931  wss 3933  {csn 4557   class class class wbr 5057  cmpt 5137   × cxp 5546  dom cdm 5548  ran crn 5549  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  supcsup 8892  0cc0 10525  +∞cpnf 10660  *cxr 10662   < clt 10663  cle 10664  [,]cicc 12729  Basecbs 16471  distcds 16562  Xscprds 16707  ∞Metcxmet 20458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-icc 12733  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-plusg 16566  df-mulr 16567  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-hom 16577  df-cco 16578  df-prds 16709  df-xmet 20466
This theorem is referenced by:  prdsxmetlem  22905  prdsmet  22907
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