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Theorem prdsdsf 23265
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 3428 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐼) → 𝑅 ∈ V)
43ralrimiva 3105 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥𝐼 𝑅 ∈ V)
54adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 𝑅 ∈ V)
6 nfcsb1v 3836 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥𝑅
76nfel1 2920 . . . . . . . . . . . . . . . 16 𝑥𝑦 / 𝑥𝑅 ∈ V
8 csbeq1a 3825 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
98eleq1d 2822 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑅 ∈ V ↔ 𝑦 / 𝑥𝑅 ∈ V))
107, 9rspc 3525 . . . . . . . . . . . . . . 15 (𝑦𝐼 → (∀𝑥𝐼 𝑅 ∈ V → 𝑦 / 𝑥𝑅 ∈ V))
115, 10mpan9 510 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → 𝑦 / 𝑥𝑅 ∈ V)
12 eqid 2737 . . . . . . . . . . . . . . 15 (𝑥𝐼𝑅) = (𝑥𝐼𝑅)
1312fvmpts 6821 . . . . . . . . . . . . . 14 ((𝑦𝐼𝑦 / 𝑥𝑅 ∈ V) → ((𝑥𝐼𝑅)‘𝑦) = 𝑦 / 𝑥𝑅)
141, 11, 13syl2anc 587 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑥𝐼𝑅)‘𝑦) = 𝑦 / 𝑥𝑅)
1514fveq2d 6721 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (dist‘((𝑥𝐼𝑅)‘𝑦)) = (dist‘𝑦 / 𝑥𝑅))
1615oveqd 7230 . . . . . . . . . . 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 771 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑓𝐵)
2517, 18, 20, 22, 5, 23, 24prdsbascl 16988 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 (𝑓𝑥) ∈ 𝑉)
26 nfcsb1v 3836 . . . . . . . . . . . . . . 15 𝑥𝑦 / 𝑥𝑉
2726nfel2 2922 . . . . . . . . . . . . . 14 𝑥(𝑓𝑦) ∈ 𝑦 / 𝑥𝑉
28 fveq2 6717 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
29 csbeq1a 3825 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦𝑉 = 𝑦 / 𝑥𝑉)
3028, 29eleq12d 2832 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝑉 ↔ (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉))
3127, 30rspc 3525 . . . . . . . . . . . . 13 (𝑦𝐼 → (∀𝑥𝐼 (𝑓𝑥) ∈ 𝑉 → (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉))
3225, 31mpan9 510 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉)
33 simprr 773 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑔𝐵)
3417, 18, 20, 22, 5, 23, 33prdsbascl 16988 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉)
3526nfel2 2922 . . . . . . . . . . . . . 14 𝑥(𝑔𝑦) ∈ 𝑦 / 𝑥𝑉
36 fveq2 6717 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑔𝑥) = (𝑔𝑦))
3736, 29eleq12d 2832 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑔𝑥) ∈ 𝑉 ↔ (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉))
3835, 37rspc 3525 . . . . . . . . . . . . 13 (𝑦𝐼 → (∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉 → (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉))
3934, 38mpan9 510 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉)
4032, 39ovresd 7375 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) = ((𝑓𝑦)(dist‘𝑦 / 𝑥𝑅)(𝑔𝑦)))
4116, 40eqtr4d 2780 . . . . . . . . . 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 3105 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉))
4443adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉))
45 nfcv 2904 . . . . . . . . . . . . . . . 16 𝑥dist
4645, 6nffv 6727 . . . . . . . . . . . . . . 15 𝑥(dist‘𝑦 / 𝑥𝑅)
4726, 26nfxp 5584 . . . . . . . . . . . . . . 15 𝑥(𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)
4846, 47nfres 5853 . . . . . . . . . . . . . 14 𝑥((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))
49 nfcv 2904 . . . . . . . . . . . . . . 15 𝑥∞Met
5049, 26nffv 6727 . . . . . . . . . . . . . 14 𝑥(∞Met‘𝑦 / 𝑥𝑉)
5148, 50nfel 2918 . . . . . . . . . . . . 13 𝑥((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)
52 prdsdsf.e . . . . . . . . . . . . . . 15 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
538fveq2d 6721 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘𝑦 / 𝑥𝑅))
5429sqxpeqd 5583 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑉 × 𝑉) = (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))
5553, 54reseq12d 5852 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)))
5652, 55syl5eq 2790 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐸 = ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)))
5729fveq2d 6721 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘𝑦 / 𝑥𝑉))
5856, 57eleq12d 2832 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐸 ∈ (∞Met‘𝑉) ↔ ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)))
5951, 58rspc 3525 . . . . . . . . . . . 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 23229 . . . . . . . . . . 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 1373 . . . . . . . . . 10 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) ∈ ℝ*)
6341, 62eqeltrd 2838 . . . . . . . . 9 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦)) ∈ ℝ*)
6463fmpttd 6932 . . . . . . . 8 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))):𝐼⟶ℝ*)
6564frnd 6553 . . . . . . 7 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ⊆ ℝ*)
66 0xr 10880 . . . . . . . . 9 0 ∈ ℝ*
6766a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 0 ∈ ℝ*)
6867snssd 4722 . . . . . . 7 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → {0} ⊆ ℝ*)
6965, 68unssd 4100 . . . . . 6 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ⊆ ℝ*)
70 supxrcl 12905 . . . . . 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 4087 . . . . . . 7 {0} ⊆ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0})
73 c0ex 10827 . . . . . . . 8 0 ∈ V
7473snss 4699 . . . . . . 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 12914 . . . . . 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 13045 . . . . 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 3112 . . 3 (𝜑 → ∀𝑓𝐵𝑔𝐵 sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞))
81 eqid 2737 . . . 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 7838 . . 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 7040 . . . 4 (𝜑 → (𝑥𝐼𝑅) ∈ V)
852ralrimiva 3105 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑅𝑍)
86 dmmptg 6105 . . . . 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 16969 . . 3 (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )))
9089feq1d 6530 . 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 1543  wcel 2110  wral 3061  Vcvv 3408  csb 3811  cun 3864  wss 3866  {csn 4541   class class class wbr 5053  cmpt 5135   × cxp 5549  dom cdm 5551  ran crn 5552  cres 5553  wf 6376  cfv 6380  (class class class)co 7213  cmpo 7215  supcsup 9056  0cc0 10729  +∞cpnf 10864  *cxr 10866   < clt 10867  cle 10868  [,]cicc 12938  Basecbs 16760  distcds 16811  Xscprds 16950  ∞Metcxmet 20348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-sup 9058  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-icc 12942  df-fz 13096  df-struct 16700  df-slot 16735  df-ndx 16745  df-base 16761  df-plusg 16815  df-mulr 16816  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-hom 16826  df-cco 16827  df-prds 16952  df-xmet 20356
This theorem is referenced by:  prdsxmetlem  23266  prdsmet  23268
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