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Theorem prdsdsf 24393
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 484 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → 𝑦𝐼)
2 prdsdsf.r . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐼) → 𝑅𝑍)
32elexd 3502 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐼) → 𝑅 ∈ V)
43ralrimiva 3144 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥𝐼 𝑅 ∈ V)
54adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 𝑅 ∈ V)
6 nfcsb1v 3933 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥𝑅
76nfel1 2920 . . . . . . . . . . . . . . . 16 𝑥𝑦 / 𝑥𝑅 ∈ V
8 csbeq1a 3922 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
98eleq1d 2824 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑅 ∈ V ↔ 𝑦 / 𝑥𝑅 ∈ V))
107, 9rspc 3610 . . . . . . . . . . . . . . 15 (𝑦𝐼 → (∀𝑥𝐼 𝑅 ∈ V → 𝑦 / 𝑥𝑅 ∈ V))
115, 10mpan9 506 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → 𝑦 / 𝑥𝑅 ∈ V)
12 eqid 2735 . . . . . . . . . . . . . . 15 (𝑥𝐼𝑅) = (𝑥𝐼𝑅)
1312fvmpts 7019 . . . . . . . . . . . . . 14 ((𝑦𝐼𝑦 / 𝑥𝑅 ∈ V) → ((𝑥𝐼𝑅)‘𝑦) = 𝑦 / 𝑥𝑅)
141, 11, 13syl2anc 584 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑥𝐼𝑅)‘𝑦) = 𝑦 / 𝑥𝑅)
1514fveq2d 6911 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (dist‘((𝑥𝐼𝑅)‘𝑦)) = (dist‘𝑦 / 𝑥𝑅))
1615oveqd 7448 . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑆𝑊)
21 prdsdsf.i . . . . . . . . . . . . . . 15 (𝜑𝐼𝑋)
2221adantr 480 . . . . . . . . . . . . . 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 17530 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 (𝑓𝑥) ∈ 𝑉)
26 nfcsb1v 3933 . . . . . . . . . . . . . . 15 𝑥𝑦 / 𝑥𝑉
2726nfel2 2922 . . . . . . . . . . . . . 14 𝑥(𝑓𝑦) ∈ 𝑦 / 𝑥𝑉
28 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
29 csbeq1a 3922 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦𝑉 = 𝑦 / 𝑥𝑉)
3028, 29eleq12d 2833 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝑉 ↔ (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉))
3127, 30rspc 3610 . . . . . . . . . . . . 13 (𝑦𝐼 → (∀𝑥𝐼 (𝑓𝑥) ∈ 𝑉 → (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉))
3225, 31mpan9 506 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (𝑓𝑦) ∈ 𝑦 / 𝑥𝑉)
33 simprr 773 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 𝑔𝐵)
3417, 18, 20, 22, 5, 23, 33prdsbascl 17530 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉)
3526nfel2 2922 . . . . . . . . . . . . . 14 𝑥(𝑔𝑦) ∈ 𝑦 / 𝑥𝑉
36 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑔𝑥) = (𝑔𝑦))
3736, 29eleq12d 2833 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑔𝑥) ∈ 𝑉 ↔ (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉))
3835, 37rspc 3610 . . . . . . . . . . . . 13 (𝑦𝐼 → (∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉 → (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉))
3934, 38mpan9 506 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → (𝑔𝑦) ∈ 𝑦 / 𝑥𝑉)
4032, 39ovresd 7600 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) = ((𝑓𝑦)(dist‘𝑦 / 𝑥𝑅)(𝑔𝑦)))
4116, 40eqtr4d 2778 . . . . . . . . . 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 3144 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉))
4443adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉))
45 nfcv 2903 . . . . . . . . . . . . . . . 16 𝑥dist
4645, 6nffv 6917 . . . . . . . . . . . . . . 15 𝑥(dist‘𝑦 / 𝑥𝑅)
4726, 26nfxp 5722 . . . . . . . . . . . . . . 15 𝑥(𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)
4846, 47nfres 6002 . . . . . . . . . . . . . 14 𝑥((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))
49 nfcv 2903 . . . . . . . . . . . . . . 15 𝑥∞Met
5049, 26nffv 6917 . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘𝑦 / 𝑥𝑅))
5429sqxpeqd 5721 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑉 × 𝑉) = (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))
5553, 54reseq12d 6001 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)))
5652, 55eqtrid 2787 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐸 = ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)))
5729fveq2d 6911 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘𝑦 / 𝑥𝑉))
5856, 57eleq12d 2833 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐸 ∈ (∞Met‘𝑉) ↔ ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)))
5951, 58rspc 3610 . . . . . . . . . . . 12 (𝑦𝐼 → (∀𝑥𝐼 𝐸 ∈ (∞Met‘𝑉) → ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉)))
6044, 59mpan9 506 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉)) ∈ (∞Met‘𝑦 / 𝑥𝑉))
61 xmetcl 24357 . . . . . . . . . . 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 1370 . . . . . . . . . 10 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)((dist‘𝑦 / 𝑥𝑅) ↾ (𝑦 / 𝑥𝑉 × 𝑦 / 𝑥𝑉))(𝑔𝑦)) ∈ ℝ*)
6341, 62eqeltrd 2839 . . . . . . . . 9 (((𝜑 ∧ (𝑓𝐵𝑔𝐵)) ∧ 𝑦𝐼) → ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦)) ∈ ℝ*)
6463fmpttd 7135 . . . . . . . 8 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))):𝐼⟶ℝ*)
6564frnd 6745 . . . . . . 7 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ⊆ ℝ*)
66 0xr 11306 . . . . . . . . 9 0 ∈ ℝ*
6766a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → 0 ∈ ℝ*)
6867snssd 4814 . . . . . . 7 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → {0} ⊆ ℝ*)
6965, 68unssd 4202 . . . . . 6 ((𝜑 ∧ (𝑓𝐵𝑔𝐵)) → (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ⊆ ℝ*)
70 supxrcl 13354 . . . . . 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 4189 . . . . . . 7 {0} ⊆ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0})
73 c0ex 11253 . . . . . . . 8 0 ∈ V
7473snss 4790 . . . . . . 7 (0 ∈ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}) ↔ {0} ⊆ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}))
7572, 74mpbir 231 . . . . . 6 0 ∈ (ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0})
76 supxrub 13363 . . . . . 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 13494 . . . . 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 3200 . . 3 (𝜑 → ∀𝑓𝐵𝑔𝐵 sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞))
81 eqid 2735 . . . 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 8092 . . 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 218 . 2 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞))
8421mptexd 7244 . . . 4 (𝜑 → (𝑥𝐼𝑅) ∈ V)
852ralrimiva 3144 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑅𝑍)
86 dmmptg 6264 . . . . 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 17511 . . 3 (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )))
9089feq1d 6721 . 2 (𝜑 → (𝐷:(𝐵 × 𝐵)⟶(0[,]+∞) ↔ (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑦𝐼 ↦ ((𝑓𝑦)(dist‘((𝑥𝐼𝑅)‘𝑦))(𝑔𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞)))
9183, 90mpbird 257 1 (𝜑𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  csb 3908  cun 3961  wss 3963  {csn 4631   class class class wbr 5148  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  cres 5691  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  supcsup 9478  0cc0 11153  +∞cpnf 11290  *cxr 11292   < clt 11293  cle 11294  [,]cicc 13387  Basecbs 17245  distcds 17307  Xscprds 17492  ∞Metcxmet 21367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-icc 13391  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-prds 17494  df-xmet 21375
This theorem is referenced by:  prdsxmetlem  24394  prdsmet  24396
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