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Mirrors > Home > MPE Home > Th. List > Mathboxes > isbnd3b | Structured version Visualization version GIF version |
Description: A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 22-Sep-2015.) |
Ref | Expression |
---|---|
isbnd3b | ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ≤ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isbnd3 35077 | . 2 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥))) | |
2 | metf 22940 | . . . . . . 7 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ) | |
3 | 2 | adantr 483 | . . . . . 6 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) → 𝑀:(𝑋 × 𝑋)⟶ℝ) |
4 | ffn 6514 | . . . . . 6 ⊢ (𝑀:(𝑋 × 𝑋)⟶ℝ → 𝑀 Fn (𝑋 × 𝑋)) | |
5 | ffnov 7278 | . . . . . . 7 ⊢ (𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥) ↔ (𝑀 Fn (𝑋 × 𝑋) ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ∈ (0[,]𝑥))) | |
6 | 5 | baib 538 | . . . . . 6 ⊢ (𝑀 Fn (𝑋 × 𝑋) → (𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ∈ (0[,]𝑥))) |
7 | 3, 4, 6 | 3syl 18 | . . . . 5 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) → (𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ∈ (0[,]𝑥))) |
8 | 0red 10644 | . . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 0 ∈ ℝ) | |
9 | simplr 767 | . . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ ℝ) | |
10 | metcl 22942 | . . . . . . . . 9 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦𝑀𝑧) ∈ ℝ) | |
11 | 10 | 3expb 1116 | . . . . . . . 8 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦𝑀𝑧) ∈ ℝ) |
12 | 11 | adantlr 713 | . . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦𝑀𝑧) ∈ ℝ) |
13 | metge0 22955 | . . . . . . . . 9 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → 0 ≤ (𝑦𝑀𝑧)) | |
14 | 13 | 3expb 1116 | . . . . . . . 8 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 0 ≤ (𝑦𝑀𝑧)) |
15 | 14 | adantlr 713 | . . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 0 ≤ (𝑦𝑀𝑧)) |
16 | elicc2 12802 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑦𝑀𝑧) ∈ (0[,]𝑥) ↔ ((𝑦𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑧) ∧ (𝑦𝑀𝑧) ≤ 𝑥))) | |
17 | df-3an 1085 | . . . . . . . . 9 ⊢ (((𝑦𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑧) ∧ (𝑦𝑀𝑧) ≤ 𝑥) ↔ (((𝑦𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑧)) ∧ (𝑦𝑀𝑧) ≤ 𝑥)) | |
18 | 16, 17 | syl6bb 289 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑦𝑀𝑧) ∈ (0[,]𝑥) ↔ (((𝑦𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑧)) ∧ (𝑦𝑀𝑧) ≤ 𝑥))) |
19 | 18 | baibd 542 | . . . . . . 7 ⊢ (((0 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ ((𝑦𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑧))) → ((𝑦𝑀𝑧) ∈ (0[,]𝑥) ↔ (𝑦𝑀𝑧) ≤ 𝑥)) |
20 | 8, 9, 12, 15, 19 | syl22anc 836 | . . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦𝑀𝑧) ∈ (0[,]𝑥) ↔ (𝑦𝑀𝑧) ≤ 𝑥)) |
21 | 20 | 2ralbidva 3198 | . . . . 5 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ∈ (0[,]𝑥) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ≤ 𝑥)) |
22 | 7, 21 | bitrd 281 | . . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) → (𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ≤ 𝑥)) |
23 | 22 | rexbidva 3296 | . . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ≤ 𝑥)) |
24 | 23 | pm5.32i 577 | . 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ≤ 𝑥)) |
25 | 1, 24 | bitri 277 | 1 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ≤ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 × cxp 5553 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 ≤ cle 10676 [,]cicc 12742 Metcmet 20531 Bndcbnd 35060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-ec 8291 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-bnd 35072 |
This theorem is referenced by: equivbnd 35083 iccbnd 35133 |
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