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Mirrors > Home > MPE Home > Th. List > metres2 | Structured version Visualization version GIF version |
Description: Lemma for metres 22541. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
metres2 | ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metxmet 22510 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
2 | xmetres2 22537 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅)) | |
3 | 1, 2 | sylan 577 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅)) |
4 | metf 22506 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | |
5 | 4 | adantr 474 | . . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
6 | simpr 479 | . . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ⊆ 𝑋) | |
7 | xpss12 5358 | . . . 4 ⊢ ((𝑅 ⊆ 𝑋 ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋)) | |
8 | 6, 7 | sylancom 584 | . . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋)) |
9 | 5, 8 | fssresd 6309 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ) |
10 | ismet2 22509 | . 2 ⊢ ((𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘𝑅) ↔ ((𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅) ∧ (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ)) | |
11 | 3, 9, 10 | sylanbrc 580 | 1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 ⊆ wss 3799 × cxp 5341 ↾ cres 5345 ⟶wf 6120 ‘cfv 6124 ℝcr 10252 ∞Metcxmet 20092 Metcmet 20093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-mulcl 10315 ax-i2m1 10321 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-xadd 12234 df-xmet 20100 df-met 20101 |
This theorem is referenced by: metres 22541 xpsmet 22558 tmsms 22663 imasf1oms 22666 prdsms 22707 remet 22964 lebnumii 23136 cmetss 23485 sstotbnd2 34116 bndss 34128 equivbnd2 34134 rrnheibor 34179 iccbnd 34182 |
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