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Mirrors > Home > MPE Home > Th. List > Mathboxes > blssp | Structured version Visualization version GIF version |
Description: A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.) |
Ref | Expression |
---|---|
blssp.2 | ⊢ 𝑁 = (𝑀 ↾ (𝑆 × 𝑆)) |
Ref | Expression |
---|---|
blssp | ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metxmet 22938 | . . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋)) | |
2 | 1 | ad2antrr 724 | . 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → 𝑀 ∈ (∞Met‘𝑋)) |
3 | simprl 769 | . . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → 𝑌 ∈ 𝑆) | |
4 | simplr 767 | . . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → 𝑆 ⊆ 𝑋) | |
5 | sseqin2 4192 | . . . 4 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∩ 𝑆) = 𝑆) | |
6 | 4, 5 | sylib 220 | . . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∩ 𝑆) = 𝑆) |
7 | 3, 6 | eleqtrrd 2916 | . 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → 𝑌 ∈ (𝑋 ∩ 𝑆)) |
8 | rpxr 12392 | . . 3 ⊢ (𝑅 ∈ ℝ+ → 𝑅 ∈ ℝ*) | |
9 | 8 | ad2antll 727 | . 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → 𝑅 ∈ ℝ*) |
10 | blssp.2 | . . 3 ⊢ 𝑁 = (𝑀 ↾ (𝑆 × 𝑆)) | |
11 | 10 | blres 23035 | . 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ (𝑋 ∩ 𝑆) ∧ 𝑅 ∈ ℝ*) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆)) |
12 | 2, 7, 9, 11 | syl3anc 1367 | 1 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3935 ⊆ wss 3936 × cxp 5548 ↾ cres 5552 ‘cfv 6350 (class class class)co 7150 ℝ*cxr 10668 ℝ+crp 12383 ∞Metcxmet 20524 Metcmet 20525 ballcbl 20526 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-mulcl 10593 ax-i2m1 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-rp 12384 df-xadd 12502 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 |
This theorem is referenced by: bndss 35058 |
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