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Mirrors > Home > MPE Home > Th. List > Mathboxes > sblpnf | Structured version Visualization version GIF version |
Description: The infinity ball in the absolute value metric is just the whole space. 𝑆 analogue of blpnf 23004. (Contributed by Steve Rodriguez, 8-Nov-2015.) |
Ref | Expression |
---|---|
sblpnf.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
sblpnf.d | ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) |
Ref | Expression |
---|---|
sblpnf | ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sblpnf.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | elpri 4547 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
3 | sblpnf.d | . . . . 5 ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) | |
4 | eqid 2798 | . . . . . . 7 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
5 | 4 | remet 23395 | . . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ) |
6 | xpeq12 5544 | . . . . . . . . 9 ⊢ ((𝑆 = ℝ ∧ 𝑆 = ℝ) → (𝑆 × 𝑆) = (ℝ × ℝ)) | |
7 | 6 | anidms 570 | . . . . . . . 8 ⊢ (𝑆 = ℝ → (𝑆 × 𝑆) = (ℝ × ℝ)) |
8 | 7 | reseq2d 5818 | . . . . . . 7 ⊢ (𝑆 = ℝ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (ℝ × ℝ))) |
9 | fveq2 6645 | . . . . . . 7 ⊢ (𝑆 = ℝ → (Met‘𝑆) = (Met‘ℝ)) | |
10 | 8, 9 | eleq12d 2884 | . . . . . 6 ⊢ (𝑆 = ℝ → (((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ↔ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ))) |
11 | 5, 10 | mpbiri 261 | . . . . 5 ⊢ (𝑆 = ℝ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆)) |
12 | 3, 11 | eqeltrid 2894 | . . . 4 ⊢ (𝑆 = ℝ → 𝐷 ∈ (Met‘𝑆)) |
13 | relco 6064 | . . . . . . . . 9 ⊢ Rel (abs ∘ − ) | |
14 | resdm 5863 | . . . . . . . . 9 ⊢ (Rel (abs ∘ − ) → ((abs ∘ − ) ↾ dom (abs ∘ − )) = (abs ∘ − )) | |
15 | 13, 14 | ax-mp 5 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ dom (abs ∘ − )) = (abs ∘ − ) |
16 | absf 14689 | . . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ | |
17 | ax-resscn 10583 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
18 | fss 6501 | . . . . . . . . . . . 12 ⊢ ((abs:ℂ⟶ℝ ∧ ℝ ⊆ ℂ) → abs:ℂ⟶ℂ) | |
19 | 16, 17, 18 | mp2an 691 | . . . . . . . . . . 11 ⊢ abs:ℂ⟶ℂ |
20 | subf 10877 | . . . . . . . . . . 11 ⊢ − :(ℂ × ℂ)⟶ℂ | |
21 | fco 6505 | . . . . . . . . . . 11 ⊢ ((abs:ℂ⟶ℂ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℂ) | |
22 | 19, 20, 21 | mp2an 691 | . . . . . . . . . 10 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℂ |
23 | 22 | fdmi 6498 | . . . . . . . . 9 ⊢ dom (abs ∘ − ) = (ℂ × ℂ) |
24 | 23 | reseq2i 5815 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ dom (abs ∘ − )) = ((abs ∘ − ) ↾ (ℂ × ℂ)) |
25 | 15, 24 | eqtr3i 2823 | . . . . . . 7 ⊢ (abs ∘ − ) = ((abs ∘ − ) ↾ (ℂ × ℂ)) |
26 | cnmet 23377 | . . . . . . 7 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
27 | 25, 26 | eqeltrri 2887 | . . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) ∈ (Met‘ℂ) |
28 | xpeq12 5544 | . . . . . . . . 9 ⊢ ((𝑆 = ℂ ∧ 𝑆 = ℂ) → (𝑆 × 𝑆) = (ℂ × ℂ)) | |
29 | 28 | anidms 570 | . . . . . . . 8 ⊢ (𝑆 = ℂ → (𝑆 × 𝑆) = (ℂ × ℂ)) |
30 | 29 | reseq2d 5818 | . . . . . . 7 ⊢ (𝑆 = ℂ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (ℂ × ℂ))) |
31 | fveq2 6645 | . . . . . . 7 ⊢ (𝑆 = ℂ → (Met‘𝑆) = (Met‘ℂ)) | |
32 | 30, 31 | eleq12d 2884 | . . . . . 6 ⊢ (𝑆 = ℂ → (((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ↔ ((abs ∘ − ) ↾ (ℂ × ℂ)) ∈ (Met‘ℂ))) |
33 | 27, 32 | mpbiri 261 | . . . . 5 ⊢ (𝑆 = ℂ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆)) |
34 | 3, 33 | eqeltrid 2894 | . . . 4 ⊢ (𝑆 = ℂ → 𝐷 ∈ (Met‘𝑆)) |
35 | 12, 34 | jaoi 854 | . . 3 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 𝐷 ∈ (Met‘𝑆)) |
36 | 1, 2, 35 | 3syl 18 | . 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑆)) |
37 | blpnf 23004 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑆) ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) | |
38 | 36, 37 | sylan 583 | 1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 {cpr 4527 × cxp 5517 dom cdm 5519 ↾ cres 5521 ∘ ccom 5523 Rel wrel 5524 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 +∞cpnf 10661 − cmin 10859 abscabs 14585 Metcmet 20077 ballcbl 20078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 |
This theorem is referenced by: dvconstbi 41038 |
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