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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 23001. (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 4583 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
3 | sblpnf.d | . . . . 5 ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) | |
4 | eqid 2821 | . . . . . . 7 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
5 | 4 | remet 23392 | . . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ) |
6 | xpeq12 5575 | . . . . . . . . 9 ⊢ ((𝑆 = ℝ ∧ 𝑆 = ℝ) → (𝑆 × 𝑆) = (ℝ × ℝ)) | |
7 | 6 | anidms 569 | . . . . . . . 8 ⊢ (𝑆 = ℝ → (𝑆 × 𝑆) = (ℝ × ℝ)) |
8 | 7 | reseq2d 5848 | . . . . . . 7 ⊢ (𝑆 = ℝ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (ℝ × ℝ))) |
9 | fveq2 6665 | . . . . . . 7 ⊢ (𝑆 = ℝ → (Met‘𝑆) = (Met‘ℝ)) | |
10 | 8, 9 | eleq12d 2907 | . . . . . 6 ⊢ (𝑆 = ℝ → (((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ↔ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ))) |
11 | 5, 10 | mpbiri 260 | . . . . 5 ⊢ (𝑆 = ℝ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆)) |
12 | 3, 11 | eqeltrid 2917 | . . . 4 ⊢ (𝑆 = ℝ → 𝐷 ∈ (Met‘𝑆)) |
13 | relco 6092 | . . . . . . . . 9 ⊢ Rel (abs ∘ − ) | |
14 | resdm 5892 | . . . . . . . . 9 ⊢ (Rel (abs ∘ − ) → ((abs ∘ − ) ↾ dom (abs ∘ − )) = (abs ∘ − )) | |
15 | 13, 14 | ax-mp 5 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ dom (abs ∘ − )) = (abs ∘ − ) |
16 | absf 14691 | . . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ | |
17 | ax-resscn 10588 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
18 | fss 6522 | . . . . . . . . . . . 12 ⊢ ((abs:ℂ⟶ℝ ∧ ℝ ⊆ ℂ) → abs:ℂ⟶ℂ) | |
19 | 16, 17, 18 | mp2an 690 | . . . . . . . . . . 11 ⊢ abs:ℂ⟶ℂ |
20 | subf 10882 | . . . . . . . . . . 11 ⊢ − :(ℂ × ℂ)⟶ℂ | |
21 | fco 6526 | . . . . . . . . . . 11 ⊢ ((abs:ℂ⟶ℂ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℂ) | |
22 | 19, 20, 21 | mp2an 690 | . . . . . . . . . 10 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℂ |
23 | 22 | fdmi 6519 | . . . . . . . . 9 ⊢ dom (abs ∘ − ) = (ℂ × ℂ) |
24 | 23 | reseq2i 5845 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ dom (abs ∘ − )) = ((abs ∘ − ) ↾ (ℂ × ℂ)) |
25 | 15, 24 | eqtr3i 2846 | . . . . . . 7 ⊢ (abs ∘ − ) = ((abs ∘ − ) ↾ (ℂ × ℂ)) |
26 | cnmet 23374 | . . . . . . 7 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
27 | 25, 26 | eqeltrri 2910 | . . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) ∈ (Met‘ℂ) |
28 | xpeq12 5575 | . . . . . . . . 9 ⊢ ((𝑆 = ℂ ∧ 𝑆 = ℂ) → (𝑆 × 𝑆) = (ℂ × ℂ)) | |
29 | 28 | anidms 569 | . . . . . . . 8 ⊢ (𝑆 = ℂ → (𝑆 × 𝑆) = (ℂ × ℂ)) |
30 | 29 | reseq2d 5848 | . . . . . . 7 ⊢ (𝑆 = ℂ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (ℂ × ℂ))) |
31 | fveq2 6665 | . . . . . . 7 ⊢ (𝑆 = ℂ → (Met‘𝑆) = (Met‘ℂ)) | |
32 | 30, 31 | eleq12d 2907 | . . . . . 6 ⊢ (𝑆 = ℂ → (((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ↔ ((abs ∘ − ) ↾ (ℂ × ℂ)) ∈ (Met‘ℂ))) |
33 | 27, 32 | mpbiri 260 | . . . . 5 ⊢ (𝑆 = ℂ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆)) |
34 | 3, 33 | eqeltrid 2917 | . . . 4 ⊢ (𝑆 = ℂ → 𝐷 ∈ (Met‘𝑆)) |
35 | 12, 34 | jaoi 853 | . . 3 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 𝐷 ∈ (Met‘𝑆)) |
36 | 1, 2, 35 | 3syl 18 | . 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑆)) |
37 | blpnf 23001 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑆) ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) | |
38 | 36, 37 | sylan 582 | 1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 {cpr 4563 × cxp 5548 dom cdm 5550 ↾ cres 5552 ∘ ccom 5554 Rel wrel 5555 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 ℝcr 10530 +∞cpnf 10666 − cmin 10864 abscabs 14587 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-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-reu 3145 df-rmo 3146 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-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 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-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 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-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 |
This theorem is referenced by: dvconstbi 40659 |
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