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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 24407. (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 4649 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
| 3 | sblpnf.d | . . . . 5 ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) | |
| 4 | eqid 2737 | . . . . . . 7 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
| 5 | 4 | remet 24811 | . . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ) |
| 6 | xpeq12 5710 | . . . . . . . . 9 ⊢ ((𝑆 = ℝ ∧ 𝑆 = ℝ) → (𝑆 × 𝑆) = (ℝ × ℝ)) | |
| 7 | 6 | anidms 566 | . . . . . . . 8 ⊢ (𝑆 = ℝ → (𝑆 × 𝑆) = (ℝ × ℝ)) |
| 8 | 7 | reseq2d 5997 | . . . . . . 7 ⊢ (𝑆 = ℝ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (ℝ × ℝ))) |
| 9 | fveq2 6906 | . . . . . . 7 ⊢ (𝑆 = ℝ → (Met‘𝑆) = (Met‘ℝ)) | |
| 10 | 8, 9 | eleq12d 2835 | . . . . . 6 ⊢ (𝑆 = ℝ → (((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ↔ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ))) |
| 11 | 5, 10 | mpbiri 258 | . . . . 5 ⊢ (𝑆 = ℝ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆)) |
| 12 | 3, 11 | eqeltrid 2845 | . . . 4 ⊢ (𝑆 = ℝ → 𝐷 ∈ (Met‘𝑆)) |
| 13 | relco 6126 | . . . . . . . . 9 ⊢ Rel (abs ∘ − ) | |
| 14 | resdm 6044 | . . . . . . . . 9 ⊢ (Rel (abs ∘ − ) → ((abs ∘ − ) ↾ dom (abs ∘ − )) = (abs ∘ − )) | |
| 15 | 13, 14 | ax-mp 5 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ dom (abs ∘ − )) = (abs ∘ − ) |
| 16 | absf 15376 | . . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ | |
| 17 | ax-resscn 11212 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
| 18 | fss 6752 | . . . . . . . . . . . 12 ⊢ ((abs:ℂ⟶ℝ ∧ ℝ ⊆ ℂ) → abs:ℂ⟶ℂ) | |
| 19 | 16, 17, 18 | mp2an 692 | . . . . . . . . . . 11 ⊢ abs:ℂ⟶ℂ |
| 20 | subf 11510 | . . . . . . . . . . 11 ⊢ − :(ℂ × ℂ)⟶ℂ | |
| 21 | fco 6760 | . . . . . . . . . . 11 ⊢ ((abs:ℂ⟶ℂ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℂ) | |
| 22 | 19, 20, 21 | mp2an 692 | . . . . . . . . . 10 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℂ |
| 23 | 22 | fdmi 6747 | . . . . . . . . 9 ⊢ dom (abs ∘ − ) = (ℂ × ℂ) |
| 24 | 23 | reseq2i 5994 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ dom (abs ∘ − )) = ((abs ∘ − ) ↾ (ℂ × ℂ)) |
| 25 | 15, 24 | eqtr3i 2767 | . . . . . . 7 ⊢ (abs ∘ − ) = ((abs ∘ − ) ↾ (ℂ × ℂ)) |
| 26 | cnmet 24792 | . . . . . . 7 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
| 27 | 25, 26 | eqeltrri 2838 | . . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) ∈ (Met‘ℂ) |
| 28 | xpeq12 5710 | . . . . . . . . 9 ⊢ ((𝑆 = ℂ ∧ 𝑆 = ℂ) → (𝑆 × 𝑆) = (ℂ × ℂ)) | |
| 29 | 28 | anidms 566 | . . . . . . . 8 ⊢ (𝑆 = ℂ → (𝑆 × 𝑆) = (ℂ × ℂ)) |
| 30 | 29 | reseq2d 5997 | . . . . . . 7 ⊢ (𝑆 = ℂ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (ℂ × ℂ))) |
| 31 | fveq2 6906 | . . . . . . 7 ⊢ (𝑆 = ℂ → (Met‘𝑆) = (Met‘ℂ)) | |
| 32 | 30, 31 | eleq12d 2835 | . . . . . 6 ⊢ (𝑆 = ℂ → (((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ↔ ((abs ∘ − ) ↾ (ℂ × ℂ)) ∈ (Met‘ℂ))) |
| 33 | 27, 32 | mpbiri 258 | . . . . 5 ⊢ (𝑆 = ℂ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆)) |
| 34 | 3, 33 | eqeltrid 2845 | . . . 4 ⊢ (𝑆 = ℂ → 𝐷 ∈ (Met‘𝑆)) |
| 35 | 12, 34 | jaoi 858 | . . 3 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 𝐷 ∈ (Met‘𝑆)) |
| 36 | 1, 2, 35 | 3syl 18 | . 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑆)) |
| 37 | blpnf 24407 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑆) ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) | |
| 38 | 36, 37 | sylan 580 | 1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 {cpr 4628 × cxp 5683 dom cdm 5685 ↾ cres 5687 ∘ ccom 5689 Rel wrel 5690 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 +∞cpnf 11292 − cmin 11492 abscabs 15273 Metcmet 21350 ballcbl 21351 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 |
| This theorem is referenced by: dvconstbi 44353 |
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