cnbl0 - Intuitionistic Logic Explorer

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Theorem cnbl0 13701

Description: Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)

**Hypothesis**
Ref	Expression
cnblcld.1	⊢ 𝐷 = (abs ∘ − )

**Assertion**
Ref	Expression
cnbl0	⊢ (𝑅 ∈ ℝ^* → (^◡abs “ (0[,)𝑅)) = (0(ball‘𝐷)𝑅))

Proof of Theorem cnbl0 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 980	. . . . . 6 ⊢ (((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) < 𝑅) ↔ (((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥)) ∧ (abs‘𝑥) < 𝑅))
2		abscl 11044	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ)
3		absge0 11053	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 0 ≤ (abs‘𝑥))
4	2, 3	jca 306	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥)))
5	4	adantl 277	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥)))
6	5	biantrurd 305	. . . . . 6 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) < 𝑅 ↔ (((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥)) ∧ (abs‘𝑥) < 𝑅)))
7	1, 6	bitr4id 199	. . . . 5 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → (((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) < 𝑅) ↔ (abs‘𝑥) < 𝑅))
8		0re 7948	. . . . . 6 ⊢ 0 ∈ ℝ
9		simpl 109	. . . . . 6 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → 𝑅 ∈ ℝ^*)
10		elico2 9924	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → ((abs‘𝑥) ∈ (0[,)𝑅) ↔ ((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) < 𝑅)))
11	8, 9, 10	sylancr 414	. . . . 5 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ (0[,)𝑅) ↔ ((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) < 𝑅)))
12		0cn 7940	. . . . . . . . 9 ⊢ 0 ∈ ℂ
13		cnblcld.1	. . . . . . . . . . 11 ⊢ 𝐷 = (abs ∘ − )
14	13	cnmetdval 13696	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘(0 − 𝑥)))
15		abssub 11094	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(0 − 𝑥)) = (abs‘(𝑥 − 0)))
16	14, 15	eqtrd 2210	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘(𝑥 − 0)))
17	12, 16	mpan 424	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (0𝐷𝑥) = (abs‘(𝑥 − 0)))
18		subid1 8167	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥)
19	18	fveq2d 5515	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (abs‘(𝑥 − 0)) = (abs‘𝑥))
20	17, 19	eqtrd 2210	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (0𝐷𝑥) = (abs‘𝑥))
21	20	adantl 277	. . . . . 6 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘𝑥))
22	21	breq1d 4010	. . . . 5 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → ((0𝐷𝑥) < 𝑅 ↔ (abs‘𝑥) < 𝑅))
23	7, 11, 22	3bitr4d 220	. . . 4 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ (0[,)𝑅) ↔ (0𝐷𝑥) < 𝑅))
24	23	pm5.32da 452	. . 3 ⊢ (𝑅 ∈ ℝ^* → ((𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,)𝑅)) ↔ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) < 𝑅)))
25		absf 11103	. . . . 5 ⊢ abs:ℂ⟶ℝ
26		ffn 5361	. . . . 5 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
27	25, 26	ax-mp 5	. . . 4 ⊢ abs Fn ℂ
28		elpreima 5631	. . . 4 ⊢ (abs Fn ℂ → (𝑥 ∈ (^◡abs “ (0[,)𝑅)) ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,)𝑅))))
29	27, 28	mp1i 10	. . 3 ⊢ (𝑅 ∈ ℝ^* → (𝑥 ∈ (^◡abs “ (0[,)𝑅)) ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,)𝑅))))
30		cnxmet 13698	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
31	13, 30	eqeltri 2250	. . . 4 ⊢ 𝐷 ∈ (∞Met‘ℂ)
32		elbl 13558	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (0(ball‘𝐷)𝑅) ↔ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) < 𝑅)))
33	31, 12, 32	mp3an12 1327	. . 3 ⊢ (𝑅 ∈ ℝ^* → (𝑥 ∈ (0(ball‘𝐷)𝑅) ↔ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) < 𝑅)))
34	24, 29, 33	3bitr4d 220	. 2 ⊢ (𝑅 ∈ ℝ^* → (𝑥 ∈ (^◡abs “ (0[,)𝑅)) ↔ 𝑥 ∈ (0(ball‘𝐷)𝑅)))
35	34	eqrdv 2175	1 ⊢ (𝑅 ∈ ℝ^* → (^◡abs “ (0[,)𝑅)) = (0(ball‘𝐷)𝑅))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ^◡ccnv 4622 “ cima 4626 ∘ ccom 4627 Fn wfn 5207 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 ℝcr 7801 0cc0 7802 ℝ^*cxr 7981 < clt 7982 ≤ cle 7983 − cmin 8118 [,)cico 9877 abscabs 10990 ∞Metcxmet 13147 ballcbl 13149

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-map 6644 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-rp 9641 df-xadd 9760 df-ico 9881 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-psmet 13154 df-xmet 13155 df-met 13156 df-bl 13157

This theorem is referenced by: (None)

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