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Theorem cnblcld 14074

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

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

**Assertion**
Ref	Expression
cnblcld	⊢ (𝑅 ∈ ℝ^* → (^◡abs “ (0[,]𝑅)) = {𝑥 ∈ ℂ ∣ (0𝐷𝑥) ≤ 𝑅})

Distinct variable groups: 𝑥,𝐷 𝑥,𝑅

**Proof of Theorem cnblcld**
Step	Hyp	Ref	Expression
1		absf 11121	. . . . 5 ⊢ abs:ℂ⟶ℝ
2		ffn 5367	. . . . 5 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
3		elpreima 5637	. . . . 5 ⊢ (abs Fn ℂ → (𝑥 ∈ (^◡abs “ (0[,]𝑅)) ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,]𝑅))))
4	1, 2, 3	mp2b 8	. . . 4 ⊢ (𝑥 ∈ (^◡abs “ (0[,]𝑅)) ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,]𝑅)))
5		df-3an 980	. . . . . . 7 ⊢ (((abs‘𝑥) ∈ ℝ^* ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≤ 𝑅) ↔ (((abs‘𝑥) ∈ ℝ^* ∧ 0 ≤ (abs‘𝑥)) ∧ (abs‘𝑥) ≤ 𝑅))
6		abscl 11062	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ)
7	6	rexrd 8009	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ^*)
8		absge0 11071	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → 0 ≤ (abs‘𝑥))
9	7, 8	jca 306	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((abs‘𝑥) ∈ ℝ^* ∧ 0 ≤ (abs‘𝑥)))
10	9	adantl 277	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ ℝ^* ∧ 0 ≤ (abs‘𝑥)))
11	10	biantrurd 305	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ≤ 𝑅 ↔ (((abs‘𝑥) ∈ ℝ^* ∧ 0 ≤ (abs‘𝑥)) ∧ (abs‘𝑥) ≤ 𝑅)))
12	5, 11	bitr4id 199	. . . . . 6 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → (((abs‘𝑥) ∈ ℝ^* ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≤ 𝑅) ↔ (abs‘𝑥) ≤ 𝑅))
13		0xr 8006	. . . . . . 7 ⊢ 0 ∈ ℝ^*
14		simpl 109	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → 𝑅 ∈ ℝ^*)
15		elicc1 9926	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → ((abs‘𝑥) ∈ (0[,]𝑅) ↔ ((abs‘𝑥) ∈ ℝ^ ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≤ 𝑅)))
16	13, 14, 15	sylancr 414	. . . . . 6 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ (0[,]𝑅) ↔ ((abs‘𝑥) ∈ ℝ^* ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≤ 𝑅)))
17		0cn 7951	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
18		cnblcld.1	. . . . . . . . . . . 12 ⊢ 𝐷 = (abs ∘ − )
19	18	cnmetdval 14068	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘(0 − 𝑥)))
20		abssub 11112	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(0 − 𝑥)) = (abs‘(𝑥 − 0)))
21	19, 20	eqtrd 2210	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘(𝑥 − 0)))
22	17, 21	mpan 424	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (0𝐷𝑥) = (abs‘(𝑥 − 0)))
23		subid1 8179	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥)
24	23	fveq2d 5521	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (abs‘(𝑥 − 0)) = (abs‘𝑥))
25	22, 24	eqtrd 2210	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (0𝐷𝑥) = (abs‘𝑥))
26	25	adantl 277	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘𝑥))
27	26	breq1d 4015	. . . . . 6 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → ((0𝐷𝑥) ≤ 𝑅 ↔ (abs‘𝑥) ≤ 𝑅))
28	12, 16, 27	3bitr4d 220	. . . . 5 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ (0[,]𝑅) ↔ (0𝐷𝑥) ≤ 𝑅))
29	28	pm5.32da 452	. . . 4 ⊢ (𝑅 ∈ ℝ^* → ((𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,]𝑅)) ↔ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) ≤ 𝑅)))
30	4, 29	bitrid 192	. . 3 ⊢ (𝑅 ∈ ℝ^* → (𝑥 ∈ (^◡abs “ (0[,]𝑅)) ↔ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) ≤ 𝑅)))
31	30	abbi2dv 2296	. 2 ⊢ (𝑅 ∈ ℝ^* → (^◡abs “ (0[,]𝑅)) = {𝑥 ∣ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) ≤ 𝑅)})
32		df-rab 2464	. 2 ⊢ {𝑥 ∈ ℂ ∣ (0𝐷𝑥) ≤ 𝑅} = {𝑥 ∣ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) ≤ 𝑅)}
33	31, 32	eqtr4di 2228	1 ⊢ (𝑅 ∈ ℝ^* → (^◡abs “ (0[,]𝑅)) = {𝑥 ∈ ℂ ∣ (0𝐷𝑥) ≤ 𝑅})

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 {cab 2163 {crab 2459 class class class wbr 4005 ^◡ccnv 4627 “ cima 4631 ∘ ccom 4632 Fn wfn 5213 ⟶wf 5214 ‘cfv 5218 (class class class)co 5877 ℂcc 7811 ℝcr 7812 0cc0 7813 ℝ^*cxr 7993 ≤ cle 7995 − cmin 8130 [,]cicc 9893 abscabs 11008

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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933

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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-rp 9656 df-icc 9897 df-seqfrec 10448 df-exp 10522 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010

This theorem is referenced by: (None)

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