atansopn - Metamath Proof Explorer

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Theorem atansopn 26896

Description: The domain of continuity of the arctangent is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)

**Hypotheses**
Ref	Expression
atansopn.d	⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
atansopn.s	⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}

**Assertion**
Ref	Expression
atansopn	⊢ 𝑆 ∈ (TopOpen‘ℂ_fld)

Distinct variable group: 𝑦,𝐷 Allowed substitution hint: 𝑆(𝑦)

**Proof of Theorem atansopn**
Step	Hyp	Ref	Expression
1		atansopn.s	. . 3 ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}
2		eqid 2736	. . . 4 ⊢ (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2)))
3	2	mptpreima 6202	. . 3 ⊢ (^◡(𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) “ 𝐷) = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}
4	1, 3	eqtr4i 2762	. 2 ⊢ 𝑆 = (^◡(𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) “ 𝐷)
5		eqid 2736	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
6	5	cnfldtopon 24747	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
7	6	a1i 11	. . . . 5 ⊢ (⊤ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
8		1cnd 11139	. . . . . 6 ⊢ (⊤ → 1 ∈ ℂ)
9	7, 7, 8	cnmptc 23627	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
10		2nn0 12454	. . . . . 6 ⊢ 2 ∈ ℕ₀
11	5	expcn 24839	. . . . . 6 ⊢ (2 ∈ ℕ₀ → (𝑦 ∈ ℂ ↦ (𝑦↑2)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
12	10, 11	mp1i 13	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦↑2)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
13	5	addcn 24831	. . . . . 6 ⊢ + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
14	13	a1i 11	. . . . 5 ⊢ (⊤ → + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
15	7, 9, 12, 14	cnmpt12f 23631	. . . 4 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
16	15	mptru 1549	. . 3 ⊢ (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
17		atansopn.d	. . . 4 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
18	17	logdmopn 26613	. . 3 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
19		cnima 23230	. . 3 ⊢ (((𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)) ∧ 𝐷 ∈ (TopOpen‘ℂ_fld)) → (^◡(𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) “ 𝐷) ∈ (TopOpen‘ℂ_fld))
20	16, 18, 19	mp2an 693	. 2 ⊢ (^◡(𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) “ 𝐷) ∈ (TopOpen‘ℂ_fld)
21	4, 20	eqeltri 2832	1 ⊢ 𝑆 ∈ (TopOpen‘ℂ_fld)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 {crab 3389 ∖ cdif 3886 ↦ cmpt 5166 ^◡ccnv 5630 “ cima 5634 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 -∞cmnf 11177 2c2 12236 ℕ₀cn0 12437 (,]cioc 13299 ↑cexp 14023 TopOpenctopn 17384 ℂ_fldccnfld 21352 TopOnctopon 22875 Cn ccn 23189 ×_t ctx 23525

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-icc 13305 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-cn 23192 df-cnp 23193 df-tx 23527 df-hmeo 23720 df-xms 24285 df-ms 24286 df-tms 24287

This theorem is referenced by: dvatan 26899

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