atansopn - Metamath Proof Explorer

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Theorem atansopn 25509

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 2821	. . . 4 ⊢ (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2)))
3	2	mptpreima 6091	. . 3 ⊢ (^◡(𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) “ 𝐷) = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}
4	1, 3	eqtr4i 2847	. 2 ⊢ 𝑆 = (^◡(𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) “ 𝐷)
5		eqid 2821	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
6	5	cnfldtopon 23390	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
7	6	a1i 11	. . . . 5 ⊢ (⊤ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
8		1cnd 10635	. . . . . 6 ⊢ (⊤ → 1 ∈ ℂ)
9	7, 7, 8	cnmptc 22269	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
10		2nn0 11913	. . . . . 6 ⊢ 2 ∈ ℕ₀
11	5	expcn 23479	. . . . . 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 23472	. . . . . 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 22273	. . . 4 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
16	15	mptru 1540	. . 3 ⊢ (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
17		atansopn.d	. . . 4 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
18	17	logdmopn 25231	. . 3 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
19		cnima 21872	. . 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 690	. 2 ⊢ (^◡(𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) “ 𝐷) ∈ (TopOpen‘ℂ_fld)
21	4, 20	eqeltri 2909	1 ⊢ 𝑆 ∈ (TopOpen‘ℂ_fld)

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 {crab 3142 ∖ cdif 3932 ↦ cmpt 5145 ^◡ccnv 5553 “ cima 5557 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 0cc0 10536 1c1 10537 + caddc 10539 -∞cmnf 10672 2c2 11691 ℕ₀cn0 11896 (,]cioc 12738 ↑cexp 13428 TopOpenctopn 16694 ℂ_fldccnfld 20544 TopOnctopon 21517 Cn ccn 21831 ×_t ctx 22167

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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616

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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-cn 21834 df-cnp 21835 df-tx 22169 df-hmeo 22362 df-xms 22929 df-ms 22930 df-tms 22931

This theorem is referenced by: dvatan 25512

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