atansopn - Metamath Proof Explorer

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Theorem atansopn 25179

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 2793	. . . 4 ⊢ (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2)))
3	2	mptpreima 5959	. . 3 ⊢ (^◡(𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) “ 𝐷) = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}
4	1, 3	eqtr4i 2820	. 2 ⊢ 𝑆 = (^◡(𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) “ 𝐷)
5		eqid 2793	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
6	5	cnfldtopon 23062	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
7	6	a1i 11	. . . . 5 ⊢ (⊤ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
8		1cnd 10471	. . . . . 6 ⊢ (⊤ → 1 ∈ ℂ)
9	7, 7, 8	cnmptc 21942	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
10		2nn0 11751	. . . . . 6 ⊢ 2 ∈ ℕ₀
11	5	expcn 23151	. . . . . 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 23144	. . . . . 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 21946	. . . 4 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
16	15	mptru 1527	. . 3 ⊢ (𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
17		atansopn.d	. . . 4 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
18	17	logdmopn 24901	. . 3 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
19		cnima 21545	. . 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 688	. 2 ⊢ (^◡(𝑦 ∈ ℂ ↦ (1 + (𝑦↑2))) “ 𝐷) ∈ (TopOpen‘ℂ_fld)
21	4, 20	eqeltri 2877	1 ⊢ 𝑆 ∈ (TopOpen‘ℂ_fld)

Colors of variables: wff setvar class

Syntax hints: = wceq 1520 ⊤wtru 1521 ∈ wcel 2079 {crab 3107 ∖ cdif 3851 ↦ cmpt 5035 ^◡ccnv 5434 “ cima 5438 ‘cfv 6217 (class class class)co 7007 ℂcc 10370 0cc0 10372 1c1 10373 + caddc 10375 -∞cmnf 10508 2c2 11529 ℕ₀cn0 11734 (,]cioc 12578 ↑cexp 13267 TopOpenctopn 16512 ℂ_fldccnfld 20215 TopOnctopon 21190 Cn ccn 21504 ×_t ctx 21840

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 ax-addf 10451 ax-mulf 10452

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-oadd 7948 df-er 8130 df-map 8249 df-ixp 8301 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-fi 8711 df-sup 8742 df-inf 8743 df-oi 8810 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-q 12187 df-rp 12229 df-xneg 12346 df-xadd 12347 df-xmul 12348 df-ioo 12581 df-ioc 12582 df-icc 12584 df-fz 12732 df-fzo 12873 df-seq 13208 df-exp 13268 df-hash 13529 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-starv 16397 df-sca 16398 df-vsca 16399 df-ip 16400 df-tset 16401 df-ple 16402 df-ds 16404 df-unif 16405 df-hom 16406 df-cco 16407 df-rest 16513 df-topn 16514 df-0g 16532 df-gsum 16533 df-topgen 16534 df-pt 16535 df-prds 16538 df-xrs 16592 df-qtop 16597 df-imas 16598 df-xps 16600 df-mre 16674 df-mrc 16675 df-acs 16677 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-submnd 17763 df-mulg 17970 df-cntz 18176 df-cmn 18623 df-psmet 20207 df-xmet 20208 df-met 20209 df-bl 20210 df-mopn 20211 df-cnfld 20216 df-top 21174 df-topon 21191 df-topsp 21213 df-bases 21226 df-cld 21299 df-cn 21507 df-cnp 21508 df-tx 21842 df-hmeo 22035 df-xms 22601 df-ms 22602 df-tms 22603

This theorem is referenced by: dvatan 25182

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