cxpcn2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cxpcn2		Structured version Visualization version GIF version

Theorem cxpcn2 26681

Description: Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.)

**Hypotheses**
Ref	Expression
cxpcn2.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
cxpcn2.k	⊢ 𝐾 = (𝐽 ↾_t ℝ⁺)

**Assertion**
Ref	Expression
cxpcn2	⊢ (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Distinct variable group: 𝑥,𝑦,𝐽 Allowed substitution hints: 𝐾(𝑥,𝑦)

**Proof of Theorem cxpcn2**
Step	Hyp	Ref	Expression
1		cxpcn2.k	. . . 4 ⊢ 𝐾 = (𝐽 ↾_t ℝ⁺)
2		cxpcn2.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
3	2	cnfldtopon 24695	. . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ)
4		rpcn 12898	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
5		ax-1 6	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺))
6		eqid 2731	. . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
7	6	ellogdm 26573	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
8	4, 5, 7	sylanbrc 583	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
9	8	ssriv 3938	. . . . 5 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
10		cnex 11084	. . . . . 6 ⊢ ℂ ∈ V
11	10	difexi 5268	. . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ V
12		restabs 23078	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0)) ∧ (ℂ ∖ (-∞(,]0)) ∈ V) → ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ↾_t ℝ⁺) = (𝐽 ↾_t ℝ⁺))
13	3, 9, 11, 12	mp3an 1463	. . . 4 ⊢ ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ↾_t ℝ⁺) = (𝐽 ↾_t ℝ⁺)
14	1, 13	eqtr4i 2757	. . 3 ⊢ 𝐾 = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ↾_t ℝ⁺)
15	3	a1i 11	. . . 4 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
16		difss 4086	. . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
17		resttopon 23074	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ (-∞(,]0)) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0))))
18	15, 16, 17	sylancl 586	. . 3 ⊢ (⊤ → (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0))))
19	9	a1i 11	. . 3 ⊢ (⊤ → ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0)))
20	3	toponrestid 22834	. . 3 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
21		ssidd 3958	. . 3 ⊢ (⊤ → ℂ ⊆ ℂ)
22		eqid 2731	. . . . 5 ⊢ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾_t (ℂ ∖ (-∞(,]0)))
23	6, 2, 22	cxpcn 26679	. . . 4 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)), 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ×_t 𝐽) Cn 𝐽)
24	23	a1i 11	. . 3 ⊢ (⊤ → (𝑥 ∈ (ℂ ∖ (-∞(,]0)), 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ×_t 𝐽) Cn 𝐽))
25	14, 18, 19, 20, 15, 21, 24	cnmpt2res 23590	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
26	25	mptru 1548	1 ⊢ (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ⊤wtru 1542 ∈ wcel 2111 Vcvv 3436 ∖ cdif 3899 ⊆ wss 3902 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 ℂcc 11001 ℝcr 11002 0cc0 11003 -∞cmnf 11141 ℝ⁺crp 12887 (,]cioc 13243 ↾_t crest 17321 TopOpenctopn 17322 ℂ_fldccnfld 21289 TopOnctopon 22823 Cn ccn 23137 ×_t ctx 23473 ↑_𝑐ccxp 26489

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081 ax-addf 11082

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-ioo 13246 df-ioc 13247 df-ico 13248 df-icc 13249 df-fz 13405 df-fzo 13552 df-fl 13693 df-mod 13771 df-seq 13906 df-exp 13966 df-fac 14178 df-bc 14207 df-hash 14235 df-shft 14971 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-limsup 15375 df-clim 15392 df-rlim 15393 df-sum 15591 df-ef 15971 df-sin 15973 df-cos 15974 df-tan 15975 df-pi 15976 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-rest 17323 df-topn 17324 df-0g 17342 df-gsum 17343 df-topgen 17344 df-pt 17345 df-prds 17348 df-xrs 17403 df-qtop 17408 df-imas 17409 df-xps 17411 df-mre 17485 df-mrc 17486 df-acs 17488 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-submnd 18689 df-mulg 18978 df-cntz 19227 df-cmn 19692 df-psmet 21281 df-xmet 21282 df-met 21283 df-bl 21284 df-mopn 21285 df-fbas 21286 df-fg 21287 df-cnfld 21290 df-top 22807 df-topon 22824 df-topsp 22846 df-bases 22859 df-cld 22932 df-ntr 22933 df-cls 22934 df-nei 23011 df-lp 23049 df-perf 23050 df-cn 23140 df-cnp 23141 df-haus 23228 df-cmp 23300 df-tx 23475 df-hmeo 23668 df-fil 23759 df-fm 23851 df-flim 23852 df-flf 23853 df-xms 24233 df-ms 24234 df-tms 24235 df-cncf 24796 df-limc 25792 df-dv 25793 df-log 26490 df-cxp 26491

This theorem is referenced by: cxpcn3 26683

W3C validator