cxpcn2 - Metamath Proof Explorer

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Theorem cxpcn2 26684

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 24698	. . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ)
4		rpcn 12903	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
5		ax-1 6	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺))
6		eqid 2733	. . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
7	6	ellogdm 26576	. . . . . . 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 3934	. . . . 5 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
10		cnex 11094	. . . . . 6 ⊢ ℂ ∈ V
11	10	difexi 5270	. . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ V
12		restabs 23081	. . . . 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 2759	. . 3 ⊢ 𝐾 = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ↾_t ℝ⁺)
15	3	a1i 11	. . . 4 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
16		difss 4085	. . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
17		resttopon 23077	. . . 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 22837	. . 3 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
21		ssidd 3954	. . 3 ⊢ (⊤ → ℂ ⊆ ℂ)
22		eqid 2733	. . . . 5 ⊢ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾_t (ℂ ∖ (-∞(,]0)))
23	6, 2, 22	cxpcn 26682	. . . 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 23593	. 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 2113 Vcvv 3437 ∖ cdif 3895 ⊆ wss 3898 ‘cfv 6486 (class class class)co 7352 ∈ cmpo 7354 ℂcc 11011 ℝcr 11012 0cc0 11013 -∞cmnf 11151 ℝ⁺crp 12892 (,]cioc 13248 ↾_t crest 17326 TopOpenctopn 17327 ℂ_fldccnfld 21293 TopOnctopon 22826 Cn ccn 23140 ×_t ctx 23476 ↑_𝑐ccxp 26492

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9538 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091 ax-addf 11092

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-fi 9302 df-sup 9333 df-inf 9334 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-ioo 13251 df-ioc 13252 df-ico 13253 df-icc 13254 df-fz 13410 df-fzo 13557 df-fl 13698 df-mod 13776 df-seq 13911 df-exp 13971 df-fac 14183 df-bc 14212 df-hash 14240 df-shft 14976 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-limsup 15380 df-clim 15397 df-rlim 15398 df-sum 15596 df-ef 15976 df-sin 15978 df-cos 15979 df-tan 15980 df-pi 15981 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-rest 17328 df-topn 17329 df-0g 17347 df-gsum 17348 df-topgen 17349 df-pt 17350 df-prds 17353 df-xrs 17408 df-qtop 17413 df-imas 17414 df-xps 17416 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19231 df-cmn 19696 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-lp 23052 df-perf 23053 df-cn 23143 df-cnp 23144 df-haus 23231 df-cmp 23303 df-tx 23478 df-hmeo 23671 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-tms 24238 df-cncf 24799 df-limc 25795 df-dv 25796 df-log 26493 df-cxp 26494

This theorem is referenced by: cxpcn3 26686

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