cxpcn2 - Metamath Proof Explorer

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Theorem cxpcn2 25487

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 23535	. . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ)
4		rpcn 12482	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
5		ax-1 6	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺))
6		eqid 2738	. . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
7	6	ellogdm 25382	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
8	4, 5, 7	sylanbrc 586	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
9	8	ssriv 3881	. . . . 5 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
10		cnex 10696	. . . . . 6 ⊢ ℂ ∈ V
11	10	difexi 5196	. . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ V
12		restabs 21916	. . . . 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 1462	. . . 4 ⊢ ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ↾_t ℝ⁺) = (𝐽 ↾_t ℝ⁺)
14	1, 13	eqtr4i 2764	. . 3 ⊢ 𝐾 = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ↾_t ℝ⁺)
15	3	a1i 11	. . . 4 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
16		difss 4022	. . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
17		resttopon 21912	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ (-∞(,]0)) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0))))
18	15, 16, 17	sylancl 589	. . 3 ⊢ (⊤ → (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0))))
19	9	a1i 11	. . 3 ⊢ (⊤ → ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0)))
20	3	toponrestid 21672	. . 3 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
21		ssidd 3900	. . 3 ⊢ (⊤ → ℂ ⊆ ℂ)
22		eqid 2738	. . . . 5 ⊢ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾_t (ℂ ∖ (-∞(,]0)))
23	6, 2, 22	cxpcn 25486	. . . 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 22428	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
26	25	mptru 1549	1 ⊢ (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 Vcvv 3398 ∖ cdif 3840 ⊆ wss 3843 ‘cfv 6339 (class class class)co 7170 ∈ cmpo 7172 ℂcc 10613 ℝcr 10614 0cc0 10615 -∞cmnf 10751 ℝ⁺crp 12472 (,]cioc 12822 ↾_t crest 16797 TopOpenctopn 16798 ℂ_fldccnfld 20217 TopOnctopon 21661 Cn ccn 21975 ×_t ctx 22311 ↑_𝑐ccxp 25299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-fi 8948 df-sup 8979 df-inf 8980 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-ioo 12825 df-ioc 12826 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-fl 13253 df-mod 13329 df-seq 13461 df-exp 13522 df-fac 13726 df-bc 13755 df-hash 13783 df-shft 14516 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-limsup 14918 df-clim 14935 df-rlim 14936 df-sum 15136 df-ef 15513 df-sin 15515 df-cos 15516 df-tan 15517 df-pi 15518 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-rest 16799 df-topn 16800 df-0g 16818 df-gsum 16819 df-topgen 16820 df-pt 16821 df-prds 16824 df-xrs 16878 df-qtop 16883 df-imas 16884 df-xps 16886 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-mulg 18343 df-cntz 18565 df-cmn 19026 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-fbas 20214 df-fg 20215 df-cnfld 20218 df-top 21645 df-topon 21662 df-topsp 21684 df-bases 21697 df-cld 21770 df-ntr 21771 df-cls 21772 df-nei 21849 df-lp 21887 df-perf 21888 df-cn 21978 df-cnp 21979 df-haus 22066 df-cmp 22138 df-tx 22313 df-hmeo 22506 df-fil 22597 df-fm 22689 df-flim 22690 df-flf 22691 df-xms 23073 df-ms 23074 df-tms 23075 df-cncf 23630 df-limc 24618 df-dv 24619 df-log 25300 df-cxp 25301

This theorem is referenced by: cxpcn3 25489

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