cxpcn2 - Metamath Proof Explorer

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Theorem cxpcn2 26723

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 24757	. . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ)
4		rpcn 12944	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
5		ax-1 6	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺))
6		eqid 2737	. . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
7	6	ellogdm 26616	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
8	4, 5, 7	sylanbrc 584	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
9	8	ssriv 3926	. . . . 5 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
10		cnex 11110	. . . . . 6 ⊢ ℂ ∈ V
11	10	difexi 5267	. . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ V
12		restabs 23140	. . . . 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 1464	. . . 4 ⊢ ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ↾_t ℝ⁺) = (𝐽 ↾_t ℝ⁺)
14	1, 13	eqtr4i 2763	. . 3 ⊢ 𝐾 = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ↾_t ℝ⁺)
15	3	a1i 11	. . . 4 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
16		difss 4077	. . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
17		resttopon 23136	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ (-∞(,]0)) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0))))
18	15, 16, 17	sylancl 587	. . 3 ⊢ (⊤ → (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0))))
19	9	a1i 11	. . 3 ⊢ (⊤ → ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0)))
20	3	toponrestid 22896	. . 3 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
21		ssidd 3946	. . 3 ⊢ (⊤ → ℂ ⊆ ℂ)
22		eqid 2737	. . . . 5 ⊢ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾_t (ℂ ∖ (-∞(,]0)))
23	6, 2, 22	cxpcn 26722	. . . 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 23652	. 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 3430 ∖ cdif 3887 ⊆ wss 3890 ‘cfv 6492 (class class class)co 7360 ∈ cmpo 7362 ℂcc 11027 ℝcr 11028 0cc0 11029 -∞cmnf 11168 ℝ⁺crp 12933 (,]cioc 13290 ↾_t crest 17374 TopOpenctopn 17375 ℂ_fldccnfld 21344 TopOnctopon 22885 Cn ccn 23199 ×_t ctx 23535 ↑_𝑐ccxp 26532

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-tan 16027 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-cmp 23362 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-limc 25843 df-dv 25844 df-log 26533 df-cxp 26534

This theorem is referenced by: cxpcn3 26725

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