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Theorem cxpcnOLD 26711

Description: Obsolete version of cxpcn 26710 as of 17-Apr-2025. (Contributed by Mario Carneiro, 1-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
cxpcnOLD.d	⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
cxpcnOLD.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
cxpcnOLD.k	⊢ 𝐾 = (𝐽 ↾_t 𝐷)

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

Distinct variable groups: 𝑥,𝑦,𝐷 𝑥,𝐽,𝑦 𝑥,𝐾,𝑦

**Proof of Theorem cxpcnOLD**
Step	Hyp	Ref	Expression
1		cxpcnOLD.d	. . . . . . 7 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
2	1	ellogdm 26604	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
3	2	simplbi 497	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
4	3	adantr 480	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ∈ ℂ)
5	1	logdmn0 26605	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
6	5	adantr 480	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ≠ 0)
7		simpr 484	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
8	4, 6, 7	cxpefd 26677	. . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → (𝑥↑_𝑐𝑦) = (exp‘(𝑦 · (log‘𝑥))))
9	8	mpoeq3ia 7436	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
10		cxpcnOLD.k	. . . . 5 ⊢ 𝐾 = (𝐽 ↾_t 𝐷)
11		cxpcnOLD.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
12	11	cnfldtopon 24726	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
13	12	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
14	3	ssriv 3937	. . . . . 6 ⊢ 𝐷 ⊆ ℂ
15		resttopon 23105	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
16	13, 14, 15	sylancl 586	. . . . 5 ⊢ (⊤ → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
17	10, 16	eqeltrid 2840	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐷))
18	17, 13	cnmpt2nd 23613	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑦) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
19		fvres 6853	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥))
20	19	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥))
21	20	mpoeq3ia 7436	. . . . . 6 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥))
22	17, 13	cnmpt1st 23612	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑥) ∈ ((𝐾 ×_t 𝐽) Cn 𝐾))
23	1	logcn 26612	. . . . . . . . 9 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
24		ssid 3956	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
25	12	toponrestid 22865	. . . . . . . . . . 11 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
26	11, 10, 25	cncfcn 24859	. . . . . . . . . 10 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (𝐾 Cn 𝐽))
27	14, 24, 26	mp2an 692	. . . . . . . . 9 ⊢ (𝐷–cn→ℂ) = (𝐾 Cn 𝐽)
28	23, 27	eleqtri 2834	. . . . . . . 8 ⊢ (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽)
29	28	a1i 11	. . . . . . 7 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽))
30	17, 13, 22, 29	cnmpt21f 23616	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
31	21, 30	eqeltrrid 2841	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
32	11	mulcn 24812	. . . . . 6 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
33	32	a1i 11	. . . . 5 ⊢ (⊤ → · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
34	17, 13, 18, 31, 33	cnmpt22f 23619	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑦 · (log‘𝑥))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
35		efcn 26409	. . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ)
36	11	cncfcn1 24860	. . . . . 6 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
37	35, 36	eleqtri 2834	. . . . 5 ⊢ exp ∈ (𝐽 Cn 𝐽)
38	37	a1i 11	. . . 4 ⊢ (⊤ → exp ∈ (𝐽 Cn 𝐽))
39	17, 13, 34, 38	cnmpt21f 23616	. . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
40	39	mptru 1548	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)
41	9, 40	eqeltri 2832	1 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ⊤wtru 1542 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 ⊆ wss 3901 ↾ cres 5626 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 ℂcc 11024 ℝcr 11025 0cc0 11026 · cmul 11031 -∞cmnf 11164 ℝ⁺crp 12905 (,]cioc 13262 expce 15984 ↾_t crest 17340 TopOpenctopn 17341 ℂ_fldccnfld 21309 TopOnctopon 22854 Cn ccn 23168 ×_t ctx 23504 –cn→ccncf 24825 logclog 26519 ↑_𝑐ccxp 26520

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-tan 15994 df-pi 15995 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824 df-log 26521 df-cxp 26522

This theorem is referenced by: (None)

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