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Theorem cpncn 25892

Description: A 𝓑C^𝑛 function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)

**Assertion**
Ref	Expression
cpncn	⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ (dom 𝐹–cn→ℂ))

**Proof of Theorem cpncn**
Step	Hyp	Ref	Expression
1		recnprss 25859	. . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
2	1	adantr 480	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑆 ⊆ ℂ)
3		simpl 482	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑆 ∈ {ℝ, ℂ})
4		0nn0 12414	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 0 ∈ ℕ₀)
6		elfvdm 6866	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁) → 𝑁 ∈ dom (𝓑C^𝑛‘𝑆))
7	6	adantl 481	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ dom (𝓑C^𝑛‘𝑆))
8		fncpn 25889	. . . . . . . . . 10 ⊢ (𝑆 ⊆ ℂ → (𝓑C^𝑛‘𝑆) Fn ℕ₀)
9		fndm 6593	. . . . . . . . . 10 ⊢ ((𝓑C^𝑛‘𝑆) Fn ℕ₀ → dom (𝓑C^𝑛‘𝑆) = ℕ₀)
10	2, 8, 9	3syl 18	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → dom (𝓑C^𝑛‘𝑆) = ℕ₀)
11	7, 10	eleqtrd 2836	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ ℕ₀)
12		nn0uz 12787	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
13	11, 12	eleqtrdi 2844	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ (ℤ_≥‘0))
14		cpnord 25891	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘0)) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘0))
15	3, 5, 13, 14	syl3anc 1373	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘0))
16		simpr 484	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁))
17	15, 16	sseldd 3932	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘0))
18		elcpn 25890	. . . . . 6 ⊢ ((𝑆 ⊆ ℂ ∧ 0 ∈ ℕ₀) → (𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘0) ↔ (𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))))
19	2, 5, 18	syl2anc 584	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → (𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘0) ↔ (𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))))
20	17, 19	mpbid 232	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → (𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ)))
21	20	simpld 494	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
22		dvn0 25880	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
23	2, 21, 22	syl2anc 584	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
24	20	simprd 495	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝑆 D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))
25	23, 24	eqeltrrd 2835	1 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ (dom 𝐹–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ⊆ wss 3899 {cpr 4580 dom cdm 5622 Fn wfn 6485 ‘cfv 6490 (class class class)co 7356 ↑_pm cpm 8762 ℂcc 11022 ℝcr 11023 0cc0 11024 ℕ₀cn0 12399 ℤ_≥cuz 12749 –cn→ccncf 24823 D^𝑛 cdvn 25819 𝓑C^𝑛ccpn 25820

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 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

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 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-icc 13266 df-fz 13422 df-fzo 13569 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-limc 25821 df-dv 25822 df-dvn 25823 df-cpn 25824

This theorem is referenced by: (None)

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