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Theorem cpncn 26104

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 26072	. . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
2	1	adantr 485	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑆 ⊆ ℂ)
3		simpl 487	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑆 ∈ {ℝ, ℂ})
4		0nn0 12523	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 0 ∈ ℕ₀)
6		elfvdm 6915	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁) → 𝑁 ∈ dom (𝓑C^𝑛‘𝑆))
7	6	adantl 486	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ dom (𝓑C^𝑛‘𝑆))
8		fncpn 26101	. . . . . . . . . 10 ⊢ (𝑆 ⊆ ℂ → (𝓑C^𝑛‘𝑆) Fn ℕ₀)
9		fndm 6638	. . . . . . . . . 10 ⊢ ((𝓑C^𝑛‘𝑆) Fn ℕ₀ → dom (𝓑C^𝑛‘𝑆) = ℕ₀)
10	2, 8, 9	3syl 19	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → dom (𝓑C^𝑛‘𝑆) = ℕ₀)
11	7, 10	eleqtrd 2865	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ ℕ₀)
12		nn0uz 12904	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
13	11, 12	eleqtrdi 2873	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ (ℤ_≥‘0))
14		cpnord 26103	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘0)) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘0))
15	3, 5, 13, 14	syl3anc 1398	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘0))
16		simpr 489	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁))
17	15, 16	sseldd 3938	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘0))
18		elcpn 26102	. . . . . 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 595	. . . . 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 235	. . . 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 499	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
22		dvn0 26092	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
23	2, 21, 22	syl2anc 595	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
24	20	simprd 500	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝑆 D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))
25	23, 24	eqeltrrd 2864	1 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ (dom 𝐹–cn→ℂ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ⊆ wss 3905 {cpr 4591 dom cdm 5661 Fn wfn 6531 ‘cfv 6536 (class class class)co 7410 ↑_pm cpm 8821 ℂcc 11102 ℝcr 11103 0cc0 11104 ℕ₀cn0 12508 ℤ_≥cuz 12866 –cn→ccncf 25044 D^𝑛 cdvn 26032 𝓑C^𝑛ccpn 26033

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-icc 13383 df-fz 13540 df-fzo 13688 df-seq 14043 df-exp 14103 df-hash 14372 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-sca 17330 df-vsca 17331 df-ip 17332 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-hom 17338 df-cco 17339 df-rest 17479 df-topn 17480 df-0g 17498 df-gsum 17499 df-topgen 17500 df-pt 17501 df-prds 17504 df-xrs 17560 df-qtop 17565 df-imas 17566 df-xps 17568 df-mre 17642 df-mrc 17643 df-acs 17645 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-submnd 18846 df-mulg 19138 df-cntz 19391 df-cmn 19856 df-psmet 21523 df-xmet 21524 df-met 21525 df-bl 21526 df-mopn 21527 df-fbas 21528 df-fg 21529 df-cnfld 21532 df-top 23060 df-topon 23077 df-topsp 23099 df-bases 23112 df-cld 23185 df-ntr 23186 df-cls 23187 df-nei 23264 df-lp 23302 df-perf 23303 df-cn 23393 df-cnp 23394 df-haus 23481 df-tx 23728 df-hmeo 23921 df-fil 24012 df-fm 24104 df-flim 24105 df-flf 24106 df-xms 24486 df-ms 24487 df-tms 24488 df-cncf 25046 df-limc 26034 df-dv 26035 df-dvn 26036 df-cpn 26037

This theorem is used by: (None)

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