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Theorem cpncn 25925

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 25893	. . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
2	1	adantr 482	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑆 ⊆ ℂ)
3		simpl 484	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑆 ∈ {ℝ, ℂ})
4		0nn0 12447	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 0 ∈ ℕ₀)
6		elfvdm 6865	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁) → 𝑁 ∈ dom (𝓑C^𝑛‘𝑆))
7	6	adantl 483	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ dom (𝓑C^𝑛‘𝑆))
8		fncpn 25922	. . . . . . . . . 10 ⊢ (𝑆 ⊆ ℂ → (𝓑C^𝑛‘𝑆) Fn ℕ₀)
9		fndm 6592	. . . . . . . . . 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 2843	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ ℕ₀)
12		nn0uz 12821	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
13	11, 12	eleqtrdi 2851	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ (ℤ_≥‘0))
14		cpnord 25924	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘0)) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘0))
15	3, 5, 13, 14	syl3anc 1380	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘0))
16		simpr 486	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁))
17	15, 16	sseldd 3918	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘0))
18		elcpn 25923	. . . . . 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 591	. . . . 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 234	. . . 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 496	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
22		dvn0 25913	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
23	2, 21, 22	syl2anc 591	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
24	20	simprd 497	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝑆 D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))
25	23, 24	eqeltrrd 2842	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 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ⊆ wss 3885 {cpr 4560 dom cdm 5621 Fn wfn 6484 ‘cfv 6489 (class class class)co 7360 ↑_pm cpm 8768 ℂcc 11031 ℝcr 11032 0cc0 11033 ℕ₀cn0 12432 ℤ_≥cuz 12783 –cn→ccncf 24865 D^𝑛 cdvn 25853 𝓑C^𝑛ccpn 25854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 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 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-icc 13300 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19287 df-cmn 19752 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-fbas 21348 df-fg 21349 df-cnfld 21352 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-lp 23123 df-perf 23124 df-cn 23214 df-cnp 23215 df-haus 23302 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-xms 24307 df-ms 24308 df-tms 24309 df-cncf 24867 df-limc 25855 df-dv 25856 df-dvn 25857 df-cpn 25858

This theorem is referenced by: (None)

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