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Theorem cpncn 24833

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 24801	. . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
2	1	adantr 484	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑆 ⊆ ℂ)
3		simpl 486	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑆 ∈ {ℝ, ℂ})
4		0nn0 12105	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 0 ∈ ℕ₀)
6		elfvdm 6749	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁) → 𝑁 ∈ dom (𝓑C^𝑛‘𝑆))
7	6	adantl 485	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ dom (𝓑C^𝑛‘𝑆))
8		fncpn 24830	. . . . . . . . . 10 ⊢ (𝑆 ⊆ ℂ → (𝓑C^𝑛‘𝑆) Fn ℕ₀)
9		fndm 6481	. . . . . . . . . 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 2840	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ ℕ₀)
12		nn0uz 12476	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
13	11, 12	eleqtrdi 2848	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ (ℤ_≥‘0))
14		cpnord 24832	. . . . . . 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 488	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁))
17	15, 16	sseldd 3902	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘0))
18		elcpn 24831	. . . . . 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 587	. . . . 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 498	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
22		dvn0 24821	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
23	2, 21, 22	syl2anc 587	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
24	20	simprd 499	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝑆 D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))
25	23, 24	eqeltrrd 2839	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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 {cpr 4543 dom cdm 5551 Fn wfn 6375 ‘cfv 6380 (class class class)co 7213 ↑_pm cpm 8509 ℂcc 10727 ℝcr 10728 0cc0 10729 ℕ₀cn0 12090 ℤ_≥cuz 12438 –cn→ccncf 23773 D^𝑛 cdvn 24761 𝓑C^𝑛ccpn 24762

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 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-limc 24763 df-dv 24764 df-dvn 24765 df-cpn 24766

This theorem is referenced by: (None)

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