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Theorem cpnres 23920

Description: The restriction of a Cⁿ function is Cⁿ. (Contributed by Mario Carneiro, 11-Feb-2015.)

**Assertion**
Ref	Expression
cpnres	⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ ((Cⁿ‘𝑆)‘𝑁))

**Proof of Theorem cpnres**
Step	Hyp	Ref	Expression
1		simpl 468	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → 𝑆 ∈ {ℝ, ℂ})
2		simpr 471	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁))
3		ssid 3773	. . . . . 6 ⊢ ℂ ⊆ ℂ
4		elfvdm 6363	. . . . . . . 8 ⊢ (𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁) → 𝑁 ∈ dom (Cⁿ‘ℂ))
5	4	adantl 467	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → 𝑁 ∈ dom (Cⁿ‘ℂ))
6		fncpn 23916	. . . . . . . . 9 ⊢ (ℂ ⊆ ℂ → (Cⁿ‘ℂ) Fn ℕ₀)
7	3, 6	ax-mp 5	. . . . . . . 8 ⊢ (Cⁿ‘ℂ) Fn ℕ₀
8		fndm 6129	. . . . . . . 8 ⊢ ((Cⁿ‘ℂ) Fn ℕ₀ → dom (Cⁿ‘ℂ) = ℕ₀)
9	7, 8	mp1i 13	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → dom (Cⁿ‘ℂ) = ℕ₀)
10	5, 9	eleqtrd 2852	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → 𝑁 ∈ ℕ₀)
11		elcpn 23917	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))))
12	3, 10, 11	sylancr 575	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → (𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))))
13	2, 12	mpbid 222	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ)))
14	13	simpld 482	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → 𝐹 ∈ (ℂ ↑_pm ℂ))
15		pmresg 8041	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
16	1, 14, 15	syl2anc 573	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
17	13	simprd 483	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))
18		cncff 22916	. . . . . 6 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ) → ((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ)
19	17, 18	syl 17	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → ((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ)
20	19	fdmd 6193	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹)
21		dvnres 23914	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑁 ∈ ℕ₀) ∧ dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
22	1, 14, 10, 20, 21	syl31anc 1479	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
23		resres 5549	. . . . . . 7 ⊢ ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ↾ dom 𝐹) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹))
24		rescom 5563	. . . . . . 7 ⊢ ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ↾ dom 𝐹) = ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆)
25	23, 24	eqtr3i 2795	. . . . . 6 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) = ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆)
26		ffn 6184	. . . . . . . 8 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ → ((ℂ D^𝑛 𝐹)‘𝑁) Fn dom 𝐹)
27		fnresdm 6139	. . . . . . . 8 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁) Fn dom 𝐹 → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) = ((ℂ D^𝑛 𝐹)‘𝑁))
28	19, 26, 27	3syl 18	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) = ((ℂ D^𝑛 𝐹)‘𝑁))
29	28	reseq1d 5532	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
30	25, 29	syl5eq 2817	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
31		inss2 3982	. . . . . 6 ⊢ (𝑆 ∩ dom 𝐹) ⊆ dom 𝐹
32		rescncf 22920	. . . . . 6 ⊢ ((𝑆 ∩ dom 𝐹) ⊆ dom 𝐹 → (((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) ∈ ((𝑆 ∩ dom 𝐹)–cn→ℂ)))
33	31, 17, 32	mpsyl 68	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) ∈ ((𝑆 ∩ dom 𝐹)–cn→ℂ))
34	30, 33	eqeltrrd 2851	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ∈ ((𝑆 ∩ dom 𝐹)–cn→ℂ))
35		dmres 5559	. . . . 5 ⊢ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹)
36	35	oveq1i 6806	. . . 4 ⊢ (dom (𝐹 ↾ 𝑆)–cn→ℂ) = ((𝑆 ∩ dom 𝐹)–cn→ℂ)
37	34, 36	syl6eleqr 2861	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))
38	22, 37	eqeltrd 2850	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))
39		recnprss 23888	. . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
40	39	adantr 466	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → 𝑆 ⊆ ℂ)
41		elcpn 23917	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐹 ↾ 𝑆) ∈ ((Cⁿ‘𝑆)‘𝑁) ↔ ((𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))))
42	40, 10, 41	syl2anc 573	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → ((𝐹 ↾ 𝑆) ∈ ((Cⁿ‘𝑆)‘𝑁) ↔ ((𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))))
43	16, 38, 42	mpbir2and 692	1 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cⁿ‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ ((Cⁿ‘𝑆)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∩ cin 3722 ⊆ wss 3723 {cpr 4319 dom cdm 5250 ↾ cres 5252 Fn wfn 6025 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 ↑_pm cpm 8014 ℂcc 10140 ℝcr 10141 ℕ₀cn0 11499 –cn→ccncf 22899 D^𝑛 cdvn 23848 Cⁿccpn 23849

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-map 8015 df-pm 8016 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fi 8477 df-sup 8508 df-inf 8509 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-z 11585 df-dec 11701 df-uz 11894 df-q 11997 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-icc 12387 df-fz 12534 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-plusg 16162 df-mulr 16163 df-starv 16164 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-rest 16291 df-topn 16292 df-topgen 16312 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-nei 21123 df-lp 21161 df-perf 21162 df-cnp 21253 df-haus 21340 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-xms 22345 df-ms 22346 df-cncf 22901 df-limc 23850 df-dv 23851 df-dvn 23852 df-cpn 23853

This theorem is referenced by: aalioulem3 24309

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