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Theorem cpnres 25101

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

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

**Proof of Theorem cpnres**
Step	Hyp	Ref	Expression
1		simpr 485	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁))
2		ssid 3943	. . . . . 6 ⊢ ℂ ⊆ ℂ
3		elfvdm 6806	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁) → 𝑁 ∈ dom (𝓑C^𝑛‘ℂ))
4	3	adantl 482	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝑁 ∈ dom (𝓑C^𝑛‘ℂ))
5		fncpn 25097	. . . . . . . . 9 ⊢ (ℂ ⊆ ℂ → (𝓑C^𝑛‘ℂ) Fn ℕ₀)
6	2, 5	ax-mp 5	. . . . . . . 8 ⊢ (𝓑C^𝑛‘ℂ) Fn ℕ₀
7		fndm 6536	. . . . . . . 8 ⊢ ((𝓑C^𝑛‘ℂ) Fn ℕ₀ → dom (𝓑C^𝑛‘ℂ) = ℕ₀)
8	6, 7	mp1i 13	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → dom (𝓑C^𝑛‘ℂ) = ℕ₀)
9	4, 8	eleqtrd 2841	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝑁 ∈ ℕ₀)
10		elcpn 25098	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))))
11	2, 9, 10	sylancr 587	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))))
12	1, 11	mpbid 231	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ)))
13	12	simpld 495	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝐹 ∈ (ℂ ↑_pm ℂ))
14		pmresg 8658	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
15	13, 14	syldan 591	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
16		simpl 483	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝑆 ∈ {ℝ, ℂ})
17	12	simprd 496	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))
18		cncff 24056	. . . . . 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 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ)
20	19	fdmd 6611	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹)
21		dvnres 25095	. . . 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	16, 13, 9, 20, 21	syl31anc 1372	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
23		resres 5904	. . . . . . 7 ⊢ ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ↾ dom 𝐹) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹))
24		rescom 5917	. . . . . . 7 ⊢ ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ↾ dom 𝐹) = ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆)
25	23, 24	eqtr3i 2768	. . . . . 6 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) = ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆)
26		ffn 6600	. . . . . . . 8 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ → ((ℂ D^𝑛 𝐹)‘𝑁) Fn dom 𝐹)
27		fnresdm 6551	. . . . . . . 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 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) = ((ℂ D^𝑛 𝐹)‘𝑁))
29	28	reseq1d 5890	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
30	25, 29	eqtrid 2790	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
31		inss2 4163	. . . . . 6 ⊢ (𝑆 ∩ dom 𝐹) ⊆ dom 𝐹
32		rescncf 24060	. . . . . 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 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) ∈ ((𝑆 ∩ dom 𝐹)–cn→ℂ))
34	30, 33	eqeltrrd 2840	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ∈ ((𝑆 ∩ dom 𝐹)–cn→ℂ))
35		dmres 5913	. . . . 5 ⊢ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹)
36	35	oveq1i 7285	. . . 4 ⊢ (dom (𝐹 ↾ 𝑆)–cn→ℂ) = ((𝑆 ∩ dom 𝐹)–cn→ℂ)
37	34, 36	eleqtrrdi 2850	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))
38	22, 37	eqeltrd 2839	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))
39		recnprss 25068	. . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
40		elcpn 25098	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐹 ↾ 𝑆) ∈ ((𝓑C^𝑛‘𝑆)‘𝑁) ↔ ((𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))))
41	39, 9, 40	syl2an2r 682	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((𝐹 ↾ 𝑆) ∈ ((𝓑C^𝑛‘𝑆)‘𝑁) ↔ ((𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))))
42	15, 38, 41	mpbir2and 710	1 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ ((𝓑C^𝑛‘𝑆)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∩ cin 3886 ⊆ wss 3887 {cpr 4563 dom cdm 5589 ↾ cres 5591 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_pm cpm 8616 ℂcc 10869 ℝcr 10870 ℕ₀cn0 12233 –cn→ccncf 24039 D^𝑛 cdvn 25028 𝓑C^𝑛ccpn 25029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-icc 13086 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-mulr 16976 df-starv 16977 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-rest 17133 df-topn 17134 df-topgen 17154 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cnp 22379 df-haus 22466 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-cncf 24041 df-limc 25030 df-dv 25031 df-dvn 25032 df-cpn 25033

This theorem is referenced by: aalioulem3 25494

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