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Theorem cpnres 26196

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 490	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁))
2		ssid 3953	. . . . . 6 ⊢ ℂ ⊆ ℂ
3		elfvdm 6915	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁) → 𝑁 ∈ dom (𝓑C^𝑛‘ℂ))
4	3	adantl 487	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝑁 ∈ dom (𝓑C^𝑛‘ℂ))
5		fncpn 26192	. . . . . . . . 9 ⊢ (ℂ ⊆ ℂ → (𝓑C^𝑛‘ℂ) Fn ℕ₀)
6	2, 5	ax-mp 5	. . . . . . . 8 ⊢ (𝓑C^𝑛‘ℂ) Fn ℕ₀
7		fndm 6638	. . . . . . . 8 ⊢ ((𝓑C^𝑛‘ℂ) Fn ℕ₀ → dom (𝓑C^𝑛‘ℂ) = ℕ₀)
8	6, 7	mp1i 14	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → dom (𝓑C^𝑛‘ℂ) = ℕ₀)
9	4, 8	eleqtrd 2862	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝑁 ∈ ℕ₀)
10		elcpn 26193	. . . . . 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 599	. . . . 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 235	. . . 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 500	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝐹 ∈ (ℂ ↑_pm ℂ))
14		pmresg 8884	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
15	13, 14	syldan 603	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
16		simpl 488	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝑆 ∈ {ℝ, ℂ})
17	12	simprd 501	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))
18		cncff 25153	. . . . . 6 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ) → ((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ)
19	17, 18	syl 18	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ)
20	19	fdmd 6716	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹)
21		dvnres 26190	. . . 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 1400	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
23		resres 5987	. . . . . . 7 ⊢ ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ↾ dom 𝐹) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹))
24		rescom 5997	. . . . . . 7 ⊢ ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ↾ dom 𝐹) = ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆)
25	23, 24	eqtr3i 2785	. . . . . 6 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) = ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆)
26		ffn 6705	. . . . . . . 8 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ → ((ℂ D^𝑛 𝐹)‘𝑁) Fn dom 𝐹)
27		fnresdm 6654	. . . . . . . 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 19	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) = ((ℂ D^𝑛 𝐹)‘𝑁))
29	28	reseq1d 5973	. . . . . 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 2807	. . . . 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 4183	. . . . . 6 ⊢ (𝑆 ∩ dom 𝐹) ⊆ dom 𝐹
32		rescncf 25157	. . . . . 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 69	. . . . 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 2861	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ∈ ((𝑆 ∩ dom 𝐹)–cn→ℂ))
35		dmres 6007	. . . . 5 ⊢ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹)
36	35	oveq1i 7426	. . . 4 ⊢ (dom (𝐹 ↾ 𝑆)–cn→ℂ) = ((𝑆 ∩ dom 𝐹)–cn→ℂ)
37	34, 36	eleqtrrdi 2871	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))
38	22, 37	eqeltrd 2860	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))
39		recnprss 26163	. . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
40		elcpn 26193	. . 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 698	. 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 726	1 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ ((𝓑C^𝑛‘𝑆)‘𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∩ cin 3898 ⊆ wss 3899 {cpr 4586 dom cdm 5655 ↾ cres 5657 Fn wfn 6530 ⟶wf 6531 ‘cfv 6535 (class class class)co 7416 ↑_pm cpm 8834 ℂcc 11147 ℝcr 11148 ℕ₀cn0 12553 –cn→ccncf 25136 D^𝑛 cdvn 26123 𝓑C^𝑛ccpn 26124

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-inf2 9627 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-er 8703 df-map 8835 df-pm 8836 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fi 9388 df-sup 9419 df-inf 9420 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-q 13023 df-rp 13068 df-xneg 13188 df-xadd 13189 df-xmul 13190 df-icc 13430 df-fz 13587 df-seq 14091 df-exp 14151 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-struct 17264 df-slot 17299 df-ndx 17311 df-base 17327 df-plusg 17380 df-mulr 17381 df-starv 17382 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-rest 17532 df-topn 17533 df-topgen 17553 df-psmet 21609 df-xmet 21610 df-met 21611 df-bl 21612 df-mopn 21613 df-fbas 21614 df-fg 21615 df-cnfld 21618 df-top 23151 df-topon 23168 df-topsp 23190 df-bases 23203 df-cld 23276 df-ntr 23277 df-cls 23278 df-nei 23355 df-lp 23393 df-perf 23394 df-cnp 23485 df-haus 23572 df-fil 24104 df-fm 24196 df-flim 24197 df-flf 24198 df-xms 24578 df-ms 24579 df-cncf 25138 df-limc 26125 df-dv 26126 df-dvn 26127 df-cpn 26128

This theorem is used by: aalioulem3 26602

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