cpncn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cpncn		Structured version Visualization version GIF version

Theorem cpncn 25687

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 25654	. . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
2	1	adantr 480	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑆 ⊆ ℂ)
3		simpl 482	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑆 ∈ {ℝ, ℂ})
4		0nn0 12492	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 0 ∈ ℕ₀)
6		elfvdm 6928	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁) → 𝑁 ∈ dom (𝓑C^𝑛‘𝑆))
7	6	adantl 481	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ dom (𝓑C^𝑛‘𝑆))
8		fncpn 25684	. . . . . . . . . 10 ⊢ (𝑆 ⊆ ℂ → (𝓑C^𝑛‘𝑆) Fn ℕ₀)
9		fndm 6652	. . . . . . . . . 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 2834	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ ℕ₀)
12		nn0uz 12869	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
13	11, 12	eleqtrdi 2842	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝑁 ∈ (ℤ_≥‘0))
14		cpnord 25686	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 0 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘0)) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘0))
15	3, 5, 13, 14	syl3anc 1370	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝓑C^𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C^𝑛‘𝑆)‘0))
16		simpr 484	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁))
17	15, 16	sseldd 3983	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘0))
18		elcpn 25685	. . . . . 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 583	. . . . 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 231	. . . 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 494	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
22		dvn0 25675	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
23	2, 21, 22	syl2anc 583	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
24	20	simprd 495	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘𝑆)‘𝑁)) → ((𝑆 D^𝑛 𝐹)‘0) ∈ (dom 𝐹–cn→ℂ))
25	23, 24	eqeltrrd 2833	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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 {cpr 4630 dom cdm 5676 Fn wfn 6538 ‘cfv 6543 (class class class)co 7412 ↑_pm cpm 8824 ℂcc 11111 ℝcr 11112 0cc0 11113 ℕ₀cn0 12477 ℤ_≥cuz 12827 –cn→ccncf 24617 D^𝑛 cdvn 25614 𝓑C^𝑛ccpn 25615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-icc 13336 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-limc 25616 df-dv 25617 df-dvn 25618 df-cpn 25619

This theorem is referenced by: (None)

W3C validator