cnpco - Metamath Proof Explorer

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Theorem cnpco 21071

Description: The composition of two continuous functions at point

is a continuous function at point

. Proposition of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

**Assertion**
Ref	Expression
cnpco	$/\$

Proof of Theorem cnpco Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnptop1 21046	. . . 4
2	1	adantr 481	. . 3 $/\$
3		cnptop2 21047	. . . 4
4	3	adantl 482	. . 3 $/\$
5		eqid 2622	. . . . 5
6	5	cnprcl 21049	. . . 4
7	6	adantr 481	. . 3 $/\$
8	2, 4, 7	3jca 1242	. 2 $/\$ $/\$ $/\$
9		eqid 2622	. . . . . 6
10		eqid 2622	. . . . . 6
11	9, 10	cnpf 21051	. . . . 5
12	11	adantl 482	. . . 4 $/\$
13	5, 9	cnpf 21051	. . . . 5
14	13	adantr 481	. . . 4 $/\$
15		fco 6058	. . . 4 $/\$
16	12, 14, 15	syl2anc 693	. . 3 $/\$
17		simplr 792	. . . . . . 7 $/\$ $/\$ $/\$
18		simprl 794	. . . . . . 7 $/\$ $/\$ $/\$
19		fvco3 6275	. . . . . . . . . 10 $/\$
20	14, 7, 19	syl2anc 693	. . . . . . . . 9 $/\$
21	20	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$
22		simprr 796	. . . . . . . 8 $/\$ $/\$ $/\$
23	21, 22	eqeltrrd 2702	. . . . . . 7 $/\$ $/\$ $/\$
24		cnpimaex 21060	. . . . . . 7 $/\$ $/\$ $/\$
25	17, 18, 23, 24	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $/\$
26		simplll 798	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		simprl 794	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		simprrl 804	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29		cnpimaex 21060	. . . . . . . 8 $/\$ $/\$ $/\$
30	26, 27, 28, 29	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		imaco 5640	. . . . . . . . . . 11
32		imass2 5501	. . . . . . . . . . 11
33	31, 32	syl5eqss 3649	. . . . . . . . . 10
34		simprrr 805	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		sstr2 3610	. . . . . . . . . 10
36	33, 34, 35	syl2imc 41	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	36	anim2d 589	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	37	reximdv 3016	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	30, 38	mpd 15	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	25, 39	rexlimddv 3035	. . . . 5 $/\$ $/\$ $/\$ $/\$
41	40	expr 643	. . . 4 $/\$ $/\$ $/\$
42	41	ralrimiva 2966	. . 3 $/\$ $/\$
43	16, 42	jca 554	. 2 $/\$ $/\$ $/\$
44	5, 10	iscnp2 21043	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
45	8, 43, 44	sylanbrc 698	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

wrex 2913

wss 3574

cuni 4436

cima 5117

ccom 5118

wf 5884

cfv 5888 (class class class)co 6650

ctop 20698

ccnp 21029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-top 20699 df-topon 20716 df-cnp 21032

This theorem is referenced by: limccnp 23655 limccnp2 23656 efrlim 24696

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