cncfco - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > cncfco		Unicode version

Theorem cncfco 15302

Description: The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)

**Hypotheses**
Ref	Expression
cncfco.4
cncfco.5

**Assertion**
Ref	Expression
cncfco

Proof of Theorem cncfco Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cncfco.5	. . . 4
2		cncff 15288	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 15288	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5495	. . 3 $/\$
8	3, 6, 7	syl2anc 411	. 2
9	1	adantr 276	. . . . 5 $/\$ $/\$
10	6	adantr 276	. . . . . 6 $/\$ $/\$
11		simprl 529	. . . . . 6 $/\$ $/\$
12	10, 11	ffvelcdmd 5777	. . . . 5 $/\$ $/\$
13		simprr 531	. . . . 5 $/\$ $/\$
14		cncfi 15289	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1271	. . . 4 $/\$ $/\$
16	4	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 535	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 110	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 15289	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1271	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 531	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelcdmd 5777	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 6034	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 4094	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 6036	. . . . . . . . . . . . . . . 16
27	26	rspcv 2904	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5711	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 411	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 276	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5711	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 411	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 6029	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5637	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 4094	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
37	36	imbi2d 230	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
38	28, 37	sylibrd 169	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
39	38	imp 124	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an32s 568	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	imim2d 54	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	anassrs 400	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	42	ralimdva 2597	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2632	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 115	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2648	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2612	. 2
50		cncfrss 15286	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 15287	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 15285	. . 3 $/\$ $/\$
55	51, 53, 54	syl2anc 411	. 2 $/\$
56	8, 49, 55	mpbir2and 950	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1395

wcel 2200

wral 2508

wrex 2509

wss 3198 class class class wbr 4084

ccom 4725

wf 5318

cfv 5322 (class class class)co 6011

cc 8018

clt 8202

cmin 8338

crp 9876

cabs 11545

ccncf 15281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4200 ax-sep 4203 ax-pow 4260 ax-pr 4295 ax-un 4526 ax-setind 4631 ax-cnex 8111 ax-resscn 8112 ax-1cn 8113 ax-1re 8114 ax-icn 8115 ax-addcl 8116 ax-addrcl 8117 ax-mulcl 8118 ax-mulrcl 8119 ax-addcom 8120 ax-mulcom 8121 ax-addass 8122 ax-mulass 8123 ax-distr 8124 ax-i2m1 8125 ax-0lt1 8126 ax-1rid 8127 ax-0id 8128 ax-rnegex 8129 ax-precex 8130 ax-cnre 8131 ax-pre-ltirr 8132 ax-pre-ltwlin 8133 ax-pre-lttrn 8134 ax-pre-apti 8135 ax-pre-ltadd 8136 ax-pre-mulgt0 8137 ax-pre-mulext 8138

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3890 df-iun 3968 df-br 4085 df-opab 4147 df-mpt 4148 df-id 4386 df-po 4389 df-iso 4390 df-xp 4727 df-rel 4728 df-cnv 4729 df-co 4730 df-dm 4731 df-rn 4732 df-res 4733 df-ima 4734 df-iota 5282 df-fun 5324 df-fn 5325 df-f 5326 df-f1 5327 df-fo 5328 df-f1o 5329 df-fv 5330 df-riota 5964 df-ov 6014 df-oprab 6015 df-mpo 6016 df-map 6812 df-pnf 8204 df-mnf 8205 df-xr 8206 df-ltxr 8207 df-le 8208 df-sub 8340 df-neg 8341 df-reap 8743 df-ap 8750 df-div 8841 df-2 9190 df-cj 11390 df-re 11391 df-im 11392 df-rsqrt 11546 df-abs 11547 df-cncf 15282

This theorem is referenced by: cncfmpt1f 15309 cdivcncfap 15315 negfcncf 15317 divcncfap 15325 sincn 15480 coscn 15481

W3C validator