cncfco - Intuitionistic Logic Explorer

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Theorem cncfco 13745

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 13731	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 13731	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5377	. . 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 5648	. . . . 5 $/\$ $/\$
13		simprr 531	. . . . 5 $/\$ $/\$
14		cncfi 13732	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1238	. . . 4 $/\$ $/\$
16	4	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 535	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 110	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 13732	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1238	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 531	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelcdmd 5648	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 5892	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 4010	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 5894	. . . . . . . . . . . . . . . 16
27	26	rspcv 2837	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5583	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 411	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 276	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5583	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 411	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 5887	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5515	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 4010	. . . . . . . . . . . . . . 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 2544	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2579	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 115	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2594	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2559	. 2
50		cncfrss 13729	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 13730	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 13728	. . 3 $/\$ $/\$
55	51, 53, 54	syl2anc 411	. 2 $/\$
56	8, 49, 55	mpbir2and 944	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148

wral 2455

wrex 2456

wss 3129 class class class wbr 4000

ccom 4627

wf 5208

cfv 5212 (class class class)co 5869

cc 7800

clt 7982

cmin 8118

crp 9640

cabs 10990

ccncf 13724

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-map 6644 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-2 8967 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-cncf 13725

This theorem is referenced by: cncfmpt1f 13751 cdivcncfap 13754 negfcncf 13756 sincn 13857 coscn 13858

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