cncfco - Intuitionistic Logic Explorer

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

Theorem cncfco 13218

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 13204	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 13204	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5353	. . 3 $/\$
8	3, 6, 7	syl2anc 409	. 2
9	1	adantr 274	. . . . 5 $/\$ $/\$
10	6	adantr 274	. . . . . 6 $/\$ $/\$
11		simprl 521	. . . . . 6 $/\$ $/\$
12	10, 11	ffvelrnd 5621	. . . . 5 $/\$ $/\$
13		simprr 522	. . . . 5 $/\$ $/\$
14		cncfi 13205	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1228	. . . 4 $/\$ $/\$
16	4	ad2antrr 480	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 525	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 109	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 13205	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1228	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 484	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 522	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelrnd 5621	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 5865	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 3992	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 5867	. . . . . . . . . . . . . . . 16
27	26	rspcv 2826	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5557	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 409	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 274	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5557	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 409	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 5860	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5490	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 3992	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
37	36	imbi2d 229	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
38	28, 37	sylibrd 168	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
39	38	imp 123	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an32s 558	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	imim2d 54	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	anassrs 398	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	42	ralimdva 2533	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2568	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 114	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2583	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2548	. 2
50		cncfrss 13202	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 13203	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 13201	. . 3 $/\$ $/\$
55	51, 53, 54	syl2anc 409	. 2 $/\$
56	8, 49, 55	mpbir2and 934	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1343

wcel 2136

wral 2444

wrex 2445

wss 3116 class class class wbr 3982

ccom 4608

wf 5184

cfv 5188 (class class class)co 5842

cc 7751

clt 7933

cmin 8069

crp 9589

cabs 10939

ccncf 13197

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-po 4274 df-iso 4275 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-map 6616 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-2 8916 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-cncf 13198

This theorem is referenced by: cncfmpt1f 13224 cdivcncfap 13227 negfcncf 13229 sincn 13330 coscn 13331

W3C validator