cncfco - Intuitionistic Logic Explorer

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Theorem cncfco 12736

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 12722	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 12722	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5283	. . 3 $/\$
8	3, 6, 7	syl2anc 408	. 2
9	1	adantr 274	. . . . 5 $/\$ $/\$
10	6	adantr 274	. . . . . 6 $/\$ $/\$
11		simprl 520	. . . . . 6 $/\$ $/\$
12	10, 11	ffvelrnd 5549	. . . . 5 $/\$ $/\$
13		simprr 521	. . . . 5 $/\$ $/\$
14		cncfi 12723	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1216	. . . 4 $/\$ $/\$
16	4	ad2antrr 479	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 524	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 109	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 12723	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1216	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 483	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 521	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelrnd 5549	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 5790	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 3934	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 5792	. . . . . . . . . . . . . . . 16
27	26	rspcv 2780	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5485	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 408	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 274	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5485	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 408	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 5785	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5418	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 3934	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
37	36	imbi2d 229	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
38	28, 37	sylibrd 168	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
39	38	imp 123	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an32s 557	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	imim2d 54	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	anassrs 397	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	42	ralimdva 2497	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2532	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 114	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2547	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2512	. 2
50		cncfrss 12720	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 12721	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 12719	. . 3 $/\$ $/\$
55	51, 53, 54	syl2anc 408	. 2 $/\$
56	8, 49, 55	mpbir2and 928	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1331

wcel 1480

wral 2414

wrex 2415

wss 3066 class class class wbr 3924

ccom 4538

wf 5114

cfv 5118 (class class class)co 5767

cc 7611

clt 7793

cmin 7926

crp 9434

cabs 10762

ccncf 12715

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-map 6537 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-2 8772 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-cncf 12716

This theorem is referenced by: cncfmpt1f 12742 cdivcncfap 12745 negfcncf 12747 sincn 12847 coscn 12848

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