cncfco - Intuitionistic Logic Explorer

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Theorem cncfco 12747

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 12733	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 12733	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5288	. . 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 5556	. . . . 5 $/\$ $/\$
13		simprr 521	. . . . 5 $/\$ $/\$
14		cncfi 12734	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1216	. . . 4 $/\$ $/\$
16	4	ad2antrr 479	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 524	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 109	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 12734	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1216	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 483	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 521	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelrnd 5556	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 5797	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 3939	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 5799	. . . . . . . . . . . . . . . 16
27	26	rspcv 2785	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5492	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 408	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 274	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5492	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 408	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 5792	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5425	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 3939	. . . . . . . . . . . . . . 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 2499	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2534	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 114	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2549	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2514	. 2
50		cncfrss 12731	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 12732	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 12730	. . 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 2416

wrex 2417

wss 3071 class class class wbr 3929

ccom 4543

wf 5119

cfv 5123 (class class class)co 5774

cc 7618

clt 7800

cmin 7933

crp 9441

cabs 10769

ccncf 12726

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 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738

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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-map 6544 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-2 8779 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-cncf 12727

This theorem is referenced by: cncfmpt1f 12753 cdivcncfap 12756 negfcncf 12758 sincn 12858 coscn 12859

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