cncfco - Intuitionistic Logic Explorer

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Theorem cncfco 12786

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 12772	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 12772	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5296	. . 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 5564	. . . . 5 $/\$ $/\$
13		simprr 522	. . . . 5 $/\$ $/\$
14		cncfi 12773	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1217	. . . 4 $/\$ $/\$
16	4	ad2antrr 480	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 525	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 109	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 12773	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1217	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 484	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 522	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelrnd 5564	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 5805	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 3947	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 5807	. . . . . . . . . . . . . . . 16
27	26	rspcv 2789	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5500	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 409	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 274	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5500	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 409	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 5800	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5433	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 3947	. . . . . . . . . . . . . . 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 2502	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2537	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 114	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2552	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2517	. 2
50		cncfrss 12770	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 12771	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 12769	. . 3 $/\$ $/\$
55	51, 53, 54	syl2anc 409	. 2 $/\$
56	8, 49, 55	mpbir2and 929	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1332

wcel 1481

wral 2417

wrex 2418

wss 3076 class class class wbr 3937

ccom 4551

wf 5127

cfv 5131 (class class class)co 5782

cc 7642

clt 7824

cmin 7957

crp 9470

cabs 10801

ccncf 12765

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-map 6552 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-2 8803 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-cncf 12766

This theorem is referenced by: cncfmpt1f 12792 cdivcncfap 12795 negfcncf 12797 sincn 12898 coscn 12899

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