cncfco - Intuitionistic Logic Explorer

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Theorem cncfco 14827

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 14813	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 14813	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5423	. . 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 5698	. . . . 5 $/\$ $/\$
13		simprr 531	. . . . 5 $/\$ $/\$
14		cncfi 14814	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1249	. . . 4 $/\$ $/\$
16	4	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 535	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 110	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 14814	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1249	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 531	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelcdmd 5698	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 5945	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 4043	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 5947	. . . . . . . . . . . . . . . 16
27	26	rspcv 2864	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5632	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 411	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 276	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5632	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 411	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 5940	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5562	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 4043	. . . . . . . . . . . . . . 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 2564	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2599	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 115	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2614	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2579	. 2
50		cncfrss 14811	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 14812	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 14810	. . 3 $/\$ $/\$
55	51, 53, 54	syl2anc 411	. 2 $/\$
56	8, 49, 55	mpbir2and 946	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2167

wral 2475

wrex 2476

wss 3157 class class class wbr 4033

ccom 4667

wf 5254

cfv 5258 (class class class)co 5922

cc 7877

clt 8061

cmin 8197

crp 9728

cabs 11162

ccncf 14806

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-po 4331 df-iso 4332 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-map 6709 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-2 9049 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-cncf 14807

This theorem is referenced by: cncfmpt1f 14834 cdivcncfap 14840 negfcncf 14842 divcncfap 14850 sincn 15005 coscn 15006

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