cncfco - Intuitionistic Logic Explorer

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Theorem cncfco 15148

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 15134	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 15134	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5456	. . 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 5734	. . . . 5 $/\$ $/\$
13		simprr 531	. . . . 5 $/\$ $/\$
14		cncfi 15135	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1250	. . . 4 $/\$ $/\$
16	4	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 535	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 110	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 15135	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1250	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 531	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelcdmd 5734	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 5985	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 4064	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 5987	. . . . . . . . . . . . . . . 16
27	26	rspcv 2877	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5668	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 411	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 276	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5668	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 411	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 5980	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5598	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 4064	. . . . . . . . . . . . . . 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 2574	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2609	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 115	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2624	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2589	. 2
50		cncfrss 15132	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 15133	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 15131	. . 3 $/\$ $/\$
55	51, 53, 54	syl2anc 411	. 2 $/\$
56	8, 49, 55	mpbir2and 947	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1373

wcel 2177

wral 2485

wrex 2486

wss 3170 class class class wbr 4054

ccom 4692

wf 5281

cfv 5285 (class class class)co 5962

cc 7953

clt 8137

cmin 8273

crp 9805

cabs 11393

ccncf 15127

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4170 ax-sep 4173 ax-pow 4229 ax-pr 4264 ax-un 4493 ax-setind 4598 ax-cnex 8046 ax-resscn 8047 ax-1cn 8048 ax-1re 8049 ax-icn 8050 ax-addcl 8051 ax-addrcl 8052 ax-mulcl 8053 ax-mulrcl 8054 ax-addcom 8055 ax-mulcom 8056 ax-addass 8057 ax-mulass 8058 ax-distr 8059 ax-i2m1 8060 ax-0lt1 8061 ax-1rid 8062 ax-0id 8063 ax-rnegex 8064 ax-precex 8065 ax-cnre 8066 ax-pre-ltirr 8067 ax-pre-ltwlin 8068 ax-pre-lttrn 8069 ax-pre-apti 8070 ax-pre-ltadd 8071 ax-pre-mulgt0 8072 ax-pre-mulext 8073

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3860 df-iun 3938 df-br 4055 df-opab 4117 df-mpt 4118 df-id 4353 df-po 4356 df-iso 4357 df-xp 4694 df-rel 4695 df-cnv 4696 df-co 4697 df-dm 4698 df-rn 4699 df-res 4700 df-ima 4701 df-iota 5246 df-fun 5287 df-fn 5288 df-f 5289 df-f1 5290 df-fo 5291 df-f1o 5292 df-fv 5293 df-riota 5917 df-ov 5965 df-oprab 5966 df-mpo 5967 df-map 6755 df-pnf 8139 df-mnf 8140 df-xr 8141 df-ltxr 8142 df-le 8143 df-sub 8275 df-neg 8276 df-reap 8678 df-ap 8685 df-div 8776 df-2 9125 df-cj 11238 df-re 11239 df-im 11240 df-rsqrt 11394 df-abs 11395 df-cncf 15128

This theorem is referenced by: cncfmpt1f 15155 cdivcncfap 15161 negfcncf 15163 divcncfap 15171 sincn 15326 coscn 15327

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