cncfco - Intuitionistic Logic Explorer

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Theorem cncfco 15582

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 15568	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 15568	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5532	. . 3 $/\$
8	3, 6, 7	syl2anc 411	. 2
9	1	adantr 276	. . . . 5 $/\$ $/\$
10	6	adantr 276	. . . . . 6 $/\$ $/\$
11		simprl 531	. . . . . 6 $/\$ $/\$
12	10, 11	ffvelcdmd 5818	. . . . 5 $/\$ $/\$
13		simprr 533	. . . . 5 $/\$ $/\$
14		cncfi 15569	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1274	. . . 4 $/\$ $/\$
16	4	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 537	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 110	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 15569	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1274	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 533	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelcdmd 5818	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 6081	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 4124	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 6083	. . . . . . . . . . . . . . . 16
27	26	rspcv 2919	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5753	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 411	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 276	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5753	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 411	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 6076	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5679	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 4124	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
37	36	imbi2d 230	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
38	28, 37	sylibrd 169	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
39	38	imp 124	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an32s 570	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	imim2d 54	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	anassrs 400	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	42	ralimdva 2611	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2646	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 115	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2662	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2626	. 2
50		cncfrss 15566	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 15567	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 15565	. . 3 $/\$ $/\$
55	51, 53, 54	syl2anc 411	. 2 $/\$
56	8, 49, 55	mpbir2and 953	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2205

wral 2522

wrex 2523

wss 3214 class class class wbr 4114

ccom 4758

wf 5353

cfv 5357 (class class class)co 6058

cc 8141

clt 8324

cmin 8460

crp 10004

cabs 11707

ccncf 15561

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-po 4422 df-iso 4423 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-map 6897 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-2 9313 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-cncf 15562

This theorem is referenced by: cncfmpt1f 15589 cdivcncfap 15595 negfcncf 15597 divcncfap 15605 sincn 15760 coscn 15761

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