cncfco - Intuitionistic Logic Explorer

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Theorem cncfco 15615

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 15601	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 15601	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5547	. . 3 $/\$
8	3, 6, 7	syl2anc 415	. 2
9	1	adantr 276	. . . . 5 $/\$ $/\$
10	6	adantr 276	. . . . . 6 $/\$ $/\$
11		simprl 535	. . . . . 6 $/\$ $/\$
12	10, 11	ffvelcdmd 5835	. . . . 5 $/\$ $/\$
13		simprr 537	. . . . 5 $/\$ $/\$
14		cncfi 15602	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1278	. . . 4 $/\$ $/\$
16	4	ad2antrr 492	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 541	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 110	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 15602	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1278	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 496	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 537	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelcdmd 5835	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 6098	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 4135	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 6100	. . . . . . . . . . . . . . . 16
27	26	rspcv 2925	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5770	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 415	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 276	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5770	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 415	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 6093	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5694	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 4135	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
37	36	imbi2d 230	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
38	28, 37	sylibrd 169	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
39	38	imp 124	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an32s 574	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	imim2d 54	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	anassrs 404	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	42	ralimdva 2617	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2652	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 115	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2668	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2632	. 2
50		cncfrss 15599	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 15600	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 15598	. . 3 $/\$ $/\$
55	51, 53, 54	syl2anc 415	. 2 $/\$
56	8, 49, 55	mpbir2and 957	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1402

wcel 2209

wral 2528

wrex 2529

wss 3220 class class class wbr 4125

ccom 4773

wf 5368

cfv 5372 (class class class)co 6075

cc 8167

clt 8350

cmin 8487

crp 10033

cabs 11741

ccncf 15594

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-po 4436 df-iso 4437 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-map 6914 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-2 9342 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-cncf 15595

This theorem is referenced by: cncfmpt1f 15622 cdivcncfap 15628 negfcncf 15630 divcncfap 15638 sincn 15793 coscn 15794

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