cncfco - Intuitionistic Logic Explorer

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Theorem cncfco 13372

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 13358	. . . 4
3	1, 2	syl 14	. . 3
4		cncfco.4	. . . 4
5		cncff 13358	. . . 4
6	4, 5	syl 14	. . 3
7		fco 5363	. . 3 $/\$
8	3, 6, 7	syl2anc 409	. 2
9	1	adantr 274	. . . . 5 $/\$ $/\$
10	6	adantr 274	. . . . . 6 $/\$ $/\$
11		simprl 526	. . . . . 6 $/\$ $/\$
12	10, 11	ffvelrnd 5632	. . . . 5 $/\$ $/\$
13		simprr 527	. . . . 5 $/\$ $/\$
14		cncfi 13359	. . . . 5 $/\$ $/\$
15	9, 12, 13, 14	syl3anc 1233	. . . 4 $/\$ $/\$
16	4	ad2antrr 485	. . . . . . 7 $/\$ $/\$ $/\$
17		simplrl 530	. . . . . . 7 $/\$ $/\$ $/\$
18		simpr 109	. . . . . . 7 $/\$ $/\$ $/\$
19		cncfi 13359	. . . . . . 7 $/\$ $/\$
20	16, 17, 18, 19	syl3anc 1233	. . . . . 6 $/\$ $/\$ $/\$
21	6	ad3antrrr 489	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
22		simprr 527	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
23	21, 22	ffvelrnd 5632	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
24		fvoveq1 5876	. . . . . . . . . . . . . . . . . 18
25	24	breq1d 3999	. . . . . . . . . . . . . . . . 17
26	25	imbrov2fvoveq 5878	. . . . . . . . . . . . . . . 16
27	26	rspcv 2830	. . . . . . . . . . . . . . 15
28	23, 27	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
29		fvco3 5567	. . . . . . . . . . . . . . . . . . 19 $/\$
30	21, 22, 29	syl2anc 409	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
31	17	adantr 274	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
32		fvco3 5567	. . . . . . . . . . . . . . . . . . 19 $/\$
33	21, 31, 32	syl2anc 409	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$
34	30, 33	oveq12d 5871	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	fveq2d 5500	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	breq1d 3999	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
37	36	imbi2d 229	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
38	28, 37	sylibrd 168	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
39	38	imp 123	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an32s 563	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	imim2d 54	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	anassrs 398	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	42	ralimdva 2537	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	reximdva 2572	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
45	44	ex 114	. . . . . 6 $/\$ $/\$ $/\$
46	20, 45	mpid 42	. . . . 5 $/\$ $/\$ $/\$
47	46	rexlimdva 2587	. . . 4 $/\$ $/\$
48	15, 47	mpd 13	. . 3 $/\$ $/\$
49	48	ralrimivva 2552	. 2
50		cncfrss 13356	. . . 4
51	4, 50	syl 14	. . 3
52		cncfrss2 13357	. . . 4
53	1, 52	syl 14	. . 3
54		elcncf2 13355	. . 3 $/\$ $/\$
55	51, 53, 54	syl2anc 409	. 2 $/\$
56	8, 49, 55	mpbir2and 939	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1348

wcel 2141

wral 2448

wrex 2449

wss 3121 class class class wbr 3989

ccom 4615

wf 5194

cfv 5198 (class class class)co 5853

cc 7772

clt 7954

cmin 8090

crp 9610

cabs 10961

ccncf 13351

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-map 6628 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-2 8937 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-cncf 13352

This theorem is referenced by: cncfmpt1f 13378 cdivcncfap 13381 negfcncf 13383 sincn 13484 coscn 13485

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