mulcn2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mulcn2		Structured version Visualization version Unicode version

Theorem mulcn2 14326

Description: Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)

**Assertion**
Ref	Expression
mulcn2	$/\$ $/\$ $/\$

Distinct variable groups:

**Proof of Theorem mulcn2**
Step	Hyp	Ref	Expression
1		rphalfcl 11858	. . . 4
2	1	3ad2ant1 1082	. . 3 $/\$ $/\$
3		abscl 14018	. . . . . 6
4	3	3ad2ant3 1084	. . . . 5 $/\$ $/\$
5		abscl 14018	. . . . . . . . . 10
6	5	3ad2ant2 1083	. . . . . . . . 9 $/\$ $/\$
7		1re 10039	. . . . . . . . 9
8		readdcl 10019	. . . . . . . . 9 $/\$
9	6, 7, 8	sylancl 694	. . . . . . . 8 $/\$ $/\$
10		absge0 14027	. . . . . . . . . 10
11		0lt1 10550	. . . . . . . . . . 11
12		addgegt0 10515	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
13	12	an4s 869	. . . . . . . . . . 11 $/\$ $/\$ $/\$
14	7, 11, 13	mpanr12 721	. . . . . . . . . 10 $/\$
15	5, 10, 14	syl2anc 693	. . . . . . . . 9
16	15	3ad2ant2 1083	. . . . . . . 8 $/\$ $/\$
17	9, 16	elrpd 11869	. . . . . . 7 $/\$ $/\$
18	2, 17	rpdivcld 11889	. . . . . 6 $/\$ $/\$
19	18	rpred 11872	. . . . 5 $/\$ $/\$
20	4, 19	readdcld 10069	. . . 4 $/\$ $/\$
21		absge0 14027	. . . . . 6
22	21	3ad2ant3 1084	. . . . 5 $/\$ $/\$
23		elrp 11834	. . . . . 6 $/\$
24		addgegt0 10515	. . . . . . 7 $/\$ $/\$ $/\$
25	24	an4s 869	. . . . . 6 $/\$ $/\$ $/\$
26	23, 25	sylan2b 492	. . . . 5 $/\$ $/\$
27	4, 22, 18, 26	syl21anc 1325	. . . 4 $/\$ $/\$
28	20, 27	elrpd 11869	. . 3 $/\$ $/\$
29	2, 28	rpdivcld 11889	. 2 $/\$ $/\$
30		simprl 794	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
31		simpl2 1065	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
32	30, 31	subcld 10392	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
33	32	abscld 14175	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
34	2	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
35	34	rpred 11872	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
36	28	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
37	33, 35, 36	ltmuldivd 11919	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
38		simprr 796	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
39		simpl3 1066	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
40	38, 39	abs2difd 14196	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
41	38	abscld 14175	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
42	4	adantr 481	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
43	41, 42	resubcld 10458	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
44	38, 39	subcld 10392	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
45	44	abscld 14175	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
46	19	adantr 481	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
47		lelttr 10128	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
48	43, 45, 46, 47	syl3anc 1326	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
49	40, 48	mpand 711	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
50	41, 42, 46	ltsubadd2d 10625	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
51	49, 50	sylibd 229	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
52	20	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
53		ltle 10126	. . . . . . . . . . . 12 $/\$
54	41, 52, 53	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
55	51, 54	syld 47	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
56	32	absge0d 14183	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
57		lemul2a 10878	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
58	57	ex 450	. . . . . . . . . . 11 $/\$ $/\$ $/\$
59	41, 52, 33, 56, 58	syl112anc 1330	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
60	33, 41	remulcld 10070	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
61	33, 52	remulcld 10070	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
62		lelttr 10128	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
63	60, 61, 35, 62	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	expd 452	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
65	55, 59, 64	3syld 60	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
66	65	com23 86	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
67	37, 66	sylbird 250	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
68	67	impd 447	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
69	32, 38	absmuld 14193	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
70	30, 31, 38	subdird 10487	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
71	70	fveq2d 6195	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
72	69, 71	eqtr3d 2658	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
73	72	breq1d 4663	. . . . . 6 $/\$ $/\$ $/\$ $/\$
74	68, 73	sylibd 229	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
75	17	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
76	45, 35, 75	ltmuldiv2d 11920	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
77	31, 38, 39	subdid 10486	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
78	77	fveq2d 6195	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
79	31, 44	absmuld 14193	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
80	78, 79	eqtr3d 2658	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
81	31	abscld 14175	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
82	81	lep1d 10955	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
83	9	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
84		abscl 14018	. . . . . . . . . . . . 13
85		absge0 14027	. . . . . . . . . . . . 13
86	84, 85	jca 554	. . . . . . . . . . . 12 $/\$
87		lemul1a 10877	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
88	87	ex 450	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
89	86, 88	syl3an3 1361	. . . . . . . . . . 11 $/\$ $/\$
90	81, 83, 44, 89	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
91	82, 90	mpd 15	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
92	80, 91	eqbrtrd 4675	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
93	31, 38	mulcld 10060	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
94	31, 39	mulcld 10060	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
95	93, 94	subcld 10392	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
96	95	abscld 14175	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
97	83, 45	remulcld 10070	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
98		lelttr 10128	. . . . . . . . 9 $/\$ $/\$ $/\$
99	96, 97, 35, 98	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	92, 99	mpand 711	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
101	76, 100	sylbird 250	. . . . . 6 $/\$ $/\$ $/\$ $/\$
102	101	adantld 483	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	74, 102	jcad 555	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
104		mulcl 10020	. . . . . 6 $/\$
105	104	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$
106		simpl1 1064	. . . . . 6 $/\$ $/\$ $/\$ $/\$
107	106	rpred 11872	. . . . 5 $/\$ $/\$ $/\$ $/\$
108		abs3lem 14078	. . . . 5 $/\$ $/\$ $/\$ $/\$
109	105, 94, 93, 107, 108	syl22anc 1327	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
110	103, 109	syld 47	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
111	110	ralrimivva 2971	. 2 $/\$ $/\$ $/\$
112		breq2 4657	. . . . . 6
113	112	anbi1d 741	. . . . 5 $/\$ $/\$
114	113	imbi1d 331	. . . 4 $/\$ $/\$
115	114	2ralbidv 2989	. . 3 $/\$ $/\$
116		breq2 4657	. . . . . 6
117	116	anbi2d 740	. . . . 5 $/\$ $/\$
118	117	imbi1d 331	. . . 4 $/\$ $/\$
119	118	2ralbidv 2989	. . 3 $/\$ $/\$
120	115, 119	rspc2ev 3324	. 2 $/\$ $/\$ $/\$ $/\$
121	29, 18, 111, 120	syl3anc 1326	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

wrex 2913 class class class wbr 4653

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cmin 10266

cdiv 10684

c2 11070

crp 11832

cabs 13974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: climmul 14363 rlimmul 14375 mulcn 22670 mulc1cncf 22708 mullimc 39848 mullimcf 39855

W3C validator