Theorem reccn2ap 11263

Description: The reciprocal function is continuous. The class T is just for convenience in writing the proof and typically would be passed in as an instance of eqid 2170. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)

Hypothesis

Ref	Expression
reccn2ap.t	|- T = (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) x. ( ( abs ` A ) / 2 ) )

Assertion

Ref	Expression
reccn2ap	|- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> E. y e. RR+ A. z e. { w e. CC | w # 0 } ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) )

Distinct variable groups:   y, w, z, A   w, B, y, z   y, T, z

Allowed substitution hint:   T( w)

Proof of Theorem reccn2ap

Step	Hyp	Ref	Expression
1		reccn2ap.t	. . 3 |- T = (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) x. ( ( abs ` A ) / 2 ) )
2		1red 7922	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 1 e. RR )
3		simp1 992	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A e. CC )
4		simp2 993	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A # 0 )
5	3, 4	absrpclapd 11139	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( abs ` A ) e. RR+ )
6		simp3 994	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> B e. RR+ )
7	5, 6	rpmulcld 9657	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR+ )
8	7	rpred 9640	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR )
9		mincl 11181	. . . . . 6 |- ( ( 1 e. RR /\ ( ( abs ` A ) x. B ) e. RR ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) e. RR )
10	2, 8, 9	syl2anc 409	. . . . 5 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) e. RR )
11	7	rpgt0d 9643	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 < ( ( abs ` A ) x. B ) )
12		0lt1 8033	. . . . . . 7 |- 0 < 1
13	11, 12	jctil 310	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( 0 < 1 /\ 0 < ( ( abs ` A ) x. B ) ) )
14		0red 7908	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 e. RR )
15		ltmininf 11185	. . . . . . 7 |- ( ( 0 e. RR /\ 1 e. RR /\ ( ( abs ` A ) x. B ) e. RR ) -> ( 0 < inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) <-> ( 0 < 1 /\ 0 < ( ( abs ` A ) x. B ) ) ) )
16	14, 2, 8, 15	syl3anc 1233	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( 0 < inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) <-> ( 0 < 1 /\ 0 < ( ( abs ` A ) x. B ) ) ) )
17	13, 16	mpbird 166	. . . . 5 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 < inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) )
18	10, 17	elrpd 9637	. . . 4 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) e. RR+ )
19	5	rphalfcld 9653	. . . 4 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) / 2 ) e. RR+ )
20	18, 19	rpmulcld 9657	. . 3 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) x. ( ( abs ` A ) / 2 ) ) e. RR+ )
21	1, 20	eqeltrid 2257	. 2 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> T e. RR+ )
22	3	adantr 274	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> A e. CC )
23		simprl 526	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> z e. { w e. CC | w # 0 } )
24		breq1 3990	. . . . . . . . . . . 12 |- ( w = z -> ( w # 0 <-> z # 0 ) )
25	24	elrab 2886	. . . . . . . . . . 11 |- ( z e. { w e. CC | w # 0 } <-> ( z e. CC /\ z # 0 ) )
26	23, 25	sylib 121	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( z e. CC /\ z # 0 ) )
27	26	simpld 111	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> z e. CC )
28	22, 27	mulcld 7927	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( A x. z ) e. CC )
29	4	adantr 274	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> A # 0 )
30	26	simprd 113	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> z # 0 )
31	22, 27, 29, 30	mulap0d 8563	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( A x. z ) # 0 )
32	22, 27, 28, 31	divsubdirapd 8734	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( A - z ) / ( A x. z ) ) = ( ( A / ( A x. z ) ) - ( z / ( A x. z ) ) ) )
33	22	mulid1d 7924	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( A x. 1 ) = A )
34	33	oveq1d 5865	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( A x. 1 ) / ( A x. z ) ) = ( A / ( A x. z ) ) )
35		1cnd 7923	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> 1 e. CC )
36	35, 27, 22, 30, 29	divcanap5d 8721	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( A x. 1 ) / ( A x. z ) ) = ( 1 / z ) )
37	34, 36	eqtr3d 2205	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( A / ( A x. z ) ) = ( 1 / z ) )
38	27	mulid1d 7924	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( z x. 1 ) = z )
39	27, 22	mulcomd 7928	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( z x. A ) = ( A x. z ) )
40	38, 39	oveq12d 5868	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( z x. 1 ) / ( z x. A ) ) = ( z / ( A x. z ) ) )
41	35, 22, 27, 29, 30	divcanap5d 8721	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( z x. 1 ) / ( z x. A ) ) = ( 1 / A ) )
42	40, 41	eqtr3d 2205	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( z / ( A x. z ) ) = ( 1 / A ) )
43	37, 42	oveq12d 5868	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( A / ( A x. z ) ) - ( z / ( A x. z ) ) ) = ( ( 1 / z ) - ( 1 / A ) ) )
44	32, 43	eqtrd 2203	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( A - z ) / ( A x. z ) ) = ( ( 1 / z ) - ( 1 / A ) ) )
45	44	fveq2d 5498	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( ( A - z ) / ( A x. z ) ) ) = ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) )
46	22, 27	subcld 8217	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( A - z ) e. CC )
47	46, 28, 31	absdivapd 11146	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( ( A - z ) / ( A x. z ) ) ) = ( ( abs ` ( A - z ) ) / ( abs ` ( A x. z ) ) ) )
48	45, 47	eqtr3d 2205	. . . . 5 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) = ( ( abs ` ( A - z ) ) / ( abs ` ( A x. z ) ) ) )
49	46	abscld 11132	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) e. RR )
50	21	adantr 274	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> T e. RR+ )
51	50	rpred 9640	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> T e. RR )
52	28	abscld 11132	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A x. z ) ) e. RR )
53	6	rpred 9640	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> B e. RR )
54	53	adantr 274	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> B e. RR )
55	52, 54	remulcld 7937	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` ( A x. z ) ) x. B ) e. RR )
56	22, 27	abssubd 11144	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) = ( abs ` ( z - A ) ) )
57		simprr 527	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( z - A ) ) < T )
58	56, 57	eqbrtrd 4009	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) < T )
59	7	adantr 274	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) x. B ) e. RR+ )
60	59	rpred 9640	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) x. B ) e. RR )
61	19	adantr 274	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) / 2 ) e. RR+ )
62	61	rpred 9640	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) / 2 ) e. RR )
63	60, 62	remulcld 7937	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) e. RR )
64		1re 7906	. . . . . . . . . . 11 |- 1 e. RR
65		min2inf 11183	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ ( ( abs ` A ) x. B ) e. RR ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) <_ ( ( abs ` A ) x. B ) )
66	64, 60, 65	sylancr 412	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) <_ ( ( abs ` A ) x. B ) )
67	10	adantr 274	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) e. RR )
68	67, 60, 61	lemul1d 9684	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) <_ ( ( abs ` A ) x. B ) <-> (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) x. ( ( abs ` A ) / 2 ) ) <_ ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) ) )
69	66, 68	mpbid 146	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) x. ( ( abs ` A ) / 2 ) ) <_ ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) )
70	1, 69	eqbrtrid 4022	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> T <_ ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) )
71	27	abscld 11132	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` z ) e. RR )
72	22	abscld 11132	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` A ) e. RR )
73	72	recnd 7935	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` A ) e. CC )
74	73	2halvesd 9110	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) / 2 ) + ( ( abs ` A ) / 2 ) ) = ( abs ` A ) )
75	72, 71	resubcld 8287	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) - ( abs ` z ) ) e. RR )
76	27, 22	subcld 8217	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( z - A ) e. CC )
77	76	abscld 11132	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( z - A ) ) e. RR )
78	56, 77	eqeltrd 2247	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) e. RR )
79	22, 27	abs2difd 11148	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) - ( abs ` z ) ) <_ ( abs ` ( A - z ) ) )
80		min1inf 11182	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 e. RR /\ ( ( abs ` A ) x. B ) e. RR ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) <_ 1 )
81	64, 60, 80	sylancr 412	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) <_ 1 )
82		1red 7922	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> 1 e. RR )
83	67, 82, 61	lemul1d 9684	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) <_ 1 <-> (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) x. ( ( abs ` A ) / 2 ) ) <_ ( 1 x. ( ( abs ` A ) / 2 ) ) ) )
84	81, 83	mpbid 146	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) x. ( ( abs ` A ) / 2 ) ) <_ ( 1 x. ( ( abs ` A ) / 2 ) ) )
85	1, 84	eqbrtrid 4022	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> T <_ ( 1 x. ( ( abs ` A ) / 2 ) ) )
86	62	recnd 7935	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) / 2 ) e. CC )
87	86	mulid2d 7925	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( 1 x. ( ( abs ` A ) / 2 ) ) = ( ( abs ` A ) / 2 ) )
88	85, 87	breqtrd 4013	. . . . . . . . . . . . . . 15 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> T <_ ( ( abs ` A ) / 2 ) )
89	78, 51, 62, 58, 88	ltletrd 8329	. . . . . . . . . . . . . 14 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) < ( ( abs ` A ) / 2 ) )
90	75, 78, 62, 79, 89	lelttrd 8031	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) - ( abs ` z ) ) < ( ( abs ` A ) / 2 ) )
91	72, 71, 62	ltsubadd2d 8449	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) - ( abs ` z ) ) < ( ( abs ` A ) / 2 ) <-> ( abs ` A ) < ( ( abs ` z ) + ( ( abs ` A ) / 2 ) ) ) )
92	90, 91	mpbid 146	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` A ) < ( ( abs ` z ) + ( ( abs ` A ) / 2 ) ) )
93	74, 92	eqbrtrd 4009	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) / 2 ) + ( ( abs ` A ) / 2 ) ) < ( ( abs ` z ) + ( ( abs ` A ) / 2 ) ) )
94	62, 71, 62	ltadd1d 8444	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) / 2 ) < ( abs ` z ) <-> ( ( ( abs ` A ) / 2 ) + ( ( abs ` A ) / 2 ) ) < ( ( abs ` z ) + ( ( abs ` A ) / 2 ) ) ) )
95	93, 94	mpbird 166	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) / 2 ) < ( abs ` z ) )
96	62, 71, 59, 95	ltmul2dd 9697	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) < ( ( ( abs ` A ) x. B ) x. ( abs ` z ) ) )
97	22, 27	absmuld 11145	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A x. z ) ) = ( ( abs ` A ) x. ( abs ` z ) ) )
98	97	oveq1d 5865	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` ( A x. z ) ) x. B ) = ( ( ( abs ` A ) x. ( abs ` z ) ) x. B ) )
99	71	recnd 7935	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` z ) e. CC )
100	54	recnd 7935	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> B e. CC )
101	73, 99, 100	mul32d 8059	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) x. ( abs ` z ) ) x. B ) = ( ( ( abs ` A ) x. B ) x. ( abs ` z ) ) )
102	98, 101	eqtrd 2203	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` ( A x. z ) ) x. B ) = ( ( ( abs ` A ) x. B ) x. ( abs ` z ) ) )
103	96, 102	breqtrrd 4015	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) < ( ( abs ` ( A x. z ) ) x. B ) )
104	51, 63, 55, 70, 103	lelttrd 8031	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> T < ( ( abs ` ( A x. z ) ) x. B ) )
105	49, 51, 55, 58, 104	lttrd 8032	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) < ( ( abs ` ( A x. z ) ) x. B ) )
106	28, 31	absrpclapd 11139	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A x. z ) ) e. RR+ )
107	49, 54, 106	ltdivmuld 9692	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` ( A - z ) ) / ( abs ` ( A x. z ) ) ) < B <-> ( abs ` ( A - z ) ) < ( ( abs ` ( A x. z ) ) x. B ) ) )
108	105, 107	mpbird 166	. . . . 5 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` ( A - z ) ) / ( abs ` ( A x. z ) ) ) < B )
109	48, 108	eqbrtrd 4009	. . . 4 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ ( z e. { w e. CC | w # 0 } /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B )
110	109	expr 373	. . 3 |- ( ( ( A e. CC /\ A # 0 /\ B e. RR+ ) /\ z e. { w e. CC | w # 0 } ) -> ( ( abs ` ( z - A ) ) < T -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) )
111	110	ralrimiva 2543	. 2 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A. z e. { w e. CC | w # 0 } ( ( abs ` ( z - A ) ) < T -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) )
112		breq2 3991	. . 3 |- ( y = T -> ( ( abs ` ( z - A ) ) < y <-> ( abs ` ( z - A ) ) < T ) )
113	112	rspceaimv 2842	. 2 |- ( ( T e. RR+ /\ A. z e. { w e. CC | w # 0 } ( ( abs ` ( z - A ) ) < T -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) ) -> E. y e. RR+ A. z e. { w e. CC | w # 0 } ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) )
114	21, 111, 113	syl2anc 409	1 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> E. y e. RR+ A. z e. { w e. CC | w # 0 } ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   {cpr 3582   class class class wbr 3987   ` cfv 5196  (class class class)co 5850  infcinf 6956   CCcc 7759   RRcr 7760   0cc0 7761   1c1 7762   + caddc 7764   x. cmul 7766   < clt 7941   <_ cle 7942   - cmin 8077   # cap 8487   / cdiv 8576   2c2 8916   RR+crp 9597   abscabs 10948

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 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  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-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-sup 6957  df-inf 6958  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-rp 9598  df-seqfrec 10389  df-exp 10463  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950

This theorem is referenced by:  divcnap  13270  cdivcncfap  13302