Theorem reccn2ap 11114

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 2140. (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 7805	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 1 e. RR )
3		simp1 982	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A e. CC )
4		simp2 983	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A # 0 )
5	3, 4	absrpclapd 10992	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( abs ` A ) e. RR+ )
6		simp3 984	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> B e. RR+ )
7	5, 6	rpmulcld 9530	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR+ )
8	7	rpred 9513	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR )
9		mincl 11034	. . . . . 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 9516	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 < ( ( abs ` A ) x. B ) )
12		0lt1 7913	. . . . . . 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 7791	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 e. RR )
15		ltmininf 11038	. . . . . . 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 1217	. . . . . 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 9510	. . . 4 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) e. RR+ )
19	5	rphalfcld 9526	. . . 4 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) / 2 ) e. RR+ )
20	18, 19	rpmulcld 9530	. . 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 2227	. 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 521	. . . . . . . . . . 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 3940	. . . . . . . . . . . 12 |- ( w = z -> ( w # 0 <-> z # 0 ) )
25	24	elrab 2844	. . . . . . . . . . 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 7810	. . . . . . . . 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 8443	. . . . . . . . 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 8614	. . . . . . . 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 7807	. . . . . . . . . . 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 5797	. . . . . . . . . 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 7806	. . . . . . . . . . 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 8601	. . . . . . . . . 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 2175	. . . . . . . . 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 7807	. . . . . . . . . . 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 7811	. . . . . . . . . . 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 5800	. . . . . . . . . 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 8601	. . . . . . . . . 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 2175	. . . . . . . . 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 5800	. . . . . . . 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 2173	. . . . . . 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 5433	. . . . . 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 8097	. . . . . . 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 10999	. . . . . 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 2175	. . . . 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 10985	. . . . . . 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 9513	. . . . . . 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 10985	. . . . . . . 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 9513	. . . . . . . . 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 7820	. . . . . . 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 10997	. . . . . . . 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 522	. . . . . . . 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 3958	. . . . . . 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 9513	. . . . . . . . 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 9513	. . . . . . . . 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 7820	. . . . . . . 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 7789	. . . . . . . . . . 11 |- 1 e. RR
65		min2inf 11036	. . . . . . . . . . 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 411	. . . . . . . . . 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 9557	. . . . . . . . . 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 3971	. . . . . . . 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 10985	. . . . . . . . . 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 10985	. . . . . . . . . . . . . 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 7818	. . . . . . . . . . . . 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 8989	. . . . . . . . . . . 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 8167	. . . . . . . . . . . . . 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 8097	. . . . . . . . . . . . . . . 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 10985	. . . . . . . . . . . . . . 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 2217	. . . . . . . . . . . . . 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 11001	. . . . . . . . . . . . . 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 11035	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 e. RR /\ ( ( abs ` A ) x. B ) e. RR ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) <_ 1 )
81	64, 60, 80	sylancr 411	. . . . . . . . . . . . . . . . . 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 7805	. . . . . . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . 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 7818	. . . . . . . . . . . . . . . . 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 7808	. . . . . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . 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 8209	. . . . . . . . . . . . . 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 7911	. . . . . . . . . . . . 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 8329	. . . . . . . . . . . . 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 3958	. . . . . . . . . . 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 8324	. . . . . . . . . . 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 9570	. . . . . . . . 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 10998	. . . . . . . . . . 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 5797	. . . . . . . . . 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 7818	. . . . . . . . . . 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 7818	. . . . . . . . . . 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 7939	. . . . . . . . . 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 2173	. . . . . . . . 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 3964	. . . . . . . 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 7911	. . . . . . 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 7912	. . . . . 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 10992	. . . . . . 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 9565	. . . . . 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 3958	. . . 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 2508	. 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 3941	. . 3 |- ( y = T -> ( ( abs ` ( z - A ) ) < y <-> ( abs ` ( z - A ) ) < T ) )
113	112	rspceaimv 2801	. 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 963   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   {cpr 3533   class class class wbr 3937   ` cfv 5131  (class class class)co 5782  infcinf 6878   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   < clt 7824   <_ cle 7825   - cmin 7957   # cap 8367   / cdiv 8456   2c2 8795   RR+crp 9470   abscabs 10801

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-sup 6879  df-inf 6880  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-rp 9471  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803

This theorem is referenced by:  divcnap  12763  cdivcncfap  12795