Theorem reccn2ap 11456

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 2193. (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 8034	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 1 e. RR )
3		simp1 999	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A e. CC )
4		simp2 1000	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A # 0 )
5	3, 4	absrpclapd 11332	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( abs ` A ) e. RR+ )
6		simp3 1001	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> B e. RR+ )
7	5, 6	rpmulcld 9779	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR+ )
8	7	rpred 9762	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR )
9		mincl 11374	. . . . . 6 |- ( ( 1 e. RR /\ ( ( abs ` A ) x. B ) e. RR ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) e. RR )
10	2, 8, 9	syl2anc 411	. . . . 5 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) e. RR )
11	7	rpgt0d 9765	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 < ( ( abs ` A ) x. B ) )
12		0lt1 8146	. . . . . . 7 |- 0 < 1
13	11, 12	jctil 312	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( 0 < 1 /\ 0 < ( ( abs ` A ) x. B ) ) )
14		0red 8020	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 e. RR )
15		ltmininf 11378	. . . . . . 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 1249	. . . . . 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 167	. . . . 5 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 < inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) )
18	10, 17	elrpd 9759	. . . 4 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) e. RR+ )
19	5	rphalfcld 9775	. . . 4 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) / 2 ) e. RR+ )
20	18, 19	rpmulcld 9779	. . 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 2280	. 2 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> T e. RR+ )
22	3	adantr 276	. . . . . . . . 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 529	. . . . . . . . . . 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 4032	. . . . . . . . . . . 12 |- ( w = z -> ( w # 0 <-> z # 0 ) )
25	24	elrab 2916	. . . . . . . . . . 11 |- ( z e. { w e. CC | w # 0 } <-> ( z e. CC /\ z # 0 ) )
26	23, 25	sylib 122	. . . . . . . . . 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 112	. . . . . . . . 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 8040	. . . . . . . . 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 276	. . . . . . . . . 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 114	. . . . . . . . . 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 8677	. . . . . . . . 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 8849	. . . . . . . 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	mulridd 8036	. . . . . . . . . . 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 5933	. . . . . . . . . 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 8035	. . . . . . . . . . 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 8836	. . . . . . . . . 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 2228	. . . . . . . . 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	mulridd 8036	. . . . . . . . . . 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 8041	. . . . . . . . . . 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 5936	. . . . . . . . . 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 8836	. . . . . . . . . 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 2228	. . . . . . . . 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 5936	. . . . . . . 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 2226	. . . . . . 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 5558	. . . . . 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 8330	. . . . . . 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 11339	. . . . . 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 2228	. . . . 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 11325	. . . . . . 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 276	. . . . . . . 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 9762	. . . . . . 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 11325	. . . . . . . 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 9762	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> B e. RR )
54	53	adantr 276	. . . . . . . 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 8050	. . . . . . 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 11337	. . . . . . . 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 531	. . . . . . . 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 4051	. . . . . . 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 276	. . . . . . . . . 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 9762	. . . . . . . . 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 276	. . . . . . . . . 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 9762	. . . . . . . . 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 8050	. . . . . . . 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 8018	. . . . . . . . . . 11 |- 1 e. RR
65		min2inf 11376	. . . . . . . . . . 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 414	. . . . . . . . . 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 276	. . . . . . . . . . 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 9806	. . . . . . . . . 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 147	. . . . . . . . 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 4064	. . . . . . . 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 11325	. . . . . . . . . 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 11325	. . . . . . . . . . . . . 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 8048	. . . . . . . . . . . . 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 9228	. . . . . . . . . . . 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 8400	. . . . . . . . . . . . . 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 8330	. . . . . . . . . . . . . . . 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 11325	. . . . . . . . . . . . . . 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 2270	. . . . . . . . . . . . . 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 11341	. . . . . . . . . . . . . 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 11375	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 e. RR /\ ( ( abs ` A ) x. B ) e. RR ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) <_ 1 )
81	64, 60, 80	sylancr 414	. . . . . . . . . . . . . . . . . 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 8034	. . . . . . . . . . . . . . . . . . 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 9806	. . . . . . . . . . . . . . . . . 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 147	. . . . . . . . . . . . . . . . 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 4064	. . . . . . . . . . . . . . . 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 8048	. . . . . . . . . . . . . . . . 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 8038	. . . . . . . . . . . . . . . 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 4055	. . . . . . . . . . . . . . 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 8442	. . . . . . . . . . . . . 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 8144	. . . . . . . . . . . . 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 8562	. . . . . . . . . . . . 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 147	. . . . . . . . . . . 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 4051	. . . . . . . . . . 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 8557	. . . . . . . . . . 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 167	. . . . . . . . . 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 9819	. . . . . . . . 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 11338	. . . . . . . . . . 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 5933	. . . . . . . . . 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 8048	. . . . . . . . . . 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 8048	. . . . . . . . . . 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 8172	. . . . . . . . . 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 2226	. . . . . . . . 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 4057	. . . . . . . 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 8144	. . . . . . 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 8145	. . . . . 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 11332	. . . . . . 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 9814	. . . . . 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 167	. . . . 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 4051	. . . 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 375	. . 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 2567	. 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 4033	. . 3 |- ( y = T -> ( ( abs ` ( z - A ) ) < y <-> ( abs ` ( z - A ) ) < T ) )
113	112	rspceaimv 2872	. 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 411	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   {cpr 3619   class class class wbr 4029   ` cfv 5254  (class class class)co 5918  infcinf 7042   CCcc 7870   RRcr 7871   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   < clt 8054   <_ cle 8055   - cmin 8190   # cap 8600   / cdiv 8691   2c2 9033   RR+crp 9719   abscabs 11141

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143

This theorem is referenced by:  divcnap  14723  cdivcncfap  14758