Theorem reccn2ap 11819

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 2229. (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 8157	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 1 e. RR )
3		simp1 1021	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A e. CC )
4		simp2 1022	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A # 0 )
5	3, 4	absrpclapd 11694	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( abs ` A ) e. RR+ )
6		simp3 1023	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> B e. RR+ )
7	5, 6	rpmulcld 9905	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR+ )
8	7	rpred 9888	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR )
9		mincl 11737	. . . . . 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 9891	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 < ( ( abs ` A ) x. B ) )
12		0lt1 8269	. . . . . . 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 8143	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 e. RR )
15		ltmininf 11741	. . . . . . 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 1271	. . . . . 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 9885	. . . 4 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) e. RR+ )
19	5	rphalfcld 9901	. . . 4 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) / 2 ) e. RR+ )
20	18, 19	rpmulcld 9905	. . 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 2316	. 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 4085	. . . . . . . . . . . 12 |- ( w = z -> ( w # 0 <-> z # 0 ) )
25	24	elrab 2959	. . . . . . . . . . 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 8163	. . . . . . . . 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 8801	. . . . . . . . 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 8973	. . . . . . . 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 8159	. . . . . . . . . . 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 6015	. . . . . . . . . 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 8158	. . . . . . . . . . 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 8960	. . . . . . . . . 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 2264	. . . . . . . . 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 8159	. . . . . . . . . . 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 8164	. . . . . . . . . . 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 6018	. . . . . . . . . 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 8960	. . . . . . . . . 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 2264	. . . . . . . . 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 6018	. . . . . . . 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 2262	. . . . . . 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 5630	. . . . . 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 8453	. . . . . . 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 11701	. . . . . 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 2264	. . . . 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 11687	. . . . . . 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 9888	. . . . . . 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 11687	. . . . . . . 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 9888	. . . . . . . . 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 8173	. . . . . . 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 11699	. . . . . . . 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 4104	. . . . . . 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 9888	. . . . . . . . 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 9888	. . . . . . . . 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 8173	. . . . . . . 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 8141	. . . . . . . . . . 11 |- 1 e. RR
65		min2inf 11739	. . . . . . . . . . 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 9932	. . . . . . . . . 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 4117	. . . . . . . 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 11687	. . . . . . . . . 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 11687	. . . . . . . . . . . . . 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 8171	. . . . . . . . . . . . 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 9353	. . . . . . . . . . . 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 8523	. . . . . . . . . . . . . 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 8453	. . . . . . . . . . . . . . . 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 11687	. . . . . . . . . . . . . . 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 2306	. . . . . . . . . . . . . 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 11703	. . . . . . . . . . . . . 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 11738	. . . . . . . . . . . . . . . . . . 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 8157	. . . . . . . . . . . . . . . . . . 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 9932	. . . . . . . . . . . . . . . . . 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 4117	. . . . . . . . . . . . . . . 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 8171	. . . . . . . . . . . . . . . . 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 8161	. . . . . . . . . . . . . . . 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 4108	. . . . . . . . . . . . . . 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 8566	. . . . . . . . . . . . . 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 8267	. . . . . . . . . . . . 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 8686	. . . . . . . . . . . . 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 4104	. . . . . . . . . . 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 8681	. . . . . . . . . . 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 9945	. . . . . . . . 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 11700	. . . . . . . . . . 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 6015	. . . . . . . . . 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 8171	. . . . . . . . . . 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 8171	. . . . . . . . . . 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 8295	. . . . . . . . . 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 2262	. . . . . . . . 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 4110	. . . . . . . 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 8267	. . . . . . 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 8268	. . . . . 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 11694	. . . . . . 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 9940	. . . . . 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 4104	. . . 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 2603	. 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 4086	. . 3 |- ( y = T -> ( ( abs ` ( z - A ) ) < y <-> ( abs ` ( z - A ) ) < T ) )
113	112	rspceaimv 2915	. 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   {cpr 3667   class class class wbr 4082   ` cfv 5317  (class class class)co 6000  infcinf 7146   CCcc 7993   RRcr 7994   0cc0 7995   1c1 7996   + caddc 7998   x. cmul 8000   < clt 8177   <_ cle 8178   - cmin 8313   # cap 8724   / cdiv 8815   2c2 9157   RR+crp 9845   abscabs 11503

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-rp 9846  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505

This theorem is referenced by:  divcnap  15233  cdivcncfap  15272