Theorem reccn2ap 11321

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 2177. (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 7972	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 1 e. RR )
3		simp1 997	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A e. CC )
4		simp2 998	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> A # 0 )
5	3, 4	absrpclapd 11197	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( abs ` A ) e. RR+ )
6		simp3 999	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> B e. RR+ )
7	5, 6	rpmulcld 9713	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR+ )
8	7	rpred 9696	. . . . . 6 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR )
9		mincl 11239	. . . . . 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 9699	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 < ( ( abs ` A ) x. B ) )
12		0lt1 8084	. . . . . . 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 7958	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> 0 e. RR )
15		ltmininf 11243	. . . . . . 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 1238	. . . . . 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 9693	. . . 4 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) e. RR+ )
19	5	rphalfcld 9709	. . . 4 |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> ( ( abs ` A ) / 2 ) e. RR+ )
20	18, 19	rpmulcld 9713	. . 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 2264	. 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 4007	. . . . . . . . . . . 12 |- ( w = z -> ( w # 0 <-> z # 0 ) )
25	24	elrab 2894	. . . . . . . . . . 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 7978	. . . . . . . . 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 8615	. . . . . . . . 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 8787	. . . . . . . 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 7974	. . . . . . . . . . 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 5890	. . . . . . . . . 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 7973	. . . . . . . . . . 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 8774	. . . . . . . . . 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 2212	. . . . . . . . 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 7974	. . . . . . . . . . 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 7979	. . . . . . . . . . 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 5893	. . . . . . . . . 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 8774	. . . . . . . . . 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 2212	. . . . . . . . 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 5893	. . . . . . . 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 2210	. . . . . . 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 5520	. . . . . 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 8268	. . . . . . 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 11204	. . . . . 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 2212	. . . . 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 11190	. . . . . . 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 9696	. . . . . . 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 11190	. . . . . . . 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 9696	. . . . . . . . 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 7988	. . . . . . 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 11202	. . . . . . . 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 4026	. . . . . . 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 9696	. . . . . . . . 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 9696	. . . . . . . . 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 7988	. . . . . . . 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 7956	. . . . . . . . . . 11 |- 1 e. RR
65		min2inf 11241	. . . . . . . . . . 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 9740	. . . . . . . . . 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 4039	. . . . . . . 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 11190	. . . . . . . . . 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 11190	. . . . . . . . . . . . . 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 7986	. . . . . . . . . . . . 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 9164	. . . . . . . . . . . 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 8338	. . . . . . . . . . . . . 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 8268	. . . . . . . . . . . . . . . 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 11190	. . . . . . . . . . . . . . 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 2254	. . . . . . . . . . . . . 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 11206	. . . . . . . . . . . . . 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 11240	. . . . . . . . . . . . . . . . . . 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 7972	. . . . . . . . . . . . . . . . . . 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 9740	. . . . . . . . . . . . . . . . . 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 4039	. . . . . . . . . . . . . . . 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 7986	. . . . . . . . . . . . . . . . 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 7976	. . . . . . . . . . . . . . . 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 4030	. . . . . . . . . . . . . . 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 8380	. . . . . . . . . . . . . 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 8082	. . . . . . . . . . . . 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 8500	. . . . . . . . . . . . 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 4026	. . . . . . . . . . 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 8495	. . . . . . . . . . 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 9753	. . . . . . . . 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 11203	. . . . . . . . . . 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 5890	. . . . . . . . . 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 7986	. . . . . . . . . . 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 7986	. . . . . . . . . . 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 8110	. . . . . . . . . 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 2210	. . . . . . . . 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 4032	. . . . . . . 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 8082	. . . . . . 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 8083	. . . . . 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 11197	. . . . . . 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 9748	. . . . . 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 4026	. . . 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 2550	. 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 4008	. . 3 |- ( y = T -> ( ( abs ` ( z - A ) ) < y <-> ( abs ` ( z - A ) ) < T ) )
113	112	rspceaimv 2850	. 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 978   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   {cpr 3594   class class class wbr 4004   ` cfv 5217  (class class class)co 5875  infcinf 6982   CCcc 7809   RRcr 7810   0cc0 7811   1c1 7812   + caddc 7814   x. cmul 7816   < clt 7992   <_ cle 7993   - cmin 8128   # cap 8538   / cdiv 8629   2c2 8970   RR+crp 9653   abscabs 11006

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-sup 6983  df-inf 6984  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-rp 9654  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008

This theorem is referenced by:  divcnap  14058  cdivcncfap  14090