Theorem cvg1nlemcxze 11493

Description: Lemma for cvg1n 11497. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)

Hypotheses

Ref	Expression
cvg1nlemcxze.c	|- ( ph -> C e. RR+ )
cvg1nlemcxze.x	|- ( ph -> X e. RR+ )
cvg1nlemcxze.z	|- ( ph -> Z e. NN )
cvg1nlemcxze.e	|- ( ph -> E e. NN )
cvg1nlemcxze.a	|- ( ph -> A e. NN )
cvg1nlemcxze.1	|- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) + A ) < E )

Assertion

Ref	Expression
cvg1nlemcxze	|- ( ph -> ( C / ( E x. Z ) ) < ( X / 2 ) )

Proof of Theorem cvg1nlemcxze

Step	Hyp	Ref	Expression
1		cvg1nlemcxze.c	. . . . . . . 8 |- ( ph -> C e. RR+ )
2	1	rpcnd 9894	. . . . . . 7 |- ( ph -> C e. CC )
3		2cnd 9183	. . . . . . 7 |- ( ph -> 2 e. CC )
4		cvg1nlemcxze.x	. . . . . . . 8 |- ( ph -> X e. RR+ )
5	4	rpcnd 9894	. . . . . . 7 |- ( ph -> X e. CC )
6	4	rpap0d 9898	. . . . . . 7 |- ( ph -> X # 0 )
7	2, 3, 5, 6	div23apd 8975	. . . . . 6 |- ( ph -> ( ( C x. 2 ) / X ) = ( ( C / X ) x. 2 ) )
8		2rp 9854	. . . . . . . . . . . . 13 |- 2 e. RR+
9	8	a1i 9	. . . . . . . . . . . 12 |- ( ph -> 2 e. RR+ )
10	1, 9	rpmulcld 9909	. . . . . . . . . . 11 |- ( ph -> ( C x. 2 ) e. RR+ )
11	10, 4	rpdivcld 9910	. . . . . . . . . 10 |- ( ph -> ( ( C x. 2 ) / X ) e. RR+ )
12		cvg1nlemcxze.z	. . . . . . . . . . 11 |- ( ph -> Z e. NN )
13	12	nnrpd 9890	. . . . . . . . . 10 |- ( ph -> Z e. RR+ )
14	11, 13	rpdivcld 9910	. . . . . . . . 9 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) e. RR+ )
15	14	rpred 9892	. . . . . . . 8 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) e. RR )
16		cvg1nlemcxze.a	. . . . . . . . . 10 |- ( ph -> A e. NN )
17	16	nnred 9123	. . . . . . . . 9 |- ( ph -> A e. RR )
18	15, 17	readdcld 8176	. . . . . . . 8 |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) + A ) e. RR )
19		cvg1nlemcxze.e	. . . . . . . . 9 |- ( ph -> E e. NN )
20	19	nnred 9123	. . . . . . . 8 |- ( ph -> E e. RR )
21	16	nnrpd 9890	. . . . . . . . 9 |- ( ph -> A e. RR+ )
22	15, 21	ltaddrpd 9926	. . . . . . . 8 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) < ( ( ( ( C x. 2 ) / X ) / Z ) + A ) )
23		cvg1nlemcxze.1	. . . . . . . 8 |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) + A ) < E )
24	15, 18, 20, 22, 23	lttrd 8272	. . . . . . 7 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) < E )
25	11	rpred 9892	. . . . . . . 8 |- ( ph -> ( ( C x. 2 ) / X ) e. RR )
26	25, 20, 13	ltdivmul2d 9945	. . . . . . 7 |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) < E <-> ( ( C x. 2 ) / X ) < ( E x. Z ) ) )
27	24, 26	mpbid 147	. . . . . 6 |- ( ph -> ( ( C x. 2 ) / X ) < ( E x. Z ) )
28	7, 27	eqbrtrrd 4107	. . . . 5 |- ( ph -> ( ( C / X ) x. 2 ) < ( E x. Z ) )
29	1	rpred 9892	. . . . . . 7 |- ( ph -> C e. RR )
30	29, 4	rerpdivcld 9924	. . . . . 6 |- ( ph -> ( C / X ) e. RR )
31	19, 12	nnmulcld 9159	. . . . . . 7 |- ( ph -> ( E x. Z ) e. NN )
32	31	nnred 9123	. . . . . 6 |- ( ph -> ( E x. Z ) e. RR )
33	30, 32, 9	ltmuldivd 9940	. . . . 5 |- ( ph -> ( ( ( C / X ) x. 2 ) < ( E x. Z ) <-> ( C / X ) < ( ( E x. Z ) / 2 ) ) )
34	28, 33	mpbid 147	. . . 4 |- ( ph -> ( C / X ) < ( ( E x. Z ) / 2 ) )
35	29, 9, 32, 4	lt2mul2divd 9961	. . . 4 |- ( ph -> ( ( C x. 2 ) < ( ( E x. Z ) x. X ) <-> ( C / X ) < ( ( E x. Z ) / 2 ) ) )
36	34, 35	mpbird 167	. . 3 |- ( ph -> ( C x. 2 ) < ( ( E x. Z ) x. X ) )
37	31	nncnd 9124	. . . 4 |- ( ph -> ( E x. Z ) e. CC )
38	37, 5	mulcomd 8168	. . 3 |- ( ph -> ( ( E x. Z ) x. X ) = ( X x. ( E x. Z ) ) )
39	36, 38	breqtrd 4109	. 2 |- ( ph -> ( C x. 2 ) < ( X x. ( E x. Z ) ) )
40	4	rpred 9892	. . 3 |- ( ph -> X e. RR )
41	31	nnrpd 9890	. . 3 |- ( ph -> ( E x. Z ) e. RR+ )
42	29, 9, 40, 41	lt2mul2divd 9961	. 2 |- ( ph -> ( ( C x. 2 ) < ( X x. ( E x. Z ) ) <-> ( C / ( E x. Z ) ) < ( X / 2 ) ) )
43	39, 42	mpbid 147	1 |- ( ph -> ( C / ( E x. Z ) ) < ( X / 2 ) )

Syntax hints:   -> wi 4   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   + caddc 8002   x. cmul 8004   < clt 8181   / cdiv 8819   NNcn 9110   2c2 9161   RR+crp 9849

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-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  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-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-rp 9850

This theorem is referenced by:  cvg1nlemres  11496