Theorem cvg1nlemcxze 10959

Description: Lemma for cvg1n 10963. 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 9669	. . . . . . 7 |- ( ph -> C e. CC )
3		2cnd 8965	. . . . . . 7 |- ( ph -> 2 e. CC )
4		cvg1nlemcxze.x	. . . . . . . 8 |- ( ph -> X e. RR+ )
5	4	rpcnd 9669	. . . . . . 7 |- ( ph -> X e. CC )
6	4	rpap0d 9673	. . . . . . 7 |- ( ph -> X # 0 )
7	2, 3, 5, 6	div23apd 8758	. . . . . 6 |- ( ph -> ( ( C x. 2 ) / X ) = ( ( C / X ) x. 2 ) )
8		2rp 9629	. . . . . . . . . . . . 13 |- 2 e. RR+
9	8	a1i 9	. . . . . . . . . . . 12 |- ( ph -> 2 e. RR+ )
10	1, 9	rpmulcld 9684	. . . . . . . . . . 11 |- ( ph -> ( C x. 2 ) e. RR+ )
11	10, 4	rpdivcld 9685	. . . . . . . . . 10 |- ( ph -> ( ( C x. 2 ) / X ) e. RR+ )
12		cvg1nlemcxze.z	. . . . . . . . . . 11 |- ( ph -> Z e. NN )
13	12	nnrpd 9665	. . . . . . . . . 10 |- ( ph -> Z e. RR+ )
14	11, 13	rpdivcld 9685	. . . . . . . . 9 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) e. RR+ )
15	14	rpred 9667	. . . . . . . 8 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) e. RR )
16		cvg1nlemcxze.a	. . . . . . . . . 10 |- ( ph -> A e. NN )
17	16	nnred 8905	. . . . . . . . 9 |- ( ph -> A e. RR )
18	15, 17	readdcld 7961	. . . . . . . 8 |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) + A ) e. RR )
19		cvg1nlemcxze.e	. . . . . . . . 9 |- ( ph -> E e. NN )
20	19	nnred 8905	. . . . . . . 8 |- ( ph -> E e. RR )
21	16	nnrpd 9665	. . . . . . . . 9 |- ( ph -> A e. RR+ )
22	15, 21	ltaddrpd 9701	. . . . . . . 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 8057	. . . . . . 7 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) < E )
25	11	rpred 9667	. . . . . . . 8 |- ( ph -> ( ( C x. 2 ) / X ) e. RR )
26	25, 20, 13	ltdivmul2d 9720	. . . . . . 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 4022	. . . . 5 |- ( ph -> ( ( C / X ) x. 2 ) < ( E x. Z ) )
29	1	rpred 9667	. . . . . . 7 |- ( ph -> C e. RR )
30	29, 4	rerpdivcld 9699	. . . . . 6 |- ( ph -> ( C / X ) e. RR )
31	19, 12	nnmulcld 8941	. . . . . . 7 |- ( ph -> ( E x. Z ) e. NN )
32	31	nnred 8905	. . . . . 6 |- ( ph -> ( E x. Z ) e. RR )
33	30, 32, 9	ltmuldivd 9715	. . . . 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 9736	. . . 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 8906	. . . 4 |- ( ph -> ( E x. Z ) e. CC )
38	37, 5	mulcomd 7953	. . 3 |- ( ph -> ( ( E x. Z ) x. X ) = ( X x. ( E x. Z ) ) )
39	36, 38	breqtrd 4024	. 2 |- ( ph -> ( C x. 2 ) < ( X x. ( E x. Z ) ) )
40	4	rpred 9667	. . 3 |- ( ph -> X e. RR )
41	31	nnrpd 9665	. . 3 |- ( ph -> ( E x. Z ) e. RR+ )
42	29, 9, 40, 41	lt2mul2divd 9736	. 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 2146   class class class wbr 3998  (class class class)co 5865   + caddc 7789   x. cmul 7791   < clt 7966   / cdiv 8602   NNcn 8892   2c2 8943   RR+crp 9624

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-rp 9625

This theorem is referenced by:  cvg1nlemres  10962