Theorem cvg1nlemcxze 11004

Description: Lemma for cvg1n 11008. 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 9711	. . . . . . 7 |- ( ph -> C e. CC )
3		2cnd 9005	. . . . . . 7 |- ( ph -> 2 e. CC )
4		cvg1nlemcxze.x	. . . . . . . 8 |- ( ph -> X e. RR+ )
5	4	rpcnd 9711	. . . . . . 7 |- ( ph -> X e. CC )
6	4	rpap0d 9715	. . . . . . 7 |- ( ph -> X # 0 )
7	2, 3, 5, 6	div23apd 8798	. . . . . 6 |- ( ph -> ( ( C x. 2 ) / X ) = ( ( C / X ) x. 2 ) )
8		2rp 9671	. . . . . . . . . . . . 13 |- 2 e. RR+
9	8	a1i 9	. . . . . . . . . . . 12 |- ( ph -> 2 e. RR+ )
10	1, 9	rpmulcld 9726	. . . . . . . . . . 11 |- ( ph -> ( C x. 2 ) e. RR+ )
11	10, 4	rpdivcld 9727	. . . . . . . . . 10 |- ( ph -> ( ( C x. 2 ) / X ) e. RR+ )
12		cvg1nlemcxze.z	. . . . . . . . . . 11 |- ( ph -> Z e. NN )
13	12	nnrpd 9707	. . . . . . . . . 10 |- ( ph -> Z e. RR+ )
14	11, 13	rpdivcld 9727	. . . . . . . . 9 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) e. RR+ )
15	14	rpred 9709	. . . . . . . 8 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) e. RR )
16		cvg1nlemcxze.a	. . . . . . . . . 10 |- ( ph -> A e. NN )
17	16	nnred 8945	. . . . . . . . 9 |- ( ph -> A e. RR )
18	15, 17	readdcld 8000	. . . . . . . 8 |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) + A ) e. RR )
19		cvg1nlemcxze.e	. . . . . . . . 9 |- ( ph -> E e. NN )
20	19	nnred 8945	. . . . . . . 8 |- ( ph -> E e. RR )
21	16	nnrpd 9707	. . . . . . . . 9 |- ( ph -> A e. RR+ )
22	15, 21	ltaddrpd 9743	. . . . . . . 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 8096	. . . . . . 7 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) < E )
25	11	rpred 9709	. . . . . . . 8 |- ( ph -> ( ( C x. 2 ) / X ) e. RR )
26	25, 20, 13	ltdivmul2d 9762	. . . . . . 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 4039	. . . . 5 |- ( ph -> ( ( C / X ) x. 2 ) < ( E x. Z ) )
29	1	rpred 9709	. . . . . . 7 |- ( ph -> C e. RR )
30	29, 4	rerpdivcld 9741	. . . . . 6 |- ( ph -> ( C / X ) e. RR )
31	19, 12	nnmulcld 8981	. . . . . . 7 |- ( ph -> ( E x. Z ) e. NN )
32	31	nnred 8945	. . . . . 6 |- ( ph -> ( E x. Z ) e. RR )
33	30, 32, 9	ltmuldivd 9757	. . . . 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 9778	. . . 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 8946	. . . 4 |- ( ph -> ( E x. Z ) e. CC )
38	37, 5	mulcomd 7992	. . 3 |- ( ph -> ( ( E x. Z ) x. X ) = ( X x. ( E x. Z ) ) )
39	36, 38	breqtrd 4041	. 2 |- ( ph -> ( C x. 2 ) < ( X x. ( E x. Z ) ) )
40	4	rpred 9709	. . 3 |- ( ph -> X e. RR )
41	31	nnrpd 9707	. . 3 |- ( ph -> ( E x. Z ) e. RR+ )
42	29, 9, 40, 41	lt2mul2divd 9778	. 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 2158   class class class wbr 4015  (class class class)co 5888   + caddc 7827   x. cmul 7829   < clt 8005   / cdiv 8642   NNcn 8932   2c2 8983   RR+crp 9666

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-rp 9667

This theorem is referenced by:  cvg1nlemres  11007