Theorem cvg1nlemcxze 10946

Description: Lemma for cvg1n 10950. 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 9655	. . . . . . 7 |- ( ph -> C e. CC )
3		2cnd 8951	. . . . . . 7 |- ( ph -> 2 e. CC )
4		cvg1nlemcxze.x	. . . . . . . 8 |- ( ph -> X e. RR+ )
5	4	rpcnd 9655	. . . . . . 7 |- ( ph -> X e. CC )
6	4	rpap0d 9659	. . . . . . 7 |- ( ph -> X # 0 )
7	2, 3, 5, 6	div23apd 8745	. . . . . 6 |- ( ph -> ( ( C x. 2 ) / X ) = ( ( C / X ) x. 2 ) )
8		2rp 9615	. . . . . . . . . . . . 13 |- 2 e. RR+
9	8	a1i 9	. . . . . . . . . . . 12 |- ( ph -> 2 e. RR+ )
10	1, 9	rpmulcld 9670	. . . . . . . . . . 11 |- ( ph -> ( C x. 2 ) e. RR+ )
11	10, 4	rpdivcld 9671	. . . . . . . . . 10 |- ( ph -> ( ( C x. 2 ) / X ) e. RR+ )
12		cvg1nlemcxze.z	. . . . . . . . . . 11 |- ( ph -> Z e. NN )
13	12	nnrpd 9651	. . . . . . . . . 10 |- ( ph -> Z e. RR+ )
14	11, 13	rpdivcld 9671	. . . . . . . . 9 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) e. RR+ )
15	14	rpred 9653	. . . . . . . 8 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) e. RR )
16		cvg1nlemcxze.a	. . . . . . . . . 10 |- ( ph -> A e. NN )
17	16	nnred 8891	. . . . . . . . 9 |- ( ph -> A e. RR )
18	15, 17	readdcld 7949	. . . . . . . 8 |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) + A ) e. RR )
19		cvg1nlemcxze.e	. . . . . . . . 9 |- ( ph -> E e. NN )
20	19	nnred 8891	. . . . . . . 8 |- ( ph -> E e. RR )
21	16	nnrpd 9651	. . . . . . . . 9 |- ( ph -> A e. RR+ )
22	15, 21	ltaddrpd 9687	. . . . . . . 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 8045	. . . . . . 7 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) < E )
25	11	rpred 9653	. . . . . . . 8 |- ( ph -> ( ( C x. 2 ) / X ) e. RR )
26	25, 20, 13	ltdivmul2d 9706	. . . . . . 7 |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) < E <-> ( ( C x. 2 ) / X ) < ( E x. Z ) ) )
27	24, 26	mpbid 146	. . . . . 6 |- ( ph -> ( ( C x. 2 ) / X ) < ( E x. Z ) )
28	7, 27	eqbrtrrd 4013	. . . . 5 |- ( ph -> ( ( C / X ) x. 2 ) < ( E x. Z ) )
29	1	rpred 9653	. . . . . . 7 |- ( ph -> C e. RR )
30	29, 4	rerpdivcld 9685	. . . . . 6 |- ( ph -> ( C / X ) e. RR )
31	19, 12	nnmulcld 8927	. . . . . . 7 |- ( ph -> ( E x. Z ) e. NN )
32	31	nnred 8891	. . . . . 6 |- ( ph -> ( E x. Z ) e. RR )
33	30, 32, 9	ltmuldivd 9701	. . . . 5 |- ( ph -> ( ( ( C / X ) x. 2 ) < ( E x. Z ) <-> ( C / X ) < ( ( E x. Z ) / 2 ) ) )
34	28, 33	mpbid 146	. . . 4 |- ( ph -> ( C / X ) < ( ( E x. Z ) / 2 ) )
35	29, 9, 32, 4	lt2mul2divd 9722	. . . 4 |- ( ph -> ( ( C x. 2 ) < ( ( E x. Z ) x. X ) <-> ( C / X ) < ( ( E x. Z ) / 2 ) ) )
36	34, 35	mpbird 166	. . 3 |- ( ph -> ( C x. 2 ) < ( ( E x. Z ) x. X ) )
37	31	nncnd 8892	. . . 4 |- ( ph -> ( E x. Z ) e. CC )
38	37, 5	mulcomd 7941	. . 3 |- ( ph -> ( ( E x. Z ) x. X ) = ( X x. ( E x. Z ) ) )
39	36, 38	breqtrd 4015	. 2 |- ( ph -> ( C x. 2 ) < ( X x. ( E x. Z ) ) )
40	4	rpred 9653	. . 3 |- ( ph -> X e. RR )
41	31	nnrpd 9651	. . 3 |- ( ph -> ( E x. Z ) e. RR+ )
42	29, 9, 40, 41	lt2mul2divd 9722	. 2 |- ( ph -> ( ( C x. 2 ) < ( X x. ( E x. Z ) ) <-> ( C / ( E x. Z ) ) < ( X / 2 ) ) )
43	39, 42	mpbid 146	1 |- ( ph -> ( C / ( E x. Z ) ) < ( X / 2 ) )

Syntax hints:   -> wi 4   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   + caddc 7777   x. cmul 7779   < clt 7954   / cdiv 8589   NNcn 8878   2c2 8929   RR+crp 9610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-rp 9611

This theorem is referenced by:  cvg1nlemres  10949