Theorem cvg1nlemcxze 11542

Description: Lemma for cvg1n 11546. 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 9932	. . . . . . 7 |- ( ph -> C e. CC )
3		2cnd 9215	. . . . . . 7 |- ( ph -> 2 e. CC )
4		cvg1nlemcxze.x	. . . . . . . 8 |- ( ph -> X e. RR+ )
5	4	rpcnd 9932	. . . . . . 7 |- ( ph -> X e. CC )
6	4	rpap0d 9936	. . . . . . 7 |- ( ph -> X # 0 )
7	2, 3, 5, 6	div23apd 9007	. . . . . 6 |- ( ph -> ( ( C x. 2 ) / X ) = ( ( C / X ) x. 2 ) )
8		2rp 9892	. . . . . . . . . . . . 13 |- 2 e. RR+
9	8	a1i 9	. . . . . . . . . . . 12 |- ( ph -> 2 e. RR+ )
10	1, 9	rpmulcld 9947	. . . . . . . . . . 11 |- ( ph -> ( C x. 2 ) e. RR+ )
11	10, 4	rpdivcld 9948	. . . . . . . . . 10 |- ( ph -> ( ( C x. 2 ) / X ) e. RR+ )
12		cvg1nlemcxze.z	. . . . . . . . . . 11 |- ( ph -> Z e. NN )
13	12	nnrpd 9928	. . . . . . . . . 10 |- ( ph -> Z e. RR+ )
14	11, 13	rpdivcld 9948	. . . . . . . . 9 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) e. RR+ )
15	14	rpred 9930	. . . . . . . 8 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) e. RR )
16		cvg1nlemcxze.a	. . . . . . . . . 10 |- ( ph -> A e. NN )
17	16	nnred 9155	. . . . . . . . 9 |- ( ph -> A e. RR )
18	15, 17	readdcld 8208	. . . . . . . 8 |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) + A ) e. RR )
19		cvg1nlemcxze.e	. . . . . . . . 9 |- ( ph -> E e. NN )
20	19	nnred 9155	. . . . . . . 8 |- ( ph -> E e. RR )
21	16	nnrpd 9928	. . . . . . . . 9 |- ( ph -> A e. RR+ )
22	15, 21	ltaddrpd 9964	. . . . . . . 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 8304	. . . . . . 7 |- ( ph -> ( ( ( C x. 2 ) / X ) / Z ) < E )
25	11	rpred 9930	. . . . . . . 8 |- ( ph -> ( ( C x. 2 ) / X ) e. RR )
26	25, 20, 13	ltdivmul2d 9983	. . . . . . 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 4112	. . . . 5 |- ( ph -> ( ( C / X ) x. 2 ) < ( E x. Z ) )
29	1	rpred 9930	. . . . . . 7 |- ( ph -> C e. RR )
30	29, 4	rerpdivcld 9962	. . . . . 6 |- ( ph -> ( C / X ) e. RR )
31	19, 12	nnmulcld 9191	. . . . . . 7 |- ( ph -> ( E x. Z ) e. NN )
32	31	nnred 9155	. . . . . 6 |- ( ph -> ( E x. Z ) e. RR )
33	30, 32, 9	ltmuldivd 9978	. . . . 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 9999	. . . 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 9156	. . . 4 |- ( ph -> ( E x. Z ) e. CC )
38	37, 5	mulcomd 8200	. . 3 |- ( ph -> ( ( E x. Z ) x. X ) = ( X x. ( E x. Z ) ) )
39	36, 38	breqtrd 4114	. 2 |- ( ph -> ( C x. 2 ) < ( X x. ( E x. Z ) ) )
40	4	rpred 9930	. . 3 |- ( ph -> X e. RR )
41	31	nnrpd 9928	. . 3 |- ( ph -> ( E x. Z ) e. RR+ )
42	29, 9, 40, 41	lt2mul2divd 9999	. 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 2202   class class class wbr 4088  (class class class)co 6017   + caddc 8034   x. cmul 8036   < clt 8213   / cdiv 8851   NNcn 9142   2c2 9193   RR+crp 9887

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-rp 9888

This theorem is referenced by:  cvg1nlemres  11545