Theorem qbtwnrelemcalc 10324

Description: Lemma for qbtwnre 10325. Calculations involved in showing the constructed rational number is less than B. (Contributed by Jim Kingdon, 14-Oct-2021.)

Hypotheses

Ref	Expression
qbtwnrelemcalc.m	|- ( ph -> M e. ZZ )
qbtwnrelemcalc.n	|- ( ph -> N e. NN )
qbtwnrelemcalc.a	|- ( ph -> A e. RR )
qbtwnrelemcalc.b	|- ( ph -> B e. RR )
qbtwnrelemcalc.lt	|- ( ph -> M < ( A x. ( 2 x. N ) ) )
qbtwnrelemcalc.1n	|- ( ph -> ( 1 / N ) < ( B - A ) )

Assertion

Ref	Expression
qbtwnrelemcalc	|- ( ph -> ( ( M + 2 ) / ( 2 x. N ) ) < B )

Proof of Theorem qbtwnrelemcalc

Step	Hyp	Ref	Expression
1		2re 9052	. . . . 5 |- 2 e. RR
2	1	a1i 9	. . . 4 |- ( ph -> 2 e. RR )
3		qbtwnrelemcalc.b	. . . . . 6 |- ( ph -> B e. RR )
4		qbtwnrelemcalc.n	. . . . . . . 8 |- ( ph -> N e. NN )
5	4	nnred 8995	. . . . . . 7 |- ( ph -> N e. RR )
6	2, 5	remulcld 8050	. . . . . 6 |- ( ph -> ( 2 x. N ) e. RR )
7	3, 6	remulcld 8050	. . . . 5 |- ( ph -> ( B x. ( 2 x. N ) ) e. RR )
8		qbtwnrelemcalc.a	. . . . . 6 |- ( ph -> A e. RR )
9	8, 6	remulcld 8050	. . . . 5 |- ( ph -> ( A x. ( 2 x. N ) ) e. RR )
10	7, 9	resubcld 8400	. . . 4 |- ( ph -> ( ( B x. ( 2 x. N ) ) - ( A x. ( 2 x. N ) ) ) e. RR )
11		qbtwnrelemcalc.m	. . . . . 6 |- ( ph -> M e. ZZ )
12	11	zred 9439	. . . . 5 |- ( ph -> M e. RR )
13	7, 12	resubcld 8400	. . . 4 |- ( ph -> ( ( B x. ( 2 x. N ) ) - M ) e. RR )
14		2t1e2 9135	. . . . . . . . 9 |- ( 2 x. 1 ) = 2
15	14	oveq1i 5928	. . . . . . . 8 |- ( ( 2 x. 1 ) / ( 2 x. N ) ) = ( 2 / ( 2 x. N ) )
16		1cnd 8035	. . . . . . . . 9 |- ( ph -> 1 e. CC )
17	5	recnd 8048	. . . . . . . . 9 |- ( ph -> N e. CC )
18	2	recnd 8048	. . . . . . . . 9 |- ( ph -> 2 e. CC )
19	4	nnap0d 9028	. . . . . . . . 9 |- ( ph -> N # 0 )
20		2ap0 9075	. . . . . . . . . 10 |- 2 # 0
21	20	a1i 9	. . . . . . . . 9 |- ( ph -> 2 # 0 )
22	16, 17, 18, 19, 21	divcanap5d 8836	. . . . . . . 8 |- ( ph -> ( ( 2 x. 1 ) / ( 2 x. N ) ) = ( 1 / N ) )
23	15, 22	eqtr3id 2240	. . . . . . 7 |- ( ph -> ( 2 / ( 2 x. N ) ) = ( 1 / N ) )
24		qbtwnrelemcalc.1n	. . . . . . 7 |- ( ph -> ( 1 / N ) < ( B - A ) )
25	23, 24	eqbrtrd 4051	. . . . . 6 |- ( ph -> ( 2 / ( 2 x. N ) ) < ( B - A ) )
26	3, 8	resubcld 8400	. . . . . . 7 |- ( ph -> ( B - A ) e. RR )
27		2rp 9724	. . . . . . . . 9 |- 2 e. RR+
28	27	a1i 9	. . . . . . . 8 |- ( ph -> 2 e. RR+ )
29	4	nnrpd 9760	. . . . . . . 8 |- ( ph -> N e. RR+ )
30	28, 29	rpmulcld 9779	. . . . . . 7 |- ( ph -> ( 2 x. N ) e. RR+ )
31	2, 26, 30	ltdivmul2d 9815	. . . . . 6 |- ( ph -> ( ( 2 / ( 2 x. N ) ) < ( B - A ) <-> 2 < ( ( B - A ) x. ( 2 x. N ) ) ) )
32	25, 31	mpbid 147	. . . . 5 |- ( ph -> 2 < ( ( B - A ) x. ( 2 x. N ) ) )
33	3	recnd 8048	. . . . . 6 |- ( ph -> B e. CC )
34	8	recnd 8048	. . . . . 6 |- ( ph -> A e. CC )
35	18, 17	mulcld 8040	. . . . . 6 |- ( ph -> ( 2 x. N ) e. CC )
36	33, 34, 35	subdird 8434	. . . . 5 |- ( ph -> ( ( B - A ) x. ( 2 x. N ) ) = ( ( B x. ( 2 x. N ) ) - ( A x. ( 2 x. N ) ) ) )
37	32, 36	breqtrd 4055	. . . 4 |- ( ph -> 2 < ( ( B x. ( 2 x. N ) ) - ( A x. ( 2 x. N ) ) ) )
38		qbtwnrelemcalc.lt	. . . . 5 |- ( ph -> M < ( A x. ( 2 x. N ) ) )
39	12, 9, 7, 38	ltsub2dd 8577	. . . 4 |- ( ph -> ( ( B x. ( 2 x. N ) ) - ( A x. ( 2 x. N ) ) ) < ( ( B x. ( 2 x. N ) ) - M ) )
40	2, 10, 13, 37, 39	lttrd 8145	. . 3 |- ( ph -> 2 < ( ( B x. ( 2 x. N ) ) - M ) )
41	12, 2, 7	ltaddsub2d 8565	. . 3 |- ( ph -> ( ( M + 2 ) < ( B x. ( 2 x. N ) ) <-> 2 < ( ( B x. ( 2 x. N ) ) - M ) ) )
42	40, 41	mpbird 167	. 2 |- ( ph -> ( M + 2 ) < ( B x. ( 2 x. N ) ) )
43	12, 2	readdcld 8049	. . 3 |- ( ph -> ( M + 2 ) e. RR )
44	43, 3, 30	ltdivmul2d 9815	. 2 |- ( ph -> ( ( ( M + 2 ) / ( 2 x. N ) ) < B <-> ( M + 2 ) < ( B x. ( 2 x. N ) ) ) )
45	42, 44	mpbird 167	1 |- ( ph -> ( ( M + 2 ) / ( 2 x. N ) ) < B )

Syntax hints:   -> wi 4   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   RRcr 7871   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   < clt 8054   - cmin 8190   # cap 8600   / cdiv 8691   NNcn 8982   2c2 9033   ZZcz 9317   RR+crp 9719

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-z 9318  df-rp 9720

This theorem is referenced by:  qbtwnre  10325