Theorem qbtwnrelemcalc 10362

Description: Lemma for qbtwnre 10363. 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 9077	. . . . 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 9020	. . . . . . 7 |- ( ph -> N e. RR )
6	2, 5	remulcld 8074	. . . . . 6 |- ( ph -> ( 2 x. N ) e. RR )
7	3, 6	remulcld 8074	. . . . 5 |- ( ph -> ( B x. ( 2 x. N ) ) e. RR )
8		qbtwnrelemcalc.a	. . . . . 6 |- ( ph -> A e. RR )
9	8, 6	remulcld 8074	. . . . 5 |- ( ph -> ( A x. ( 2 x. N ) ) e. RR )
10	7, 9	resubcld 8424	. . . 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 9465	. . . . 5 |- ( ph -> M e. RR )
13	7, 12	resubcld 8424	. . . 4 |- ( ph -> ( ( B x. ( 2 x. N ) ) - M ) e. RR )
14		2t1e2 9161	. . . . . . . . 9 |- ( 2 x. 1 ) = 2
15	14	oveq1i 5935	. . . . . . . 8 |- ( ( 2 x. 1 ) / ( 2 x. N ) ) = ( 2 / ( 2 x. N ) )
16		1cnd 8059	. . . . . . . . 9 |- ( ph -> 1 e. CC )
17	5	recnd 8072	. . . . . . . . 9 |- ( ph -> N e. CC )
18	2	recnd 8072	. . . . . . . . 9 |- ( ph -> 2 e. CC )
19	4	nnap0d 9053	. . . . . . . . 9 |- ( ph -> N # 0 )
20		2ap0 9100	. . . . . . . . . 10 |- 2 # 0
21	20	a1i 9	. . . . . . . . 9 |- ( ph -> 2 # 0 )
22	16, 17, 18, 19, 21	divcanap5d 8861	. . . . . . . 8 |- ( ph -> ( ( 2 x. 1 ) / ( 2 x. N ) ) = ( 1 / N ) )
23	15, 22	eqtr3id 2243	. . . . . . 7 |- ( ph -> ( 2 / ( 2 x. N ) ) = ( 1 / N ) )
24		qbtwnrelemcalc.1n	. . . . . . 7 |- ( ph -> ( 1 / N ) < ( B - A ) )
25	23, 24	eqbrtrd 4056	. . . . . 6 |- ( ph -> ( 2 / ( 2 x. N ) ) < ( B - A ) )
26	3, 8	resubcld 8424	. . . . . . 7 |- ( ph -> ( B - A ) e. RR )
27		2rp 9750	. . . . . . . . 9 |- 2 e. RR+
28	27	a1i 9	. . . . . . . 8 |- ( ph -> 2 e. RR+ )
29	4	nnrpd 9786	. . . . . . . 8 |- ( ph -> N e. RR+ )
30	28, 29	rpmulcld 9805	. . . . . . 7 |- ( ph -> ( 2 x. N ) e. RR+ )
31	2, 26, 30	ltdivmul2d 9841	. . . . . 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 8072	. . . . . 6 |- ( ph -> B e. CC )
34	8	recnd 8072	. . . . . 6 |- ( ph -> A e. CC )
35	18, 17	mulcld 8064	. . . . . 6 |- ( ph -> ( 2 x. N ) e. CC )
36	33, 34, 35	subdird 8458	. . . . 5 |- ( ph -> ( ( B - A ) x. ( 2 x. N ) ) = ( ( B x. ( 2 x. N ) ) - ( A x. ( 2 x. N ) ) ) )
37	32, 36	breqtrd 4060	. . . 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 8602	. . . 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 8169	. . 3 |- ( ph -> 2 < ( ( B x. ( 2 x. N ) ) - M ) )
41	12, 2, 7	ltaddsub2d 8590	. . 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 8073	. . 3 |- ( ph -> ( M + 2 ) e. RR )
44	43, 3, 30	ltdivmul2d 9841	. 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 2167   class class class wbr 4034  (class class class)co 5925   RRcr 7895   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   < clt 8078   - cmin 8214   # cap 8625   / cdiv 8716   NNcn 9007   2c2 9058   ZZcz 9343   RR+crp 9745

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-z 9344  df-rp 9746

This theorem is referenced by:  qbtwnre  10363