Theorem qbtwnrelemcalc 9632

Description: Lemma for qbtwnre 9633. 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 8463	. . . . 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 8407	. . . . . . 7 |- ( ph -> N e. RR )
6	2, 5	remulcld 7497	. . . . . 6 |- ( ph -> ( 2 x. N ) e. RR )
7	3, 6	remulcld 7497	. . . . 5 |- ( ph -> ( B x. ( 2 x. N ) ) e. RR )
8		qbtwnrelemcalc.a	. . . . . 6 |- ( ph -> A e. RR )
9	8, 6	remulcld 7497	. . . . 5 |- ( ph -> ( A x. ( 2 x. N ) ) e. RR )
10	7, 9	resubcld 7838	. . . 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 8838	. . . . 5 |- ( ph -> M e. RR )
13	7, 12	resubcld 7838	. . . 4 |- ( ph -> ( ( B x. ( 2 x. N ) ) - M ) e. RR )
14		2t1e2 8539	. . . . . . . . 9 |- ( 2 x. 1 ) = 2
15	14	oveq1i 5644	. . . . . . . 8 |- ( ( 2 x. 1 ) / ( 2 x. N ) ) = ( 2 / ( 2 x. N ) )
16		1cnd 7483	. . . . . . . . 9 |- ( ph -> 1 e. CC )
17	5	recnd 7495	. . . . . . . . 9 |- ( ph -> N e. CC )
18	2	recnd 7495	. . . . . . . . 9 |- ( ph -> 2 e. CC )
19	4	nnap0d 8439	. . . . . . . . 9 |- ( ph -> N # 0 )
20		2ap0 8486	. . . . . . . . . 10 |- 2 # 0
21	20	a1i 9	. . . . . . . . 9 |- ( ph -> 2 # 0 )
22	16, 17, 18, 19, 21	divcanap5d 8256	. . . . . . . 8 |- ( ph -> ( ( 2 x. 1 ) / ( 2 x. N ) ) = ( 1 / N ) )
23	15, 22	syl5eqr 2134	. . . . . . 7 |- ( ph -> ( 2 / ( 2 x. N ) ) = ( 1 / N ) )
24		qbtwnrelemcalc.1n	. . . . . . 7 |- ( ph -> ( 1 / N ) < ( B - A ) )
25	23, 24	eqbrtrd 3857	. . . . . 6 |- ( ph -> ( 2 / ( 2 x. N ) ) < ( B - A ) )
26	3, 8	resubcld 7838	. . . . . . 7 |- ( ph -> ( B - A ) e. RR )
27		2rp 9108	. . . . . . . . 9 |- 2 e. RR+
28	27	a1i 9	. . . . . . . 8 |- ( ph -> 2 e. RR+ )
29	4	nnrpd 9141	. . . . . . . 8 |- ( ph -> N e. RR+ )
30	28, 29	rpmulcld 9159	. . . . . . 7 |- ( ph -> ( 2 x. N ) e. RR+ )
31	2, 26, 30	ltdivmul2d 9195	. . . . . 6 |- ( ph -> ( ( 2 / ( 2 x. N ) ) < ( B - A ) <-> 2 < ( ( B - A ) x. ( 2 x. N ) ) ) )
32	25, 31	mpbid 145	. . . . 5 |- ( ph -> 2 < ( ( B - A ) x. ( 2 x. N ) ) )
33	3	recnd 7495	. . . . . 6 |- ( ph -> B e. CC )
34	8	recnd 7495	. . . . . 6 |- ( ph -> A e. CC )
35	18, 17	mulcld 7487	. . . . . 6 |- ( ph -> ( 2 x. N ) e. CC )
36	33, 34, 35	subdird 7872	. . . . 5 |- ( ph -> ( ( B - A ) x. ( 2 x. N ) ) = ( ( B x. ( 2 x. N ) ) - ( A x. ( 2 x. N ) ) ) )
37	32, 36	breqtrd 3861	. . . 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 8011	. . . 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 7588	. . 3 |- ( ph -> 2 < ( ( B x. ( 2 x. N ) ) - M ) )
41	12, 2, 7	ltaddsub2d 7999	. . 3 |- ( ph -> ( ( M + 2 ) < ( B x. ( 2 x. N ) ) <-> 2 < ( ( B x. ( 2 x. N ) ) - M ) ) )
42	40, 41	mpbird 165	. 2 |- ( ph -> ( M + 2 ) < ( B x. ( 2 x. N ) ) )
43	12, 2	readdcld 7496	. . 3 |- ( ph -> ( M + 2 ) e. RR )
44	43, 3, 30	ltdivmul2d 9195	. 2 |- ( ph -> ( ( ( M + 2 ) / ( 2 x. N ) ) < B <-> ( M + 2 ) < ( B x. ( 2 x. N ) ) ) )
45	42, 44	mpbird 165	1 |- ( ph -> ( ( M + 2 ) / ( 2 x. N ) ) < B )

Syntax hints:   -> wi 4   e. wcel 1438   class class class wbr 3837  (class class class)co 5634   RRcr 7328   0cc0 7329   1c1 7330   + caddc 7332   x. cmul 7334   < clt 7501   - cmin 7632   # cap 8034   / cdiv 8113   NNcn 8394   2c2 8444   ZZcz 8720   RR+crp 9103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-z 8721  df-rp 9104

This theorem is referenced by:  qbtwnre  9633