Theorem qbtwnrelemcalc 10487

Description: Lemma for qbtwnre 10488. 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 9191	. . . . 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 9134	. . . . . . 7 |- ( ph -> N e. RR )
6	2, 5	remulcld 8188	. . . . . 6 |- ( ph -> ( 2 x. N ) e. RR )
7	3, 6	remulcld 8188	. . . . 5 |- ( ph -> ( B x. ( 2 x. N ) ) e. RR )
8		qbtwnrelemcalc.a	. . . . . 6 |- ( ph -> A e. RR )
9	8, 6	remulcld 8188	. . . . 5 |- ( ph -> ( A x. ( 2 x. N ) ) e. RR )
10	7, 9	resubcld 8538	. . . 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 9580	. . . . 5 |- ( ph -> M e. RR )
13	7, 12	resubcld 8538	. . . 4 |- ( ph -> ( ( B x. ( 2 x. N ) ) - M ) e. RR )
14		2t1e2 9275	. . . . . . . . 9 |- ( 2 x. 1 ) = 2
15	14	oveq1i 6017	. . . . . . . 8 |- ( ( 2 x. 1 ) / ( 2 x. N ) ) = ( 2 / ( 2 x. N ) )
16		1cnd 8173	. . . . . . . . 9 |- ( ph -> 1 e. CC )
17	5	recnd 8186	. . . . . . . . 9 |- ( ph -> N e. CC )
18	2	recnd 8186	. . . . . . . . 9 |- ( ph -> 2 e. CC )
19	4	nnap0d 9167	. . . . . . . . 9 |- ( ph -> N # 0 )
20		2ap0 9214	. . . . . . . . . 10 |- 2 # 0
21	20	a1i 9	. . . . . . . . 9 |- ( ph -> 2 # 0 )
22	16, 17, 18, 19, 21	divcanap5d 8975	. . . . . . . 8 |- ( ph -> ( ( 2 x. 1 ) / ( 2 x. N ) ) = ( 1 / N ) )
23	15, 22	eqtr3id 2276	. . . . . . 7 |- ( ph -> ( 2 / ( 2 x. N ) ) = ( 1 / N ) )
24		qbtwnrelemcalc.1n	. . . . . . 7 |- ( ph -> ( 1 / N ) < ( B - A ) )
25	23, 24	eqbrtrd 4105	. . . . . 6 |- ( ph -> ( 2 / ( 2 x. N ) ) < ( B - A ) )
26	3, 8	resubcld 8538	. . . . . . 7 |- ( ph -> ( B - A ) e. RR )
27		2rp 9866	. . . . . . . . 9 |- 2 e. RR+
28	27	a1i 9	. . . . . . . 8 |- ( ph -> 2 e. RR+ )
29	4	nnrpd 9902	. . . . . . . 8 |- ( ph -> N e. RR+ )
30	28, 29	rpmulcld 9921	. . . . . . 7 |- ( ph -> ( 2 x. N ) e. RR+ )
31	2, 26, 30	ltdivmul2d 9957	. . . . . 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 8186	. . . . . 6 |- ( ph -> B e. CC )
34	8	recnd 8186	. . . . . 6 |- ( ph -> A e. CC )
35	18, 17	mulcld 8178	. . . . . 6 |- ( ph -> ( 2 x. N ) e. CC )
36	33, 34, 35	subdird 8572	. . . . 5 |- ( ph -> ( ( B - A ) x. ( 2 x. N ) ) = ( ( B x. ( 2 x. N ) ) - ( A x. ( 2 x. N ) ) ) )
37	32, 36	breqtrd 4109	. . . 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 8716	. . . 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 8283	. . 3 |- ( ph -> 2 < ( ( B x. ( 2 x. N ) ) - M ) )
41	12, 2, 7	ltaddsub2d 8704	. . 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 8187	. . 3 |- ( ph -> ( M + 2 ) e. RR )
44	43, 3, 30	ltdivmul2d 9957	. 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 2200   class class class wbr 4083  (class class class)co 6007   RRcr 8009   0cc0 8010   1c1 8011   + caddc 8013   x. cmul 8015   < clt 8192   - cmin 8328   # cap 8739   / cdiv 8830   NNcn 9121   2c2 9172   ZZcz 9457   RR+crp 9861

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-z 9458  df-rp 9862

This theorem is referenced by:  qbtwnre  10488