Theorem qbtwnrelemcalc 10212

Description: Lemma for qbtwnre 10213. 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 8948	. . . . 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 8891	. . . . . . 7 |- ( ph -> N e. RR )
6	2, 5	remulcld 7950	. . . . . 6 |- ( ph -> ( 2 x. N ) e. RR )
7	3, 6	remulcld 7950	. . . . 5 |- ( ph -> ( B x. ( 2 x. N ) ) e. RR )
8		qbtwnrelemcalc.a	. . . . . 6 |- ( ph -> A e. RR )
9	8, 6	remulcld 7950	. . . . 5 |- ( ph -> ( A x. ( 2 x. N ) ) e. RR )
10	7, 9	resubcld 8300	. . . 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 9334	. . . . 5 |- ( ph -> M e. RR )
13	7, 12	resubcld 8300	. . . 4 |- ( ph -> ( ( B x. ( 2 x. N ) ) - M ) e. RR )
14		2t1e2 9031	. . . . . . . . 9 |- ( 2 x. 1 ) = 2
15	14	oveq1i 5863	. . . . . . . 8 |- ( ( 2 x. 1 ) / ( 2 x. N ) ) = ( 2 / ( 2 x. N ) )
16		1cnd 7936	. . . . . . . . 9 |- ( ph -> 1 e. CC )
17	5	recnd 7948	. . . . . . . . 9 |- ( ph -> N e. CC )
18	2	recnd 7948	. . . . . . . . 9 |- ( ph -> 2 e. CC )
19	4	nnap0d 8924	. . . . . . . . 9 |- ( ph -> N # 0 )
20		2ap0 8971	. . . . . . . . . 10 |- 2 # 0
21	20	a1i 9	. . . . . . . . 9 |- ( ph -> 2 # 0 )
22	16, 17, 18, 19, 21	divcanap5d 8734	. . . . . . . 8 |- ( ph -> ( ( 2 x. 1 ) / ( 2 x. N ) ) = ( 1 / N ) )
23	15, 22	eqtr3id 2217	. . . . . . 7 |- ( ph -> ( 2 / ( 2 x. N ) ) = ( 1 / N ) )
24		qbtwnrelemcalc.1n	. . . . . . 7 |- ( ph -> ( 1 / N ) < ( B - A ) )
25	23, 24	eqbrtrd 4011	. . . . . 6 |- ( ph -> ( 2 / ( 2 x. N ) ) < ( B - A ) )
26	3, 8	resubcld 8300	. . . . . . 7 |- ( ph -> ( B - A ) e. RR )
27		2rp 9615	. . . . . . . . 9 |- 2 e. RR+
28	27	a1i 9	. . . . . . . 8 |- ( ph -> 2 e. RR+ )
29	4	nnrpd 9651	. . . . . . . 8 |- ( ph -> N e. RR+ )
30	28, 29	rpmulcld 9670	. . . . . . 7 |- ( ph -> ( 2 x. N ) e. RR+ )
31	2, 26, 30	ltdivmul2d 9706	. . . . . 6 |- ( ph -> ( ( 2 / ( 2 x. N ) ) < ( B - A ) <-> 2 < ( ( B - A ) x. ( 2 x. N ) ) ) )
32	25, 31	mpbid 146	. . . . 5 |- ( ph -> 2 < ( ( B - A ) x. ( 2 x. N ) ) )
33	3	recnd 7948	. . . . . 6 |- ( ph -> B e. CC )
34	8	recnd 7948	. . . . . 6 |- ( ph -> A e. CC )
35	18, 17	mulcld 7940	. . . . . 6 |- ( ph -> ( 2 x. N ) e. CC )
36	33, 34, 35	subdird 8334	. . . . 5 |- ( ph -> ( ( B - A ) x. ( 2 x. N ) ) = ( ( B x. ( 2 x. N ) ) - ( A x. ( 2 x. N ) ) ) )
37	32, 36	breqtrd 4015	. . . 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 8477	. . . 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 8045	. . 3 |- ( ph -> 2 < ( ( B x. ( 2 x. N ) ) - M ) )
41	12, 2, 7	ltaddsub2d 8465	. . 3 |- ( ph -> ( ( M + 2 ) < ( B x. ( 2 x. N ) ) <-> 2 < ( ( B x. ( 2 x. N ) ) - M ) ) )
42	40, 41	mpbird 166	. 2 |- ( ph -> ( M + 2 ) < ( B x. ( 2 x. N ) ) )
43	12, 2	readdcld 7949	. . 3 |- ( ph -> ( M + 2 ) e. RR )
44	43, 3, 30	ltdivmul2d 9706	. 2 |- ( ph -> ( ( ( M + 2 ) / ( 2 x. N ) ) < B <-> ( M + 2 ) < ( B x. ( 2 x. N ) ) ) )
45	42, 44	mpbird 166	1 |- ( ph -> ( ( M + 2 ) / ( 2 x. N ) ) < B )

Syntax hints:   -> wi 4   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   < clt 7954   - cmin 8090   # cap 8500   / cdiv 8589   NNcn 8878   2c2 8929   ZZcz 9212   RR+crp 9610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-z 9213  df-rp 9611

This theorem is referenced by:  qbtwnre  10213