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Mirrors > Home > ILE Home > Th. List > qbtwnrelemcalc | Unicode version |
Description: Lemma for qbtwnre 10034. Calculations involved in showing the constructed rational number is less than . (Contributed by Jim Kingdon, 14-Oct-2021.) |
Ref | Expression |
---|---|
qbtwnrelemcalc.m | |
qbtwnrelemcalc.n | |
qbtwnrelemcalc.a | |
qbtwnrelemcalc.b | |
qbtwnrelemcalc.lt | |
qbtwnrelemcalc.1n |
Ref | Expression |
---|---|
qbtwnrelemcalc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8790 | . . . . 5 | |
2 | 1 | a1i 9 | . . . 4 |
3 | qbtwnrelemcalc.b | . . . . . 6 | |
4 | qbtwnrelemcalc.n | . . . . . . . 8 | |
5 | 4 | nnred 8733 | . . . . . . 7 |
6 | 2, 5 | remulcld 7796 | . . . . . 6 |
7 | 3, 6 | remulcld 7796 | . . . . 5 |
8 | qbtwnrelemcalc.a | . . . . . 6 | |
9 | 8, 6 | remulcld 7796 | . . . . 5 |
10 | 7, 9 | resubcld 8143 | . . . 4 |
11 | qbtwnrelemcalc.m | . . . . . 6 | |
12 | 11 | zred 9173 | . . . . 5 |
13 | 7, 12 | resubcld 8143 | . . . 4 |
14 | 2t1e2 8873 | . . . . . . . . 9 | |
15 | 14 | oveq1i 5784 | . . . . . . . 8 |
16 | 1cnd 7782 | . . . . . . . . 9 | |
17 | 5 | recnd 7794 | . . . . . . . . 9 |
18 | 2 | recnd 7794 | . . . . . . . . 9 |
19 | 4 | nnap0d 8766 | . . . . . . . . 9 # |
20 | 2ap0 8813 | . . . . . . . . . 10 # | |
21 | 20 | a1i 9 | . . . . . . . . 9 # |
22 | 16, 17, 18, 19, 21 | divcanap5d 8577 | . . . . . . . 8 |
23 | 15, 22 | syl5eqr 2186 | . . . . . . 7 |
24 | qbtwnrelemcalc.1n | . . . . . . 7 | |
25 | 23, 24 | eqbrtrd 3950 | . . . . . 6 |
26 | 3, 8 | resubcld 8143 | . . . . . . 7 |
27 | 2rp 9446 | . . . . . . . . 9 | |
28 | 27 | a1i 9 | . . . . . . . 8 |
29 | 4 | nnrpd 9482 | . . . . . . . 8 |
30 | 28, 29 | rpmulcld 9500 | . . . . . . 7 |
31 | 2, 26, 30 | ltdivmul2d 9536 | . . . . . 6 |
32 | 25, 31 | mpbid 146 | . . . . 5 |
33 | 3 | recnd 7794 | . . . . . 6 |
34 | 8 | recnd 7794 | . . . . . 6 |
35 | 18, 17 | mulcld 7786 | . . . . . 6 |
36 | 33, 34, 35 | subdird 8177 | . . . . 5 |
37 | 32, 36 | breqtrd 3954 | . . . 4 |
38 | qbtwnrelemcalc.lt | . . . . 5 | |
39 | 12, 9, 7, 38 | ltsub2dd 8320 | . . . 4 |
40 | 2, 10, 13, 37, 39 | lttrd 7888 | . . 3 |
41 | 12, 2, 7 | ltaddsub2d 8308 | . . 3 |
42 | 40, 41 | mpbird 166 | . 2 |
43 | 12, 2 | readdcld 7795 | . . 3 |
44 | 43, 3, 30 | ltdivmul2d 9536 | . 2 |
45 | 42, 44 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7619 cc0 7620 c1 7621 caddc 7623 cmul 7625 clt 7800 cmin 7933 # cap 8343 cdiv 8432 cn 8720 c2 8771 cz 9054 crp 9441 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-z 9055 df-rp 9442 |
This theorem is referenced by: qbtwnre 10034 |
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