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Mirrors > Home > ILE Home > Th. List > qbtwnrelemcalc | Unicode version |
Description: Lemma for qbtwnre 10282. Calculations involved in showing the
constructed
rational number is less than ![]() |
Ref | Expression |
---|---|
qbtwnrelemcalc.m |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
qbtwnrelemcalc.n |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
qbtwnrelemcalc.a |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
qbtwnrelemcalc.b |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
qbtwnrelemcalc.lt |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
qbtwnrelemcalc.1n |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
qbtwnrelemcalc |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 9014 |
. . . . 5
![]() ![]() ![]() ![]() | |
2 | 1 | a1i 9 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | qbtwnrelemcalc.b |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | qbtwnrelemcalc.n |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | nnred 8957 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 2, 5 | remulcld 8013 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 3, 6 | remulcld 8013 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | qbtwnrelemcalc.a |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8, 6 | remulcld 8013 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 7, 9 | resubcld 8363 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | qbtwnrelemcalc.m |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 11 | zred 9400 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 7, 12 | resubcld 8363 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 2t1e2 9097 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 14 | oveq1i 5902 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 1cnd 7998 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 5 | recnd 8011 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 2 | recnd 8011 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 4 | nnap0d 8990 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 2ap0 9037 |
. . . . . . . . . 10
![]() ![]() ![]() | |
21 | 20 | a1i 9 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 16, 17, 18, 19, 21 | divcanap5d 8799 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 15, 22 | eqtr3id 2236 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | qbtwnrelemcalc.1n |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 23, 24 | eqbrtrd 4040 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 3, 8 | resubcld 8363 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 2rp 9683 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
28 | 27 | a1i 9 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 4 | nnrpd 9719 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 28, 29 | rpmulcld 9738 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 2, 26, 30 | ltdivmul2d 9774 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 25, 31 | mpbid 147 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 3 | recnd 8011 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 8 | recnd 8011 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 18, 17 | mulcld 8003 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 33, 34, 35 | subdird 8397 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 32, 36 | breqtrd 4044 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | qbtwnrelemcalc.lt |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
39 | 12, 9, 7, 38 | ltsub2dd 8540 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 2, 10, 13, 37, 39 | lttrd 8108 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 12, 2, 7 | ltaddsub2d 8528 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 40, 41 | mpbird 167 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 12, 2 | readdcld 8012 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 43, 3, 30 | ltdivmul2d 9774 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 42, 44 | mpbird 167 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-mulrcl 7935 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-mulass 7939 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-1rid 7943 ax-0id 7944 ax-rnegex 7945 ax-precex 7946 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-apti 7951 ax-pre-ltadd 7952 ax-pre-mulgt0 7953 ax-pre-mulext 7954 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5234 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 df-sub 8155 df-neg 8156 df-reap 8557 df-ap 8564 df-div 8655 df-inn 8945 df-2 9003 df-z 9279 df-rp 9679 |
This theorem is referenced by: qbtwnre 10282 |
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