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Mirrors > Home > ILE Home > Th. List > numltc | Unicode version |
Description: Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numlt.1 |
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numlt.2 |
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numlt.3 |
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numltc.3 |
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numltc.4 |
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numltc.5 |
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numltc.6 |
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Ref | Expression |
---|---|
numltc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numlt.1 |
. . . . 5
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2 | numlt.2 |
. . . . 5
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3 | numltc.3 |
. . . . 5
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4 | numltc.5 |
. . . . 5
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5 | 1, 2, 3, 1, 4 | numlt 9439 |
. . . 4
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6 | 1 | nnrei 8959 |
. . . . . . 7
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7 | 6 | recni 8000 |
. . . . . 6
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8 | 2 | nn0rei 9218 |
. . . . . . 7
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9 | 8 | recni 8000 |
. . . . . 6
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10 | ax-1cn 7935 |
. . . . . 6
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11 | 7, 9, 10 | adddii 7998 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 7 | mulid1i 7990 |
. . . . . 6
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13 | 12 | oveq2i 5908 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 11, 13 | eqtri 2210 |
. . . 4
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15 | 5, 14 | breqtrri 4045 |
. . 3
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16 | numltc.6 |
. . . . 5
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17 | numlt.3 |
. . . . . 6
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18 | nn0ltp1le 9346 |
. . . . . 6
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19 | 2, 17, 18 | mp2an 426 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 16, 19 | mpbi 145 |
. . . 4
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21 | 1 | nngt0i 8980 |
. . . . 5
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22 | peano2re 8124 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 8, 22 | ax-mp 5 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 17 | nn0rei 9218 |
. . . . . 6
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25 | 23, 24, 6 | lemul2i 8913 |
. . . . 5
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26 | 21, 25 | ax-mp 5 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 20, 26 | mpbi 145 |
. . 3
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28 | 6, 8 | remulcli 8002 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 3 | nn0rei 9218 |
. . . . 5
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30 | 28, 29 | readdcli 8001 |
. . . 4
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31 | 6, 23 | remulcli 8002 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 6, 24 | remulcli 8002 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 30, 31, 32 | ltletri 8095 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 15, 27, 33 | mp2an 426 |
. 2
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35 | numltc.4 |
. . 3
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36 | 32, 35 | nn0addge1i 9255 |
. 2
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37 | 35 | nn0rei 9218 |
. . . 4
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38 | 32, 37 | readdcli 8001 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 30, 32, 38 | ltletri 8095 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 34, 36, 39 | mp2an 426 |
1
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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 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 |
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-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 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-inn 8951 df-n0 9208 df-z 9285 |
This theorem is referenced by: decltc 9443 numlti 9451 |
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