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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 | |
numlt.2 | |
numlt.3 | |
numltc.3 | |
numltc.4 | |
numltc.5 | |
numltc.6 |
Ref | Expression |
---|---|
numltc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numlt.1 | . . . . 5 | |
2 | numlt.2 | . . . . 5 | |
3 | numltc.3 | . . . . 5 | |
4 | numltc.5 | . . . . 5 | |
5 | 1, 2, 3, 1, 4 | numlt 8501 | . . . 4 |
6 | 1 | nnrei 8048 | . . . . . . 7 |
7 | 6 | recni 7131 | . . . . . 6 |
8 | 2 | nn0rei 8299 | . . . . . . 7 |
9 | 8 | recni 7131 | . . . . . 6 |
10 | ax-1cn 7069 | . . . . . 6 | |
11 | 7, 9, 10 | adddii 7129 | . . . . 5 |
12 | 7 | mulid1i 7121 | . . . . . 6 |
13 | 12 | oveq2i 5543 | . . . . 5 |
14 | 11, 13 | eqtri 2101 | . . . 4 |
15 | 5, 14 | breqtrri 3810 | . . 3 |
16 | numltc.6 | . . . . 5 | |
17 | numlt.3 | . . . . . 6 | |
18 | nn0ltp1le 8413 | . . . . . 6 | |
19 | 2, 17, 18 | mp2an 416 | . . . . 5 |
20 | 16, 19 | mpbi 143 | . . . 4 |
21 | 1 | nngt0i 8069 | . . . . 5 |
22 | peano2re 7244 | . . . . . . 7 | |
23 | 8, 22 | ax-mp 7 | . . . . . 6 |
24 | 17 | nn0rei 8299 | . . . . . 6 |
25 | 23, 24, 6 | lemul2i 8003 | . . . . 5 |
26 | 21, 25 | ax-mp 7 | . . . 4 |
27 | 20, 26 | mpbi 143 | . . 3 |
28 | 6, 8 | remulcli 7133 | . . . . 5 |
29 | 3 | nn0rei 8299 | . . . . 5 |
30 | 28, 29 | readdcli 7132 | . . . 4 |
31 | 6, 23 | remulcli 7133 | . . . 4 |
32 | 6, 24 | remulcli 7133 | . . . 4 |
33 | 30, 31, 32 | ltletri 7217 | . . 3 |
34 | 15, 27, 33 | mp2an 416 | . 2 |
35 | numltc.4 | . . 3 | |
36 | 32, 35 | nn0addge1i 8336 | . 2 |
37 | 35 | nn0rei 8299 | . . . 4 |
38 | 32, 37 | readdcli 7132 | . . 3 |
39 | 30, 32, 38 | ltletri 7217 | . 2 |
40 | 34, 36, 39 | mp2an 416 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 103 wcel 1433 class class class wbr 3785 (class class class)co 5532 cr 6980 cc0 6981 c1 6982 caddc 6984 cmul 6986 clt 7153 cle 7154 cn 8039 cn0 8288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 |
This theorem is referenced by: decltc 8505 numlti 8513 |
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