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Mirrors > Home > MPE Home > Th. List > decle | Structured version Visualization version GIF version |
Description: Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
decle.1 | โข ๐ด โ โ0 |
decle.2 | โข ๐ต โ โ0 |
decle.3 | โข ๐ถ โ โ0 |
decle.4 | โข ๐ต โค ๐ถ |
Ref | Expression |
---|---|
decle | โข ;๐ด๐ต โค ;๐ด๐ถ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decle.4 | . . 3 โข ๐ต โค ๐ถ | |
2 | decle.2 | . . . . 5 โข ๐ต โ โ0 | |
3 | 2 | nn0rei 12290 | . . . 4 โข ๐ต โ โ |
4 | decle.3 | . . . . 5 โข ๐ถ โ โ0 | |
5 | 4 | nn0rei 12290 | . . . 4 โข ๐ถ โ โ |
6 | 10nn0 12501 | . . . . . 6 โข ;10 โ โ0 | |
7 | decle.1 | . . . . . 6 โข ๐ด โ โ0 | |
8 | 6, 7 | nn0mulcli 12317 | . . . . 5 โข (;10 ยท ๐ด) โ โ0 |
9 | 8 | nn0rei 12290 | . . . 4 โข (;10 ยท ๐ด) โ โ |
10 | 3, 5, 9 | leadd2i 11577 | . . 3 โข (๐ต โค ๐ถ โ ((;10 ยท ๐ด) + ๐ต) โค ((;10 ยท ๐ด) + ๐ถ)) |
11 | 1, 10 | mpbi 229 | . 2 โข ((;10 ยท ๐ด) + ๐ต) โค ((;10 ยท ๐ด) + ๐ถ) |
12 | dfdec10 12486 | . 2 โข ;๐ด๐ต = ((;10 ยท ๐ด) + ๐ต) | |
13 | dfdec10 12486 | . 2 โข ;๐ด๐ถ = ((;10 ยท ๐ด) + ๐ถ) | |
14 | 11, 12, 13 | 3brtr4i 5111 | 1 โข ;๐ด๐ต โค ;๐ด๐ถ |
Colors of variables: wff setvar class |
Syntax hints: โ wcel 2104 class class class wbr 5081 (class class class)co 7307 0cc0 10917 1c1 10918 + caddc 10920 ยท cmul 10922 โค cle 11056 โ0cn0 12279 ;cdc 12483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-ov 7310 df-om 7745 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-dec 12484 |
This theorem is referenced by: (None) |
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