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Mirrors > Home > MPE Home > Th. List > decsubi | Structured version Visualization version GIF version |
Description: Difference between a numeral ๐ and a nonnegative integer ๐ (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decaddi.1 | โข ๐ด โ โ0 |
decaddi.2 | โข ๐ต โ โ0 |
decaddi.3 | โข ๐ โ โ0 |
decaddi.4 | โข ๐ = ;๐ด๐ต |
decaddci.5 | โข (๐ด + 1) = ๐ท |
decsubi.5 | โข (๐ต โ ๐) = ๐ถ |
Ref | Expression |
---|---|
decsubi | โข (๐ โ ๐) = ;๐ด๐ถ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12643 | . . . . 5 โข ;10 โ โ0 | |
2 | decaddi.1 | . . . . 5 โข ๐ด โ โ0 | |
3 | 1, 2 | nn0mulcli 12458 | . . . 4 โข (;10 ยท ๐ด) โ โ0 |
4 | 3 | nn0cni 12432 | . . 3 โข (;10 ยท ๐ด) โ โ |
5 | decaddi.2 | . . . 4 โข ๐ต โ โ0 | |
6 | 5 | nn0cni 12432 | . . 3 โข ๐ต โ โ |
7 | decaddi.3 | . . . 4 โข ๐ โ โ0 | |
8 | 7 | nn0cni 12432 | . . 3 โข ๐ โ โ |
9 | 4, 6, 8 | addsubassi 11499 | . 2 โข (((;10 ยท ๐ด) + ๐ต) โ ๐) = ((;10 ยท ๐ด) + (๐ต โ ๐)) |
10 | decaddi.4 | . . . 4 โข ๐ = ;๐ด๐ต | |
11 | dfdec10 12628 | . . . 4 โข ;๐ด๐ต = ((;10 ยท ๐ด) + ๐ต) | |
12 | 10, 11 | eqtri 2765 | . . 3 โข ๐ = ((;10 ยท ๐ด) + ๐ต) |
13 | 12 | oveq1i 7372 | . 2 โข (๐ โ ๐) = (((;10 ยท ๐ด) + ๐ต) โ ๐) |
14 | dfdec10 12628 | . . 3 โข ;๐ด๐ถ = ((;10 ยท ๐ด) + ๐ถ) | |
15 | decsubi.5 | . . . . 5 โข (๐ต โ ๐) = ๐ถ | |
16 | 15 | eqcomi 2746 | . . . 4 โข ๐ถ = (๐ต โ ๐) |
17 | 16 | oveq2i 7373 | . . 3 โข ((;10 ยท ๐ด) + ๐ถ) = ((;10 ยท ๐ด) + (๐ต โ ๐)) |
18 | 14, 17 | eqtri 2765 | . 2 โข ;๐ด๐ถ = ((;10 ยท ๐ด) + (๐ต โ ๐)) |
19 | 9, 13, 18 | 3eqtr4i 2775 | 1 โข (๐ โ ๐) = ;๐ด๐ถ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7362 0cc0 11058 1c1 11059 + caddc 11061 ยท cmul 11063 โ cmin 11392 โ0cn0 12420 ;cdc 12625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-ltxr 11201 df-sub 11394 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-dec 12626 |
This theorem is referenced by: fmtno5 45823 m5prm 45864 m7prm 45866 m11nprm 45867 341fppr2 46000 ackval41 46855 |
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