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Mirrors > Home > MPE Home > Th. List > decadd | Structured version Visualization version GIF version |
Description: Add two numerals ๐ and ๐ (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decma.a | โข ๐ด โ โ0 |
decma.b | โข ๐ต โ โ0 |
decma.c | โข ๐ถ โ โ0 |
decma.d | โข ๐ท โ โ0 |
decma.m | โข ๐ = ;๐ด๐ต |
decma.n | โข ๐ = ;๐ถ๐ท |
decadd.e | โข (๐ด + ๐ถ) = ๐ธ |
decadd.f | โข (๐ต + ๐ท) = ๐น |
Ref | Expression |
---|---|
decadd | โข (๐ + ๐) = ;๐ธ๐น |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12726 | . . 3 โข ;10 โ โ0 | |
2 | decma.a | . . 3 โข ๐ด โ โ0 | |
3 | decma.b | . . 3 โข ๐ต โ โ0 | |
4 | decma.c | . . 3 โข ๐ถ โ โ0 | |
5 | decma.d | . . 3 โข ๐ท โ โ0 | |
6 | decma.m | . . . 4 โข ๐ = ;๐ด๐ต | |
7 | dfdec10 12711 | . . . 4 โข ;๐ด๐ต = ((;10 ยท ๐ด) + ๐ต) | |
8 | 6, 7 | eqtri 2756 | . . 3 โข ๐ = ((;10 ยท ๐ด) + ๐ต) |
9 | decma.n | . . . 4 โข ๐ = ;๐ถ๐ท | |
10 | dfdec10 12711 | . . . 4 โข ;๐ถ๐ท = ((;10 ยท ๐ถ) + ๐ท) | |
11 | 9, 10 | eqtri 2756 | . . 3 โข ๐ = ((;10 ยท ๐ถ) + ๐ท) |
12 | decadd.e | . . 3 โข (๐ด + ๐ถ) = ๐ธ | |
13 | decadd.f | . . 3 โข (๐ต + ๐ท) = ๐น | |
14 | 1, 2, 3, 4, 5, 8, 11, 12, 13 | numadd 12755 | . 2 โข (๐ + ๐) = ((;10 ยท ๐ธ) + ๐น) |
15 | dfdec10 12711 | . 2 โข ;๐ธ๐น = ((;10 ยท ๐ธ) + ๐น) | |
16 | 14, 15 | eqtr4i 2759 | 1 โข (๐ + ๐) = ;๐ธ๐น |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 โ wcel 2099 (class class class)co 7420 0cc0 11139 1c1 11140 + caddc 11142 ยท cmul 11144 โ0cn0 12503 ;cdc 12708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-ltxr 11284 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-dec 12709 |
This theorem is referenced by: decaddm10 12767 decaddi 12768 10p10e20 12803 dec5dvds2 17034 2exp16 17060 37prm 17090 43prm 17091 317prm 17095 631prm 17096 1259lem2 17101 1259lem3 17102 1259lem4 17103 2503lem1 17106 2503lem2 17107 4001lem1 17110 4001lem2 17111 4001lem3 17112 log2ublem3 26893 log2ub 26894 1kp2ke3k 30269 hgt750lemd 34280 hgt750lem2 34284 12gcd5e1 41474 3lexlogpow5ineq1 41525 decpmul 41862 sqdeccom12 41863 sq3deccom12 41864 ex-decpmul 41868 resqrtvalex 43075 imsqrtvalex 43076 fmtno5lem4 46896 257prm 46901 fmtno4prmfac 46912 fmtno4nprmfac193 46914 fmtno5faclem3 46921 fmtno5fac 46922 |
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