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Mirrors > Home > MPE Home > Th. List > decaddc | Structured version Visualization version GIF version |
Description: Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decma.a | ⊢ 𝐴 ∈ ℕ0 |
decma.b | ⊢ 𝐵 ∈ ℕ0 |
decma.c | ⊢ 𝐶 ∈ ℕ0 |
decma.d | ⊢ 𝐷 ∈ ℕ0 |
decma.m | ⊢ 𝑀 = ;𝐴𝐵 |
decma.n | ⊢ 𝑁 = ;𝐶𝐷 |
decaddc.e | ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 |
decaddc.f | ⊢ 𝐹 ∈ ℕ0 |
decaddc.2 | ⊢ (𝐵 + 𝐷) = ;1𝐹 |
Ref | Expression |
---|---|
decaddc | ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 11922 | . . 3 ⊢ ;10 ∈ ℕ0 | |
2 | decma.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
3 | decma.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
4 | decma.c | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
5 | decma.d | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
6 | decma.m | . . . 4 ⊢ 𝑀 = ;𝐴𝐵 | |
7 | dfdec10 11907 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
8 | 6, 7 | eqtri 2796 | . . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵) |
9 | decma.n | . . . 4 ⊢ 𝑁 = ;𝐶𝐷 | |
10 | dfdec10 11907 | . . . 4 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
11 | 9, 10 | eqtri 2796 | . . 3 ⊢ 𝑁 = ((;10 · 𝐶) + 𝐷) |
12 | decaddc.f | . . 3 ⊢ 𝐹 ∈ ℕ0 | |
13 | decaddc.e | . . 3 ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 | |
14 | decaddc.2 | . . . 4 ⊢ (𝐵 + 𝐷) = ;1𝐹 | |
15 | dfdec10 11907 | . . . 4 ⊢ ;1𝐹 = ((;10 · 1) + 𝐹) | |
16 | 14, 15 | eqtri 2796 | . . 3 ⊢ (𝐵 + 𝐷) = ((;10 · 1) + 𝐹) |
17 | 1, 2, 3, 4, 5, 8, 11, 12, 13, 16 | numaddc 11953 | . 2 ⊢ (𝑀 + 𝑁) = ((;10 · 𝐸) + 𝐹) |
18 | dfdec10 11907 | . 2 ⊢ ;𝐸𝐹 = ((;10 · 𝐸) + 𝐹) | |
19 | 17, 18 | eqtr4i 2799 | 1 ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2048 (class class class)co 6970 0cc0 10327 1c1 10328 + caddc 10330 · cmul 10332 ℕ0cn0 11700 ;cdc 11904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-ltxr 10471 df-sub 10664 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-dec 11905 |
This theorem is referenced by: decaddc2 11961 decaddci 11966 2exp16 16270 prmlem2 16299 37prm 16300 1259lem1 16310 1259lem4 16313 2503lem2 16317 4001lem1 16320 threehalves 30326 1mhdrd 30327 hgt750lem2 31532 fmtno5lem4 43026 fmtno4nprmfac193 43044 fmtno5fac 43052 |
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