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Mirrors > Home > MPE Home > Th. List > decaddc2 | 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 |
decaddc2.t | ⊢ (𝐵 + 𝐷) = ;10 |
Ref | Expression |
---|---|
decaddc2 | ⊢ (𝑀 + 𝑁) = ;𝐸0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decma.a | . 2 ⊢ 𝐴 ∈ ℕ0 | |
2 | decma.b | . 2 ⊢ 𝐵 ∈ ℕ0 | |
3 | decma.c | . 2 ⊢ 𝐶 ∈ ℕ0 | |
4 | decma.d | . 2 ⊢ 𝐷 ∈ ℕ0 | |
5 | decma.m | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | decma.n | . 2 ⊢ 𝑁 = ;𝐶𝐷 | |
7 | decaddc.e | . 2 ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 | |
8 | 0nn0 11659 | . 2 ⊢ 0 ∈ ℕ0 | |
9 | decaddc2.t | . 2 ⊢ (𝐵 + 𝐷) = ;10 | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | decaddc 11901 | 1 ⊢ (𝑀 + 𝑁) = ;𝐸0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2106 (class class class)co 6922 0cc0 10272 1c1 10273 + caddc 10275 ℕ0cn0 11642 ;cdc 11845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-dec 11846 |
This theorem is referenced by: 1259lem4 16239 2503lem2 16243 4001lem1 16246 4001lem2 16247 log2ublem3 25127 log2ub 25128 ex-decpmul 38140 |
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