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Mirrors > Home > MPE Home > Th. List > decaddci | Structured version Visualization version GIF version |
Description: Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
decaddi.1 | ⊢ 𝐴 ∈ ℕ0 |
decaddi.2 | ⊢ 𝐵 ∈ ℕ0 |
decaddi.3 | ⊢ 𝑁 ∈ ℕ0 |
decaddi.4 | ⊢ 𝑀 = ;𝐴𝐵 |
decaddci.5 | ⊢ (𝐴 + 1) = 𝐷 |
decaddci.6 | ⊢ 𝐶 ∈ ℕ0 |
decaddci.7 | ⊢ (𝐵 + 𝑁) = ;1𝐶 |
Ref | Expression |
---|---|
decaddci | ⊢ (𝑀 + 𝑁) = ;𝐷𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decaddi.1 | . 2 ⊢ 𝐴 ∈ ℕ0 | |
2 | decaddi.2 | . 2 ⊢ 𝐵 ∈ ℕ0 | |
3 | 0nn0 12538 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | decaddi.3 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
5 | decaddi.4 | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | 4 | dec0h 12752 | . 2 ⊢ 𝑁 = ;0𝑁 |
7 | 1 | nn0cni 12535 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
8 | 7 | addridi 11445 | . . . 4 ⊢ (𝐴 + 0) = 𝐴 |
9 | 8 | oveq1i 7440 | . . 3 ⊢ ((𝐴 + 0) + 1) = (𝐴 + 1) |
10 | decaddci.5 | . . 3 ⊢ (𝐴 + 1) = 𝐷 | |
11 | 9, 10 | eqtri 2762 | . 2 ⊢ ((𝐴 + 0) + 1) = 𝐷 |
12 | decaddci.6 | . 2 ⊢ 𝐶 ∈ ℕ0 | |
13 | decaddci.7 | . 2 ⊢ (𝐵 + 𝑁) = ;1𝐶 | |
14 | 1, 2, 3, 4, 5, 6, 11, 12, 13 | decaddc 12785 | 1 ⊢ (𝑀 + 𝑁) = ;𝐷𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 (class class class)co 7430 0cc0 11152 1c1 11153 + caddc 11155 ℕ0cn0 12523 ;cdc 12730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-sub 11491 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-dec 12731 |
This theorem is referenced by: decaddci2 12792 6t4e24 12836 7t3e21 12840 7t5e35 12842 7t6e42 12843 8t3e24 12846 8t4e32 12847 8t7e56 12850 8t8e64 12851 9t3e27 12853 9t4e36 12854 9t5e45 12855 9t6e54 12856 9t7e63 12857 9t8e72 12858 9t9e81 12859 2exp8 17122 2exp11 17123 prmlem2 17153 43prm 17155 83prm 17156 317prm 17159 631prm 17160 1259lem1 17164 1259lem2 17165 1259lem3 17166 1259lem4 17167 1259lem5 17168 2503lem1 17170 2503lem2 17171 2503lem3 17172 4001lem1 17174 4001lem2 17175 4001lem4 17177 log2ublem3 27005 log2ub 27006 ex-exp 30478 hgt750lem2 34645 3exp7 42034 3lexlogpow5ineq1 42035 resqrtvalex 43634 imsqrtvalex 43635 fmtno5lem1 47477 fmtno5lem4 47480 257prm 47485 fmtno4nprmfac193 47498 fmtno5fac 47506 127prm 47523 2exp340mod341 47657 ackval3012 48541 |
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