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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 12491 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | decaddi.3 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
5 | decaddi.4 | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | 4 | dec0h 12703 | . 2 ⊢ 𝑁 = ;0𝑁 |
7 | 1 | nn0cni 12488 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
8 | 7 | addridi 11405 | . . . 4 ⊢ (𝐴 + 0) = 𝐴 |
9 | 8 | oveq1i 7415 | . . 3 ⊢ ((𝐴 + 0) + 1) = (𝐴 + 1) |
10 | decaddci.5 | . . 3 ⊢ (𝐴 + 1) = 𝐷 | |
11 | 9, 10 | eqtri 2754 | . 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 12736 | 1 ⊢ (𝑀 + 𝑁) = ;𝐷𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7405 0cc0 11112 1c1 11113 + caddc 11115 ℕ0cn0 12476 ;cdc 12681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-sub 11450 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-dec 12682 |
This theorem is referenced by: decaddci2 12743 6t4e24 12787 7t3e21 12791 7t5e35 12793 7t6e42 12794 8t3e24 12797 8t4e32 12798 8t7e56 12801 8t8e64 12802 9t3e27 12804 9t4e36 12805 9t5e45 12806 9t6e54 12807 9t7e63 12808 9t8e72 12809 9t9e81 12810 2exp8 17031 2exp11 17032 prmlem2 17062 43prm 17064 83prm 17065 317prm 17068 631prm 17069 1259lem1 17073 1259lem2 17074 1259lem3 17075 1259lem4 17076 1259lem5 17077 2503lem1 17079 2503lem2 17080 2503lem3 17081 4001lem1 17083 4001lem2 17084 4001lem4 17086 log2ublem3 26835 log2ub 26836 ex-exp 30212 hgt750lem2 34193 3exp7 41434 3lexlogpow5ineq1 41435 resqrtvalex 42972 imsqrtvalex 42973 fmtno5lem1 46793 fmtno5lem4 46796 257prm 46801 fmtno4nprmfac193 46814 fmtno5fac 46822 127prm 46839 2exp340mod341 46973 ackval3012 47653 |
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