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Mirrors > Home > MPE Home > Th. List > decaddi | 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 |
decaddi.5 | ⊢ (𝐵 + 𝑁) = 𝐶 |
Ref | Expression |
---|---|
decaddi | ⊢ (𝑀 + 𝑁) = ;𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decaddi.1 | . 2 ⊢ 𝐴 ∈ ℕ0 | |
2 | decaddi.2 | . 2 ⊢ 𝐵 ∈ ℕ0 | |
3 | 0nn0 11913 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | decaddi.3 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
5 | decaddi.4 | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | 4 | dec0h 12121 | . 2 ⊢ 𝑁 = ;0𝑁 |
7 | 1 | nn0cni 11910 | . . 3 ⊢ 𝐴 ∈ ℂ |
8 | 7 | addid1i 10827 | . 2 ⊢ (𝐴 + 0) = 𝐴 |
9 | decaddi.5 | . 2 ⊢ (𝐵 + 𝑁) = 𝐶 | |
10 | 1, 2, 3, 4, 5, 6, 8, 9 | decadd 12153 | 1 ⊢ (𝑀 + 𝑁) = ;𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 0cc0 10537 + caddc 10540 ℕ0cn0 11898 ;cdc 12099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-dec 12100 |
This theorem is referenced by: 4t4e16 12198 6t3e18 12204 7t4e28 12210 7t7e49 12213 2exp16 16424 17prm 16450 23prm 16452 prmlem2 16453 37prm 16454 83prm 16456 139prm 16457 163prm 16458 317prm 16459 631prm 16460 1259lem1 16464 1259lem2 16465 1259lem3 16466 1259lem4 16467 1259lem5 16468 1259prm 16469 2503lem1 16470 2503lem2 16471 2503lem3 16472 4001lem1 16474 4001lem2 16475 4001lem4 16477 4001prm 16478 log2ublem3 25526 log2ub 25527 birthday 25532 ex-fac 28230 hgt750lem2 31923 decaddcom 39190 sqn5i 39191 sqdeccom12 39195 sq3deccom12 39196 235t711 39197 ex-decpmul 39198 fmtno5lem1 43735 fmtno5lem2 43736 fmtno5lem4 43738 257prm 43743 fmtno4prmfac 43754 fmtno4nprmfac193 43756 fmtno5faclem1 43761 fmtno5faclem2 43762 fmtno5faclem3 43763 139prmALT 43779 127prm 43783 2exp11 43785 11t31e341 43917 |
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