![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > decaddci2 | 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 |
decaddci2.6 | ⊢ (B + 𝑁) = 10 |
Ref | Expression |
---|---|
decaddci2 | ⊢ (𝑀 + 𝑁) = ;𝐷0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decaddi.1 | . 2 ⊢ A ∈ ℕ0 | |
2 | decaddi.2 | . 2 ⊢ B ∈ ℕ0 | |
3 | decaddi.3 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
4 | decaddi.4 | . 2 ⊢ 𝑀 = ;AB | |
5 | decaddci.5 | . 2 ⊢ (A + 1) = 𝐷 | |
6 | 0nn0 7972 | . 2 ⊢ 0 ∈ ℕ0 | |
7 | decaddci2.6 | . . 3 ⊢ (B + 𝑁) = 10 | |
8 | dec10 8173 | . . 3 ⊢ 10 = ;10 | |
9 | 7, 8 | eqtri 2057 | . 2 ⊢ (B + 𝑁) = ;10 |
10 | 1, 2, 3, 4, 5, 6, 9 | decaddci 8188 | 1 ⊢ (𝑀 + 𝑁) = ;𝐷0 |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∈ wcel 1390 (class class class)co 5455 0cc0 6711 1c1 6712 + caddc 6714 10c10 7752 ℕ0cn0 7957 ;cdc 8144 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-i2m1 6788 ax-1rid 6790 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-sub 6981 df-inn 7696 df-2 7753 df-3 7754 df-4 7755 df-5 7756 df-6 7757 df-7 7758 df-8 7759 df-9 7760 df-10 7761 df-n0 7958 df-dec 8145 |
This theorem is referenced by: 5t4e20 8218 6t5e30 8223 8t5e40 8234 |
Copyright terms: Public domain | W3C validator |