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Mirrors > Home > ILE Home > Th. List > decaddci | 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 8960 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | decaddi.3 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
5 | decaddi.4 | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | 4 | dec0h 9171 | . 2 ⊢ 𝑁 = ;0𝑁 |
7 | 1 | nn0cni 8957 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
8 | 7 | addid1i 7872 | . . . 4 ⊢ (𝐴 + 0) = 𝐴 |
9 | 8 | oveq1i 5752 | . . 3 ⊢ ((𝐴 + 0) + 1) = (𝐴 + 1) |
10 | decaddci.5 | . . 3 ⊢ (𝐴 + 1) = 𝐷 | |
11 | 9, 10 | eqtri 2138 | . 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 9204 | 1 ⊢ (𝑀 + 𝑁) = ;𝐷𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∈ wcel 1465 (class class class)co 5742 0cc0 7588 1c1 7589 + caddc 7591 ℕ0cn0 8945 ;cdc 9150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-9 8754 df-n0 8946 df-dec 9151 |
This theorem is referenced by: decaddci2 9211 6t4e24 9255 7t3e21 9259 7t5e35 9261 7t6e42 9262 8t3e24 9265 8t4e32 9266 8t7e56 9269 8t8e64 9270 9t3e27 9272 9t4e36 9273 9t5e45 9274 9t6e54 9275 9t7e63 9276 9t8e72 9277 9t9e81 9278 ex-exp 12866 |
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