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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 8303 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | decaddi.3 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
5 | decaddi.4 | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | 4 | dec0h 8498 | . 2 ⊢ 𝑁 = ;0𝑁 |
7 | 1 | nn0cni 8300 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
8 | 7 | addid1i 7250 | . . . 4 ⊢ (𝐴 + 0) = 𝐴 |
9 | 8 | oveq1i 5542 | . . 3 ⊢ ((𝐴 + 0) + 1) = (𝐴 + 1) |
10 | decaddci.5 | . . 3 ⊢ (𝐴 + 1) = 𝐷 | |
11 | 9, 10 | eqtri 2101 | . 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 8531 | 1 ⊢ (𝑀 + 𝑁) = ;𝐷𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 (class class class)co 5532 0cc0 6981 1c1 6982 + caddc 6984 ℕ0cn0 8288 ;cdc 8477 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-5 8101 df-6 8102 df-7 8103 df-8 8104 df-9 8105 df-n0 8289 df-dec 8478 |
This theorem is referenced by: decaddci2 8538 6t4e24 8582 7t3e21 8586 7t5e35 8588 7t6e42 8589 8t3e24 8592 8t4e32 8593 8t7e56 8596 8t8e64 8597 9t3e27 8599 9t4e36 8600 9t5e45 8601 9t6e54 8602 9t7e63 8603 9t8e72 8604 9t9e81 8605 |
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