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Mirrors > Home > ILE Home > Th. List > decaddc | GIF version |
Description: Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decma.a | ⊢ 𝐴 ∈ ℕ0 |
decma.b | ⊢ 𝐵 ∈ ℕ0 |
decma.c | ⊢ 𝐶 ∈ ℕ0 |
decma.d | ⊢ 𝐷 ∈ ℕ0 |
decma.m | ⊢ 𝑀 = ;𝐴𝐵 |
decma.n | ⊢ 𝑁 = ;𝐶𝐷 |
decaddc.e | ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 |
decaddc.f | ⊢ 𝐹 ∈ ℕ0 |
decaddc.2 | ⊢ (𝐵 + 𝐷) = ;1𝐹 |
Ref | Expression |
---|---|
decaddc | ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 9399 | . . 3 ⊢ ;10 ∈ ℕ0 | |
2 | decma.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
3 | decma.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
4 | decma.c | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
5 | decma.d | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
6 | decma.m | . . . 4 ⊢ 𝑀 = ;𝐴𝐵 | |
7 | dfdec10 9385 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
8 | 6, 7 | eqtri 2198 | . . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵) |
9 | decma.n | . . . 4 ⊢ 𝑁 = ;𝐶𝐷 | |
10 | dfdec10 9385 | . . . 4 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
11 | 9, 10 | eqtri 2198 | . . 3 ⊢ 𝑁 = ((;10 · 𝐶) + 𝐷) |
12 | decaddc.f | . . 3 ⊢ 𝐹 ∈ ℕ0 | |
13 | decaddc.e | . . 3 ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 | |
14 | decaddc.2 | . . . 4 ⊢ (𝐵 + 𝐷) = ;1𝐹 | |
15 | dfdec10 9385 | . . . 4 ⊢ ;1𝐹 = ((;10 · 1) + 𝐹) | |
16 | 14, 15 | eqtri 2198 | . . 3 ⊢ (𝐵 + 𝐷) = ((;10 · 1) + 𝐹) |
17 | 1, 2, 3, 4, 5, 8, 11, 12, 13, 16 | numaddc 9429 | . 2 ⊢ (𝑀 + 𝑁) = ((;10 · 𝐸) + 𝐹) |
18 | dfdec10 9385 | . 2 ⊢ ;𝐸𝐹 = ((;10 · 𝐸) + 𝐹) | |
19 | 17, 18 | eqtr4i 2201 | 1 ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 (class class class)co 5874 0cc0 7810 1c1 7811 + caddc 7813 · cmul 7815 ℕ0cn0 9174 ;cdc 9382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-sub 8128 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-5 8979 df-6 8980 df-7 8981 df-8 8982 df-9 8983 df-n0 9175 df-dec 9383 |
This theorem is referenced by: decaddc2 9437 decaddci 9442 |
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