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Mirrors > Home > ILE Home > Th. List > numsucc | GIF version |
Description: The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numsucc.1 | ⊢ 𝑌 ∈ ℕ0 |
numsucc.2 | ⊢ 𝑇 = (𝑌 + 1) |
numsucc.3 | ⊢ 𝐴 ∈ ℕ0 |
numsucc.4 | ⊢ (𝐴 + 1) = 𝐵 |
numsucc.5 | ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝑌) |
Ref | Expression |
---|---|
numsucc | ⊢ (𝑁 + 1) = ((𝑇 · 𝐵) + 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numsucc.2 | . . . . . . 7 ⊢ 𝑇 = (𝑌 + 1) | |
2 | numsucc.1 | . . . . . . . 8 ⊢ 𝑌 ∈ ℕ0 | |
3 | 1nn0 9151 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
4 | 2, 3 | nn0addcli 9172 | . . . . . . 7 ⊢ (𝑌 + 1) ∈ ℕ0 |
5 | 1, 4 | eqeltri 2243 | . . . . . 6 ⊢ 𝑇 ∈ ℕ0 |
6 | 5 | nn0cni 9147 | . . . . 5 ⊢ 𝑇 ∈ ℂ |
7 | 6 | mulid1i 7922 | . . . 4 ⊢ (𝑇 · 1) = 𝑇 |
8 | 7 | oveq2i 5864 | . . 3 ⊢ ((𝑇 · 𝐴) + (𝑇 · 1)) = ((𝑇 · 𝐴) + 𝑇) |
9 | numsucc.3 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
10 | 9 | nn0cni 9147 | . . . 4 ⊢ 𝐴 ∈ ℂ |
11 | ax-1cn 7867 | . . . 4 ⊢ 1 ∈ ℂ | |
12 | 6, 10, 11 | adddii 7930 | . . 3 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + (𝑇 · 1)) |
13 | 1 | eqcomi 2174 | . . . 4 ⊢ (𝑌 + 1) = 𝑇 |
14 | numsucc.5 | . . . 4 ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝑌) | |
15 | 5, 9, 2, 13, 14 | numsuc 9356 | . . 3 ⊢ (𝑁 + 1) = ((𝑇 · 𝐴) + 𝑇) |
16 | 8, 12, 15 | 3eqtr4ri 2202 | . 2 ⊢ (𝑁 + 1) = (𝑇 · (𝐴 + 1)) |
17 | numsucc.4 | . . 3 ⊢ (𝐴 + 1) = 𝐵 | |
18 | 17 | oveq2i 5864 | . 2 ⊢ (𝑇 · (𝐴 + 1)) = (𝑇 · 𝐵) |
19 | 9, 3 | nn0addcli 9172 | . . . 4 ⊢ (𝐴 + 1) ∈ ℕ0 |
20 | 17, 19 | eqeltrri 2244 | . . 3 ⊢ 𝐵 ∈ ℕ0 |
21 | 5, 20 | num0u 9353 | . 2 ⊢ (𝑇 · 𝐵) = ((𝑇 · 𝐵) + 0) |
22 | 16, 18, 21 | 3eqtri 2195 | 1 ⊢ (𝑁 + 1) = ((𝑇 · 𝐵) + 0) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 (class class class)co 5853 0cc0 7774 1c1 7775 + caddc 7777 · cmul 7779 ℕ0cn0 9135 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-sub 8092 df-inn 8879 df-n0 9136 |
This theorem is referenced by: decsucc 9383 |
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