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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 9017 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
4 | 2, 3 | nn0addcli 9038 | . . . . . . 7 ⊢ (𝑌 + 1) ∈ ℕ0 |
5 | 1, 4 | eqeltri 2213 | . . . . . 6 ⊢ 𝑇 ∈ ℕ0 |
6 | 5 | nn0cni 9013 | . . . . 5 ⊢ 𝑇 ∈ ℂ |
7 | 6 | mulid1i 7792 | . . . 4 ⊢ (𝑇 · 1) = 𝑇 |
8 | 7 | oveq2i 5793 | . . 3 ⊢ ((𝑇 · 𝐴) + (𝑇 · 1)) = ((𝑇 · 𝐴) + 𝑇) |
9 | numsucc.3 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
10 | 9 | nn0cni 9013 | . . . 4 ⊢ 𝐴 ∈ ℂ |
11 | ax-1cn 7737 | . . . 4 ⊢ 1 ∈ ℂ | |
12 | 6, 10, 11 | adddii 7800 | . . 3 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + (𝑇 · 1)) |
13 | 1 | eqcomi 2144 | . . . 4 ⊢ (𝑌 + 1) = 𝑇 |
14 | numsucc.5 | . . . 4 ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝑌) | |
15 | 5, 9, 2, 13, 14 | numsuc 9219 | . . 3 ⊢ (𝑁 + 1) = ((𝑇 · 𝐴) + 𝑇) |
16 | 8, 12, 15 | 3eqtr4ri 2172 | . 2 ⊢ (𝑁 + 1) = (𝑇 · (𝐴 + 1)) |
17 | numsucc.4 | . . 3 ⊢ (𝐴 + 1) = 𝐵 | |
18 | 17 | oveq2i 5793 | . 2 ⊢ (𝑇 · (𝐴 + 1)) = (𝑇 · 𝐵) |
19 | 9, 3 | nn0addcli 9038 | . . . 4 ⊢ (𝐴 + 1) ∈ ℕ0 |
20 | 17, 19 | eqeltrri 2214 | . . 3 ⊢ 𝐵 ∈ ℕ0 |
21 | 5, 20 | num0u 9216 | . 2 ⊢ (𝑇 · 𝐵) = ((𝑇 · 𝐵) + 0) |
22 | 16, 18, 21 | 3eqtri 2165 | 1 ⊢ (𝑁 + 1) = ((𝑇 · 𝐵) + 0) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 (class class class)co 5782 0cc0 7644 1c1 7645 + caddc 7647 · cmul 7649 ℕ0cn0 9001 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-inn 8745 df-n0 9002 |
This theorem is referenced by: decsucc 9246 |
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