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Mirrors > Home > MPE Home > Th. List > numsucc | Structured version Visualization version 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 11761 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
4 | 2, 3 | nn0addcli 11782 | . . . . . . 7 ⊢ (𝑌 + 1) ∈ ℕ0 |
5 | 1, 4 | eqeltri 2879 | . . . . . 6 ⊢ 𝑇 ∈ ℕ0 |
6 | 5 | nn0cni 11757 | . . . . 5 ⊢ 𝑇 ∈ ℂ |
7 | 6 | mulid1i 10491 | . . . 4 ⊢ (𝑇 · 1) = 𝑇 |
8 | 7 | oveq2i 7027 | . . 3 ⊢ ((𝑇 · 𝐴) + (𝑇 · 1)) = ((𝑇 · 𝐴) + 𝑇) |
9 | numsucc.3 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
10 | 9 | nn0cni 11757 | . . . 4 ⊢ 𝐴 ∈ ℂ |
11 | ax-1cn 10441 | . . . 4 ⊢ 1 ∈ ℂ | |
12 | 6, 10, 11 | adddii 10499 | . . 3 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + (𝑇 · 1)) |
13 | 1 | eqcomi 2804 | . . . 4 ⊢ (𝑌 + 1) = 𝑇 |
14 | numsucc.5 | . . . 4 ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝑌) | |
15 | 5, 9, 2, 13, 14 | numsuc 11961 | . . 3 ⊢ (𝑁 + 1) = ((𝑇 · 𝐴) + 𝑇) |
16 | 8, 12, 15 | 3eqtr4ri 2830 | . 2 ⊢ (𝑁 + 1) = (𝑇 · (𝐴 + 1)) |
17 | numsucc.4 | . . 3 ⊢ (𝐴 + 1) = 𝐵 | |
18 | 17 | oveq2i 7027 | . 2 ⊢ (𝑇 · (𝐴 + 1)) = (𝑇 · 𝐵) |
19 | 9, 3 | nn0addcli 11782 | . . . 4 ⊢ (𝐴 + 1) ∈ ℕ0 |
20 | 17, 19 | eqeltrri 2880 | . . 3 ⊢ 𝐵 ∈ ℕ0 |
21 | 5, 20 | num0u 11958 | . 2 ⊢ (𝑇 · 𝐵) = ((𝑇 · 𝐵) + 0) |
22 | 16, 18, 21 | 3eqtri 2823 | 1 ⊢ (𝑁 + 1) = ((𝑇 · 𝐵) + 0) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2081 (class class class)co 7016 0cc0 10383 1c1 10384 + caddc 10386 · cmul 10388 ℕ0cn0 11745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-ltxr 10526 df-nn 11487 df-n0 11746 |
This theorem is referenced by: decsucc 11988 |
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