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Mirrors > Home > MPE Home > Th. List > numsuc | Structured version Visualization version GIF version |
Description: The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numnncl.1 | ⊢ 𝑇 ∈ ℕ0 |
numnncl.2 | ⊢ 𝐴 ∈ ℕ0 |
numcl.2 | ⊢ 𝐵 ∈ ℕ0 |
numsuc.4 | ⊢ (𝐵 + 1) = 𝐶 |
numsuc.5 | ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) |
Ref | Expression |
---|---|
numsuc | ⊢ (𝑁 + 1) = ((𝑇 · 𝐴) + 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numsuc.5 | . . 3 ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) | |
2 | 1 | oveq1i 7201 | . 2 ⊢ (𝑁 + 1) = (((𝑇 · 𝐴) + 𝐵) + 1) |
3 | numnncl.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 | |
4 | numnncl.2 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
5 | 3, 4 | nn0mulcli 12093 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ0 |
6 | 5 | nn0cni 12067 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℂ |
7 | numcl.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
8 | 7 | nn0cni 12067 | . . 3 ⊢ 𝐵 ∈ ℂ |
9 | ax-1cn 10752 | . . 3 ⊢ 1 ∈ ℂ | |
10 | 6, 8, 9 | addassi 10808 | . 2 ⊢ (((𝑇 · 𝐴) + 𝐵) + 1) = ((𝑇 · 𝐴) + (𝐵 + 1)) |
11 | numsuc.4 | . . 3 ⊢ (𝐵 + 1) = 𝐶 | |
12 | 11 | oveq2i 7202 | . 2 ⊢ ((𝑇 · 𝐴) + (𝐵 + 1)) = ((𝑇 · 𝐴) + 𝐶) |
13 | 2, 10, 12 | 3eqtri 2763 | 1 ⊢ (𝑁 + 1) = ((𝑇 · 𝐴) + 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 (class class class)co 7191 1c1 10695 + caddc 10697 · cmul 10699 ℕ0cn0 12055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-nn 11796 df-n0 12056 |
This theorem is referenced by: decsuc 12289 numsucc 12298 decbin3 12400 |
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