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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 7430 | . 2 ⊢ (𝑁 + 1) = (((𝑇 · 𝐴) + 𝐵) + 1) |
3 | numnncl.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 | |
4 | numnncl.2 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
5 | 3, 4 | nn0mulcli 12540 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ0 |
6 | 5 | nn0cni 12514 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℂ |
7 | numcl.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
8 | 7 | nn0cni 12514 | . . 3 ⊢ 𝐵 ∈ ℂ |
9 | ax-1cn 11196 | . . 3 ⊢ 1 ∈ ℂ | |
10 | 6, 8, 9 | addassi 11254 | . 2 ⊢ (((𝑇 · 𝐴) + 𝐵) + 1) = ((𝑇 · 𝐴) + (𝐵 + 1)) |
11 | numsuc.4 | . . 3 ⊢ (𝐵 + 1) = 𝐶 | |
12 | 11 | oveq2i 7431 | . 2 ⊢ ((𝑇 · 𝐴) + (𝐵 + 1)) = ((𝑇 · 𝐴) + 𝐶) |
13 | 2, 10, 12 | 3eqtri 2760 | 1 ⊢ (𝑁 + 1) = ((𝑇 · 𝐴) + 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7420 1c1 11139 + caddc 11141 · cmul 11143 ℕ0cn0 12502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-ltxr 11283 df-nn 12243 df-n0 12503 |
This theorem is referenced by: decsuc 12738 numsucc 12747 decbin3 12849 |
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