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Mirrors > Home > MPE Home > Th. List > num0u | Structured version Visualization version GIF version |
Description: Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numnncl.1 | ⊢ 𝑇 ∈ ℕ0 |
numnncl.2 | ⊢ 𝐴 ∈ ℕ0 |
Ref | Expression |
---|---|
num0u | ⊢ (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numnncl.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 | |
2 | numnncl.2 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | 1, 2 | nn0mulcli 12201 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ0 |
4 | 3 | nn0cni 12175 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℂ |
5 | 4 | addid1i 11092 | . 2 ⊢ ((𝑇 · 𝐴) + 0) = (𝑇 · 𝐴) |
6 | 5 | eqcomi 2747 | 1 ⊢ (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2108 (class class class)co 7255 0cc0 10802 + caddc 10805 · cmul 10807 ℕ0cn0 12163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-ltxr 10945 df-nn 11904 df-n0 12164 |
This theorem is referenced by: dec0u 12387 numsucc 12406 nummul1c 12415 decmul1 12430 |
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