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Mirrors > Home > ILE Home > Th. List > num0u | 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 8774 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ0 |
4 | 3 | nn0cni 8748 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℂ |
5 | 4 | addid1i 7687 | . 2 ⊢ ((𝑇 · 𝐴) + 0) = (𝑇 · 𝐴) |
6 | 5 | eqcomi 2093 | 1 ⊢ (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0) |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 ∈ wcel 1439 (class class class)co 5668 0cc0 7413 + caddc 7416 · cmul 7418 ℕ0cn0 8736 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-cnre 7519 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-br 3854 df-opab 3908 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-iota 4995 df-fun 5032 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-sub 7718 df-inn 8486 df-n0 8737 |
This theorem is referenced by: dec0u 8960 numsucc 8979 nummul1c 8988 |
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