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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 9232 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ0 |
4 | 3 | nn0cni 9206 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℂ |
5 | 4 | addid1i 8117 | . 2 ⊢ ((𝑇 · 𝐴) + 0) = (𝑇 · 𝐴) |
6 | 5 | eqcomi 2193 | 1 ⊢ (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 (class class class)co 5891 0cc0 7829 + caddc 7832 · cmul 7834 ℕ0cn0 9194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-cnre 7940 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5233 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-sub 8148 df-inn 8938 df-n0 9195 |
This theorem is referenced by: dec0u 9422 numsucc 9441 nummul1c 9450 |
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