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Mirrors > Home > ILE Home > Th. List > num0h | GIF version |
Description: Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numnncl.1 | ⊢ 𝑇 ∈ ℕ0 |
numnncl.2 | ⊢ 𝐴 ∈ ℕ0 |
Ref | Expression |
---|---|
num0h | ⊢ 𝐴 = ((𝑇 · 0) + 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numnncl.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 | |
2 | 1 | nn0cni 8739 | . . . 4 ⊢ 𝑇 ∈ ℂ |
3 | 2 | mul01i 7923 | . . 3 ⊢ (𝑇 · 0) = 0 |
4 | 3 | oveq1i 5676 | . 2 ⊢ ((𝑇 · 0) + 𝐴) = (0 + 𝐴) |
5 | numnncl.2 | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
6 | 5 | nn0cni 8739 | . . 3 ⊢ 𝐴 ∈ ℂ |
7 | 6 | addid2i 7679 | . 2 ⊢ (0 + 𝐴) = 𝐴 |
8 | 4, 7 | eqtr2i 2110 | 1 ⊢ 𝐴 = ((𝑇 · 0) + 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 ∈ wcel 1439 (class class class)co 5666 0cc0 7404 + caddc 7407 · cmul 7409 ℕ0cn0 8727 |
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 3963 ax-pow 4015 ax-pr 4045 ax-setind 4366 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-distr 7503 ax-i2m1 7504 ax-0id 7507 ax-rnegex 7508 ax-cnre 7510 |
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 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-sub 7709 df-inn 8477 df-n0 8728 |
This theorem is referenced by: dec0h 8952 numlti 8967 nummul1c 8979 |
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