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Mirrors > Home > ILE Home > Th. List > dec0h | GIF version |
Description: Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) |
Ref | Expression |
---|---|
dec0u.1 | ⊢ 𝐴 ∈ ℕ0 |
Ref | Expression |
---|---|
dec0h | ⊢ 𝐴 = ;0𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 8206 | . . 3 ⊢ 10 ∈ ℕ0 | |
2 | dec0u.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
3 | 1, 2 | num0h 8377 | . 2 ⊢ 𝐴 = ((10 · 0) + 𝐴) |
4 | df-dec 8369 | . 2 ⊢ ;0𝐴 = ((10 · 0) + 𝐴) | |
5 | 3, 4 | eqtr4i 2063 | 1 ⊢ 𝐴 = ;0𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 (class class class)co 5512 0cc0 6889 + caddc 6892 · cmul 6894 10c10 7972 ℕ0cn0 8181 ;cdc 8368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-5 7976 df-6 7977 df-7 7978 df-8 7979 df-9 7980 df-10 7981 df-n0 8182 df-dec 8369 |
This theorem is referenced by: decaddi 8411 decaddci 8412 |
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