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Mirrors > Home > MPE Home > Th. List > dec0u | Structured version Visualization version GIF version |
Description: Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
dec0u.1 | ⊢ 𝐴 ∈ ℕ0 |
Ref | Expression |
---|---|
dec0u | ⊢ (;10 · 𝐴) = ;𝐴0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 11923 | . . 3 ⊢ ;10 ∈ ℕ0 | |
2 | dec0u.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
3 | 1, 2 | num0u 11916 | . 2 ⊢ (;10 · 𝐴) = ((;10 · 𝐴) + 0) |
4 | dfdec10 11908 | . 2 ⊢ ;𝐴0 = ((;10 · 𝐴) + 0) | |
5 | 3, 4 | eqtr4i 2799 | 1 ⊢ (;10 · 𝐴) = ;𝐴0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 (class class class)co 6970 0cc0 10329 1c1 10330 + caddc 10332 · cmul 10334 ℕ0cn0 11701 ;cdc 11905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-om 7391 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-pnf 10470 df-mnf 10471 df-ltxr 10473 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-dec 11906 |
This theorem is referenced by: decmul10add 11976 5t5e25 12010 6t6e36 12015 8t6e48 12026 sq10 13433 2503lem1 16320 4001lem1 16324 4001lem3 16326 bclbnd 25552 bposlem8 25563 dfdec100 30293 dpmul100 30320 dpmul1000 30322 dpmul 30336 dpmul4 30337 decpmul 38606 sqdeccom12 38607 41prothprmlem1 43150 |
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