dec0h - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dec0h		Structured version Visualization version GIF version

Theorem dec0h 12703

Description: Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)

**Hypothesis**
Ref	Expression
dec0u.1	⊢ 𝐴 ∈ ℕ₀

**Assertion**
Ref	Expression
dec0h	⊢ 𝐴 = ;0𝐴

**Proof of Theorem dec0h**
Step	Hyp	Ref	Expression
1		10nn0 12699	. . 3 ⊢ ;10 ∈ ℕ₀
2		dec0u.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
3	1, 2	num0h 12693	. 2 ⊢ 𝐴 = ((;10 · 0) + 𝐴)
4		dfdec10 12684	. 2 ⊢ ;0𝐴 = ((;10 · 0) + 𝐴)
5	3, 4	eqtr4i 2763	1 ⊢ 𝐴 = ;0𝐴

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2106 (class class class)co 7411 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 ℕ₀cn0 12476 cdc 12681

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-dec 12682

This theorem is referenced by: declei 12717 decrmanc 12738 decrmac 12739 decaddi 12741 decaddci 12742 decmulnc 12748 dec5dvds2 17002 2exp16 17028 37prm 17058 43prm 17059 83prm 17060 139prm 17061 163prm 17062 317prm 17063 631prm 17064 1259lem1 17068 1259lem2 17069 1259lem3 17070 1259lem4 17071 1259lem5 17072 2503lem1 17074 2503lem2 17075 2503lem3 17076 2503prm 17077 4001lem1 17078 4001lem2 17079 4001lem3 17080 4001lem4 17081 log2ublem3 26677 log2ub 26678 1mhdrd 32337 hgt750lem2 33950 12gcd5e1 41174 60gcd7e1 41176 420gcd8e4 41177 60lcm7e420 41181 420lcm8e840 41182 3exp7 41224 3lexlogpow5ineq1 41225 3lexlogpow5ineq5 41231 aks4d1p1 41247 ex-decpmul 41508 wallispi2lem2 45087 139prmALT 46563 127prm 46566 nfermltl2rev 46710